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THE LOW LYING ZEROS OF A GL(4) AND A GL(6) FAMILY OF
L-FUNCTIONS
EDUARDO DUEN˜EZ AND STEVEN J. MILLER
Abstract. We investigate the large weight (k → ∞) limiting statistics for the
low lying zeros of a GL(4) and a GL(6) family of L-functions, {L(s, φ × f) :
f ∈ Hk} and {L(s, φ× sym2f) : f ∈ Hk}; here φ is a fixed even Hecke-Maass
cusp form and Hk is a Hecke eigenbasis for the space Hk of holomorphic cusp
forms of weight k for the full modular group. Katz and Sarnak conjecture that
the behavior of zeros near the central point should be well modeled by the
behavior of eigenvalues near 1 of a classical compact group. By studying the 1-
and 2-level densities, we find evidence of underlying symplectic and SO(even)
symmetry, respectively. This should be contrasted with previous results of
Iwaniec-Luo-Sarnak for the families {L(s, f) : f ∈ Hk} and {L(s, sym2f) :
f ∈ Hk}, where they find evidence of orthogonal and symplectic symmetry,
respectively. The present examples suggest a relation between the symmetry
type of a family and that of its twistings, which will be further studied in a
subsequent paper. Both the GL(4) and the GL(6) families above have all even
functional equations, and neither is naturally split from an orthogonal family.
A folklore conjecture states that such families must be symplectic, which is
true for the first family but false for the second. Thus the theory of low lying
zeros is more than just a theory of signs of functional equations. An analysis
of these families suggest that it is the second moment of the Satake parameters
that determines the symmetry group.
Contents
1. Introduction 2
2. Preliminaries 4
2.1. 1- and 2-Level Densities 4
2.2. Cusp Forms 5
2.3. Summation Formulas 6
3. Fφ×sym2Hk = {φ× sym2f : f ∈ Hk} 7
3.1. Definition, Gamma Factors, Functional Equation 7
3.2. 1-Level Density 12
3.3. 2-Level Density 15
4. Fφ×Hk = {φ× f : f ∈ Hk} 18
4.1. Logarithmic Derivative, Gamma Factors, Functional Equation 19
4.2. Explicit Formula 19
4.3. Relation of aφ×f to λφ and λf . 20
2000 Mathematics Subject Classification. 11M26 (primary), 11M41, 15A52 (secondary).
Key words and phrases. Low lying zeros, n-level density, random matrix theory, cuspidal
newforms, Maass forms.
We thank Wenzhi Luo and Peter Sarnak for suggesting this problem, as well as for many
enlightening conversations, and Jim Cogdell for valuable comments in writing the Appendix to
this article. The first-named author was partly supported by EPSRC Grant N09176.
1

2 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
5. Conclusion 21
Appendix A. Gamma factors and signs of functional equations 22
References 24
1. Introduction
Assuming GRH, the non-trivial zeros of any L-function lie on its critical line,
and therefore it is possible to investigate the statistics of its normalized zeros. The
general philosophy, born out of many examples and proven cases in function fields
[CFKRS, KS1, KS2, KeSn, ILS], is that the statistical behavior of eigenvalues of
random matrices (resp., random matrix ensembles) is similar to that of the critical
zeros of L-functions (resp., families of L-functions).
The global n-level correlations of high zeros of primitive automorphic cuspidal
L-functions, assuming a certain technical restriction, have been found to agree with
the corresponding statistics of the eigenvalues of complex hermitian matrices (the
Gaussian Unitary Ensemble, or GUE) [Mon, Hej, RS]. If the technical restriction
mentioned above were to be removed, the results on n-level correlations would im-
ply that the distributions of the normalized neighbor spacings between consecutive
critical zeros of an L-function and between GUE eigenvalues coincide, as has been
numerically observed [Od, Ru1]. The same correlations describe the global sta-
tistical behavior of the eigenvalues of other matrix ensembles, most notably of the
classical compact groups (orthogonal, unitary, symplectic). Being insensitive to the
effect of finitely many zeros, these correlations miss the behavior of the low lying
zeros, the zeros near the central point s = 1/2.
Katz and Sarnak [KS1, KS2] showed that there is another statistic that can
distinguish between the classical compact groups. It is the n-level density, and it
depends only on eigenvalues near 1. In a number of cases [FI, Gu¨, HM, HR2, ILS,
Mil2, Ro, Ru2, Yo], the behavior of the low lying zeros of families of L-functions is
found to be in agreement with that of the eigenvalues near 1 for random matrices in
one of the classical compact groups: unitary, symplectic, and orthogonal (which is
further split into SO(even) and SO(odd)). This correspondence allows us, at least
conjecturally, to assign a definite symmetry type to each family of L-functions.
Let φ be a fixed even Hecke-Maass cusp form and Hk a Hecke eigenbasis for
the space of holomorphic cusp forms of (even) weight k for the full modular group.
Iwaniec-Luo-Sarnak [ILS] proved that as k → ∞, the family {f : f ∈ Hk} has
SO(even) or SO(odd) symmetry (depending on whether k/2 is even or odd), and
the family {sym2f : f ∈ Hk} has symplectic symmetry. We consider the twisted
families Fφ×Hk = {φ × f : f ∈ Hk} and Fφ×sym2Hk = {φ × sym2f : f ∈ Hk};
the family {φ × sym2f} arose in the work of Luo-Sarnak [LS], where it is shown
to be intimately connected with the relation between the quantum and classical
fluctuations of observables on the modular surface. In both families, all functional
equations are even. We show that the first family has symplectic symmetry, and
the second SO(even). Explicitly, our main results are
Theorem 1.1. Let φ be a fixed even Hecke-Maass cusp form. As k →∞, for test
functions whose Fourier transform has small but computable support, the 1-level
density of the family Fφ×Hk only agrees with symplectic matrices, suggesting that
the underlying symmetry of this family is symplectic (and uniquely so).

THE LOW LYING ZEROS OF TWO FAMILIES OF L-FUNCTIONS 3
Theorem 1.2. Let φ be a fixed even Hecke-Maass cusp form. As k →∞, for test
functions g with supp(ĝ) ⊂ (− 524 , 524 ), the 1-level density of the family Fφ×sym2Hk
only agrees with SO(even), O and SO(odd) matrices. For small but computable
support, the 2-level density only agrees with SO(even) matrices, suggesting that the
underlying symmetry of this family is SO(even) (and uniquely so).
For families where the signs of the functional equations are all even and there
is no corresponding family with odd functional equations, a “folklore” conjecture
(for example, see page 2877 of [KeSn]) states that the symmetry is symplectic, pre-
sumably based on the observation that SO(even) and SO(odd) symmetries in the
examples known to date arise from splitting orthogonal families according to the
sign of the functional equations. A priori the symmetry type of a family with all
functional equations even is either symplectic or SO(even). All L-functions from
elements of Fφ×Hk and Fφ×sym2Hk have even functional equations, and neither fam-
ily seems to naturally arise from splitting sign within a full orthogonal family. By
calculating the 1-level density we quickly see the symmetry of the first is symplec-
tic (as predicted); however, the second family has orthogonal symmetry (we cannot
distinguish between SO(even),O and SO(odd) due to the small-support restriction
on the allowable test functions). By calculating the 2-level density for the second
family, we can discard O and SO(odd). Thus our calculations are only consistent
with the symmetry being SO(even). As our purpose is to show that the theory of
low lying zeros is more than just a theory of signs of functional equations, we do not
concern ourselves with obtaining optimal bounds in terms of support, instead sim-
plifying the arguments but still distinguishing the various classical compact group
candidates.
In studying the symmetry groups of Fφ×Hk and Fφ×sym2Hk , we see that twist-
ing a family with orthogonal (respectively, symplectic) symmetry by a fixed GL(2)
form flips the symmetry to symplectic (respectively, orthogonal). The effect on
the symmetry group by GL(n) twisting (by a fixed form, or by a second family)
in some cases will be described in a subsequent paper ([DM]). The main result
is that, for any family F satisfying certain technical conditions, we can attach a
symmetry constant cF , with cF = 0 (1,−1) if the family is unitary (symplectic,
orthogonal). For such families F and G, the family F ×G (Rankin-Selberg convolu-
tion) has symmetry constant cF×G = cF · cG (compare with [KS1]). In other words,
the symmetry of a product of families is the product of the family symmetries.
This is consistent with earlier results and should be compared, for instance, with
Rubinstein’s work [Ru2] on twisting the symplectic family of quadratic Dirichlet
characters by a fixed GL(n) form.
We assume the Generalized Riemann Hypothesis for all L-functions encountered.
Mostly GRH is used for interpretation purposes (i.e., if GRH is true than the non-
trivial zeros lie on the critical line, and we may interpret the n-level correlations and
densities as spacing statistics between ordered zeros), though in a few places GRH
is assumed to simplify the derivation of needed bounds (though these bounds can
be derived unconditionally at the cost of a more careful analysis). In §2 we review
the necessary preliminaries. We concentrate on the more difficult GL(6) family in
§3, and merely sketch the changes needed to handle the GL(4) family in §4; for
completeness the details of the calculation of the gamma factors and signs of the
functional equations are given in Appendix A. In §5 we analyze our results for these
two families. The evidence suggests that the theory of low lying zeros is not just a

4 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
theory of signs of functional equations, but rather more about the second moment
of the Satake parameters. In this regard it is similar to the universality Rudnick and
Sarnak [RS] found for the n-level correlations of high zeros of a primitive L-function
L(s, π) (π a cuspidal automorphic representation); their results are a consequence
of the universality of the second moments of the Satake parameters aπ(p).
2. Preliminaries
2.1. 1- and 2-Level Densities. Let g be an even Schwartz test function on R
whose Fourier transform
ĝ(y) =
∫ ∞
−∞
g(x)e−2πixydx (2.1)
has compact support. Let F be a finite family, all of whose L-functions satisfy
GRH. We define the 1-level density associated to F by
D1,F(g) =
1
|F|
∑
f∈F
∑
j
g
(
log cf
2π γ
(j)
f
)
, (2.2)
where 12 + iγ
(j)
f runs through the non-trivial zeros of L(s, f). Here cf is the analytic
conductor of f , and gives the natural scale for the low zeros. Since g is Schwartz,
only low lying zeros (i.e., zeros within a distance ≪ 1log cf of the central point)
contribute significantly. Thus the 1-level density is a local statistic which can
potentially help identify the symmetry type of the family.
Remark 2.1. For technical convenience, as in [ILS, Ro] we will modify (2.2) by
weighting each f by a factor wf which varies slowly with f . These factors simplify
applying the Petersson formula, and (see [ILS]) can be removed at the cost of
additional book-keeping.
Based in part on the function-field analysis where G(F) is the monodromy group
associated to the family F , it is conjectured that for each reasonable irreducible
family of L-functions there is an associated symmetry group G(F) (typically one
of the following five subgroups of unitary matrices: unitary U , symplectic USp,
orthogonal O, SO(even), SO(odd)), and that the distribution of critical zeros near 12
mirrors the distribution of eigenvalues near 1. The five groups have distinguishable
1-level densities.
To evaluate (2.2), one applies the explicit formula, converting sums over zeros
to sums over primes. Unfortunately, these prime sums can often only be evaluated
for small support. If one allows test functions with supp(ĝ) ⊂ (−δ, δ), then for
any δ > 0 the orthogonal, symplectic and unitary symmetries can be mutually
distinguished via their 1-level density. However, if δ ≤ 1 then the 1-level densities
of the three orthogonal types O, SO(even) and SO(odd) cannot be distinguished
from one another.
In order to distinguish between the three orthogonal symmetry types we study
the 2-level density of the family, defined as follows. Let g(x) = g1(x1)g2(x2), each
ĝi of compact support. Then
D2,F(g) =
1
|F|
∑
f∈F
∑
j1 6=±j2
g1
(
log cf
2π γ
(j1)
f
)
g2
(
log cf
2π γ
(j2)
f
)
, (2.3)

THE LOW LYING ZEROS OF TWO FAMILIES OF L-FUNCTIONS 5
Miller [Mil1] observed that an advantage of studying the 2-level density is that,
even for arbitrarily small support, the three orthogonal types of symmetry are
mutually distinguishable (see [Mil2] where it is used to discern the symmetry group
of families of elliptic curves). An analogous definition holds for the n-level density;
as the signs of our families are constant, our arguments can easily be extended to
determining the n-level density (though the support decreases with n).
By [KS1], the n-level densities for the classical compact groups are
Wn,SO(even)(x) = det(K1(xi, xj))i,j≤n
Wn,SO(odd)(x) = det(K−1(xi, xj))i,j≤n +
∑n
k=1 δ(xk) det(K−1(xi, xj))i,j 6=k
Wn,O(x) = 12Wn,SO(even)(x) + 12Wn,SO(odd)(x)
Wn,U(x) = det(K0(xi, xj))i,j≤n
Wn,USp(x) = det(K−1(xi, xj))i,j≤n,
(2.4)
where K(y) = sinπyπy , Kǫ(x, y) = K(x − y) + ǫK(x + y) for ǫ = 0,±1 and δ(x)
is the Dirac delta functional; see [HM] for a more tractable formula for the nth
centered moments for test functions whose Fourier transforms have support suitably
restricted. It is often more convenient to work with the Fourier transforms of the
densities. For the 1-level densities we have
Ŵ1,SO(even)(u) = δ(u) + 12I(u)
Ŵ1,SO(odd)(u) = δ(u)− 12I(u) + 1
Ŵ1,O(u) = δ(u) + 12
Ŵ1,U (u) = δ(u)
Ŵ1,USp(u) = δ(u)− 12I(u),
(2.5)
where I(u) is the characteristic function of [−1, 1]. The three orthogonal densi-
ties are indistinguishable for test functions of small support. Explicitly, for test
functions g such that supp(ĝ) ⊂ (−1, 1), we have
∫
ĝ(u)Ŵ1,SO(even)(u)du = ĝ(u) + 12g(0)∫
ĝ(u)Ŵ1,SO(odd)(u)du = ĝ(u) + 12g(0)∫
ĝ(u)Ŵ1,O(u)du = ĝ(u) + 12g(0)∫
ĝ(u)Ŵ1,U(u)du = ĝ(u)∫
ĝ(u)Ŵ1,USp(u)du = ĝ(u)− 12g(0).
(2.6)
We record the effect of the Fourier transform of the 2-level density kernel on our
test functions. Let c(G) = 0 (respectively 12 , 1) for G = SO(even) (respectively O,
SO(odd)). For even functions ĝ1(u1)ĝ2(u2) supported in |u1|+ |u2| < 1,
∫ ∫
ĝ1(u1)ĝ2(u2)Ŵ2,G(u)du1du2 =
[
ĝ1(0) + 12g1(0)
][
ĝ2(0) + 12g2(0)
]
+ 2
∫
|u|ĝ1(u)ĝ2(u)du
− 2ĝ1g2(0)− g1(0)g2(0)
+ c(G)g1(0)g2(0).
Thus, for arbitrarily small support, the 2-level density distinguishes the three or-
thogonal groups (see [Mil1] for the calculation).
2.2. Cusp Forms. We quickly review some facts about cusp forms; see [Iw2, ILS]
for details. Let Sk be the space of holomorphic cusp forms of weight k (an even

6 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
positive integer) and level 1 (that is, for the full modular group Γ = SL(2,Z)). Let
Hk be a basis of Hecke eigenforms. Then
dimSk = |Hk| =
k
12
+O(1). (2.7)
Any f ∈ Hk has a Fourier expansion
f(z) =
∞∑
n=1
af (n)e(nz), (2.8)
and we shall henceforth assume f is normalized so that af (1) = 1. Two other useful
normalizations for the coefficients are
λf (n) = af (n)n−
k−1
2 (2.9)
ψf (n) =
√
Γ(k − 1)
(4πn)k−1
1
‖f‖ af (n), (2.10)
with ‖f‖ the Petersson L2-norm of f . As mentioned in Remark 2.1, ψf (n) will
lead to a weighted sum which simplifies the application of the Petersson formula.
Essential in our investigations will be the multiplicativity properties of the Fourier
coefficients.
Lemma 2.2. Let f be a cuspidal Hecke eigenform of level 1. Then
λf (m)λf (n) =
∑
d|(m,n)
λf
(mn
d2
)
. (2.11)
In particular, we have
Corollary 2.3. Let (m,n) = 1, and p be a prime. Then
λf (m)λf (n) = λf (mn)
λf (p)2 = λf (p2) + 1. (2.12)
2.3. Summation Formulas. We recall some standard formulas for summing Fourier
coefficients over our families and test functions over primes.
Definition 2.4 (Diagonal Symbol).
∆k(m,n) =
∑
f∈Hk
ψf (m)ψf (n)
δ(m,n) =
{
1 if m = n
0 otherwise.
(2.13)
We rephrase the results from [ILS] in our language. By their equations 2.8,
2.52-2.54, and recalling that |Hk| = k12 +O(1), we find
∆k(m,n) =
ζ(2)
|Hk|+O(1)
∑
f∈Hk
λf (m)λf (n)
L(1, sym2f) . (2.14)
Lemma 2.5 (Petersson Formula). For (m,n) = 1, m and n of at most b factors,
∆k(m,n) = δ(m,n) +Ob
(
m 14n 14 logmn
k 56
)
. (2.15)

THE LOW LYING ZEROS OF TWO FAMILIES OF L-FUNCTIONS 7
For m,n as above and 12π√mn ≤ k,
∆k(m,n) = δ(m,n) +O
(√mn
2k
)
. (2.16)
Note (2.15) and (2.16) are Corollaries 2.2 and 2.3 of [ILS].
The Petersson formula allows us to easily evaluate certain weighted sums of the
Fourier coefficients. As ‖f‖ is related to L(1, sym2f), the natural weights are the
harmonic weights ωf = ζ(2)/L(1, φ × sym2f). These weights are almost constant
(see (3.34)), and following [ILS] we may remove these weights in the applications
below. See §3.2.1 for more details. We call terms with m = n diagonal terms;
the remaining terms are called non-diagonal. For small support, the non-diagonal
terms will not contribute.
Remark 2.6. There exist explicit formulas, involving Bessel functions and Kloost-
erman sums, for the error terms in the expansion of ∆k(m,n). For the families
studied in [ILS], by analyzing these terms’ contributions they are able to work with
test functions with support greater than [−1, 1], and hence distinguish SO(even)
from SO(odd); see also [HM] where these terms are handled for the n-level densi-
ties. Increasing the support has applications to non-vanishing results at the central
point. We will not be able to obtain such large support for our families; however,
by studying the 2-level density, we can still distinguish SO(even) from SO(odd).
The following are immediate applications of the Prime Number Theorem:
Lemma 2.7. Let F̂ be an even Schwartz function of compact support. Then for
any positive integer a,
∑
p
F̂
(
a log p
logR
)
log p
logR
1
p =
1
2aF (0) +O
(
1
logR
)
(2.17)
∑
p
F̂
(
log p
logR
)
4 log2 p
log2R
1
p = 2
∫ ∞
−∞
|u|F̂ (u)du+O
(
1
logR
)
. (2.18)
3. Fφ×sym2Hk = {φ× sym2f : f ∈ Hk}
We provide evidence that the underlying symmetry of the family Fφ×sym2Hk =
{φ × sym2f : f ∈ Hk} is SO(even). In §3.1 we calculate the needed quantities to
investigate the distribution of the low lying zeros. In §3.2 we calculate the one-
level density for test functions whose Fourier transform has small support, proving
the first half of Theorem 1.2. Although the support is contained in (−1, 1), the
evidence is enough to discard the possibility of symplectic (or unitary) symmetry;
however the 1-level density in this range cannot distinguish between O, SO(even)
and SO(odd) (even though the even functional equations suggest, of course, that
SO(even) is the type). To rectify this deficiency, in §3.3 we calculate the 2-level
density for small support; this suffices to eliminate O and SO(odd) and will complete
the proof of Theorem 1.2.
3.1. Definition, Gamma Factors, Functional Equation. We use the notation
of §2.2 for the holomorphic Hecke eigenform f and its Hecke eigenvalues λf (n).
Let φ be a fixed even Hecke-Maass cuspidal eigenform with Laplacian eigenvalue
λφ = 14 + t2φ for the full modular group Γ = SL(2,Z). We normalize φ so that
aφ(1) = 1, and denote by λφ(n) the corresponding Hecke eigenvalues.

8 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
For any unramified prime p, the Satake parameters (of the principal-series rep-
resentation of GL2(Qp)) associated to f are two complex numbers αp, α˜p = α−1p
satisfying
λf (pν) =
ν∑
ℓ=1
αℓpα˜ν−ℓp . (3.1)
Since f is of level 1, every prime is unramified. By the work of Deligne, |αp| = 1 —
the local representation is tempered— so that in fact α˜p is the complex conjugate αp
of αp. Thus λf (p) = αp+α−1p alone determines αp, α−1p . By (3.1), all of the λf (pν)
are algebraically expressible in terms of λf (p) (formula (3.1) is indeed equivalent
to the multiplicativity of the Fourier coefficients).
The Maass form φ has Satake parameters βp, β˜p = β−1p . The Ramanujan conjec-
ture states that |βp| = 1; while this is still open for Maass forms, powerful bounds
towards Ramanujan are available. Kim and Shahidi [KiSh] proved the crucial (for
us) bound |βp|, |β˜p| ≤ p
5
34 . Observe that 534 < 16 , which has many important con-
sequences (see Section 8 of [KiSh]), and is perhaps not coincidentally all we need
below. The exponent has been recently improved by Kim and Sarnak to 764 (see
Appendix Two of [K]).
Denote by sym2f be the Gelbart-Jacquet (symmetric-square) lift to (an auto-
morphic cuspidal representation of) GL(3) of the cusp form f [GeJa]. Its Fourier
coefficients are [Bu1, Bu2]
asym2f (m1,m2) =
∑
d|(m1,m2)
λsym2f
(m1
d , 1
)
λsym2f
(m2
d , 1
)
µ(d), (3.2)
where µ is the Mo¨bius function and
λsym2f (r, 1) =
∑
s2t=r
λf (t2). (3.3)
The symmetric-square L-function of f is then
L(s, sym2f) =
∞∑
m=1
λsym2f (m, 1)m−s. (3.4)
If, as before, αp, α−1p are the Satake parameters of f , then the parameters σp(j)
(j = 1, 2, 3) of sym2f at any prime p are the numbers α2p, 1, α−2p .
Denoting by λφ(r) the rth Hecke eigenvalue of φ, the Rankin-Selberg convolution
L(s, φ× sym2f) is the Dirichlet series
L(s, φ× sym2f) =
∑
m1,m2≥1
λφ(m1)λf (m2)aF (m1,m2)(m1m22)−s
=
∑
m
λφ,sym2f (m)m−s, (3.5)
where
λφ×sym2f (m) =
∑
m1m22=m
λφ(m1)λf (m2)aF (m1,m2). (3.6)
In fact, also by the work of Kim-Shahidi [KiSh] (and the appendix by Bushnell-
Henniart), L(s, φ × sym2f) is an automorphic L-function L(s, π) (for a suitable
an automorphic representation π of GL(6).) This ensures the standard proper-
ties (entire of order one, bounded in vertical strips, and functional equation) for

THE LOW LYING ZEROS OF TWO FAMILIES OF L-FUNCTIONS 9
L(s, φ× sym2f). In particular, L(s, φ× sym2f) conjecturally satisfies the Riemann
Hypothesis in the usual sense: L(s, φ×sym2f) = 0 and 0 ≤ ℑs ≤ 1 implies ℑs = 12 .
The Satake parameters δp(j) (j = 1, . . . 6) of πp are the six numbers α±2p β±1p
and β±1p . Furthermore, each L(s, φ× sym2f) has an even functional equation. The
proof of this assertion is given in Appendix A.
For ℜs large, the logarithmic derivative of L(s, φ×sym2f) is given by the Dirich-
let series
L′
L
(
s, φ× sym2f
)
=
∞∑
m=0
Λ(m)aφ×sym2f (m)m−s, (3.7)
where Λ(m) is von Mangoldt’s function and
aφ×sym2f (pν) =
6∑
j=1
δp(j)ν . (3.8)
Define now the archimedean (gamma) factor
L∞(s, φ, sym2f) := ΓR(s+ k − 1 + itφ)ΓR(s+ k − 1− itφ)×
× ΓR(s+ k + itφ)ΓR(s+ k − itφ)ΓR(s+ 1 + itφ)ΓR(s+ 1− itφ), (3.9)
where
ΓR(s) := π−
s
2Γ
(s
2
)
. (3.10)
The completed L-function
Λ(s, φ× sym2f) := L∞(s, φ, sym2f)L(s, φ× sym2f) (3.11)
for (3.5) satisfies the functional equation
Λ(s, φ× sym2f) = Λ(1− s, φ× sym2f). (3.12)
As the functional equation is even, we expect to observe either SO(even) or sym-
plectic symmetry.
Following Rudnick and Sarnak [RS], we define the six archimedean parameters
µj (j = 1, . . . , 6) by the requirement that 12 + µj is one of
k ± 1
2
± itφ or
3
2
± itφ. (3.13)
3.1.1. Explicit Formula. A smooth form of the explicit formula for L(s, φ× sym2f)
is as follows (see [RS] for a proof). Let g ∈ C∞c (R) be an even Schwartz function
whose Fourier transform
gˆ(y) =
∫ ∞
−∞
g(x)e−2πixydx (3.14)
is compactly supported. LetR > 0 and write the non-trivial zeros of L(s, φ×sym2f)
as ρj = 12 +iγj; we have j ∈ Z−{0} as the functional equation is even. Note γj ∈ R
is equivalent to GRH. Then
∑
j
g
( γj
2π logR
)
=
A
logR − 2
∑
p
∞∑
ν=1
gˆ
(ν log p
logR
) aφ×sym2f (pν) log p
pν/2 logR , (3.15)
where
A =
∫ ∞
−∞
6∑
j=1
(
Γ′R
ΓR
(
µj +
1
2
+
2πix
logR
)
+
Γ′R
ΓR
(
µj +
1
2
+
2πix
logR
))
g(x)dx. (3.16)

10 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
3.1.2. Gamma Factor Contribution. Recall ΓR(s) = π−s/2Γ(s/2). Thus
Γ′R(s)
ΓR(s)
= − log π
2
+
1
2
Γ′( s2 )
Γ( s2 )
. (3.17)
Let r = 2πx/ logR. Then the sum in (3.16) equals
−6 logπ + 1
2
6∑
j=1
[
Γ′
Γ
(
1
4
+
µj
2
+
ir
2
)
+
Γ′
Γ
(
1
4
+
µj
2
− ir
2
)]
, (3.18)
where µj = k ± 12 ± itφ (for four values) and 32 ± itφ (for the other two). We use
(see [ILS] or [GR] 8.363.3) that for a, b ∈ R, a > 0,
Γ′
Γ
(a+ bi) + Γ
′
Γ
(a− bi) = 2Γ
′
Γ
(a) +O(a−2b2), (3.19)
and for α ≥ 14 ,
Γ′
Γ
(
α+ 1
4
)
= logα+O(1). (3.20)
Thus, in the Γ′/Γ factors, the µj = 32 ± itφ terms are O(1) with respect to k. Set
a+ = 12 and a− = 0. Matched in complex-conjugate pairs, the other eight terms
give
Γ′
Γ
(k
2
+ a± + i
(
± tφ
2
+ r
))
+
Γ′
Γ
(k
2
+ a± − i
(
± tφ
2
+ r
))
= 2
Γ′
Γ
(k
2
+ a±
)
+O
( |tφ|2 + r2
k2
)
(3.21)
and
Γ′
Γ
(k
2
+ a±
)
= log
(k
2
+ a± −
1
4
)
+O(1) = log k +O(1). (3.22)
Note for k ≥ 2, the condition of having the argument greater than 14 is trivially
met. The main term in the sum in (3.16) is simply 12 ·4 ·2 logk = 4 log k. The main
contribution of the term AlogR in (3.15) is
4 log k
logR
∫ ∞
−∞
g(x)dx = log k
4
logR · gˆ(0). (3.23)
In the k-aspect, φ × sym2f looks like a GL(4) object. A natural choice for the
analytic conductors is therefore k4. With this scaling of the zeros, the test function
on the the left-hand side of (3.23) is evaluated at points which have mean average
spacing one near the central point (Riemann’s classical critical zero-counting for-
mula). As the quotient depends only on the logarithm of the conductor, as k →∞
the choice of any fixed constant multiple of k4 for the conductor will give the same
answer (see [ILS]). We have proved
Lemma 3.1. For L(s, φ× sym2f), up to lower order terms the contribution from
the Γ-factors in the explicit formula equals ĝ(0), and the analytic conductor equals
k4. Assuming GRH, the non-trivial zeros of L(s, φ×sym2f) are 12 + iγ
(j)
φ×sym2f with

THE LOW LYING ZEROS OF TWO FAMILIES OF L-FUNCTIONS 11
γ(j)φ×sym2f ∈ R. Taking R = k4, the explicit formula (3.15) becomes
∑
j
g
(
γ(j)φ×sym2f
2π logR
)
=
ĝ (0)− 2
∑
p
∞∑
ν=1
gˆ
(ν log p
logR
) aφ×sym2f (pν) log p
pν/2 logR +O
(
1
logR
)
. (3.24)
The appearance of the term ĝ (0) =
∫
g(x)dx on the right-hand side of (3.24)
naturally corresponds to the (expected) term
∫
ĝ (ξ) δ(ξ)dξ due to the delta mass
at the origin in the Fourier transform of the 1-level density (see (2.6)). The second
term (double sum) above will eventually be matched to
∫
ĝ (ξ) 12η(ξ)dξ, and this
will exclude symplectic as a possibility.
Remark 3.2. It is fortunate for us that the analytic conductors of L(s, φ× sym2f)
depend weakly on f . Specifically, as the only dependence on f is through its weight
k, one scaling works for all elements of our family. Oscillating conductors in a family
can sometimes be handled (one recourse is to use the average log conductor as in
[Si, Yo]; another approach is a more careful analysis and sieving, as in [Mil2] where
the conductors are monotone).
3.1.3. Relation of aφ×sym2f to λf and λφ. To evaluate the double sum in (3.24),
we express aφ×sym2f (pν) in terms of λf , λφ. Note
aφ×sym2f (pν) = α2νp βνp + α−2νp βνp + α2νp β−νp + α−2νp β−νp + βνp + β−νp
= (α2νp + 1 + α−2νp )(βνp + β−νp ). (3.25)
Case ν = 1: We have
aφ×sym2f (p) = (α2p + 1 + α−2p )(βp + β−1p ) = λf (p2)λφ(p). (3.26)
Case ν ≥ 2: We have
α2νp + 1 + α−2νp = (α2νp + α2(ν−1)p + · · ·+ α−2(ν−1)p + α−2νp )
− (α2(ν−1)p + · · ·+ α−2(ν−1)p ) + 1, (3.27)
and
βνp + β−νp = βνp + βν−2p + · · ·+ β−(ν−2)p + β−νp
− (βν−2p + · · ·+ β−(ν−2)p ), (3.28)
yielding
aφ×sym2f (pν) = (α2νp + 1 + α−2νp )(βνp + β−νp )
= (λf (p2ν)− λf (p2(ν−1)) + 1)(λφ(pν)− λφ(pν−2)).
(3.29)
Of course λφ(pν−2) = 1 when ν = 2.

12 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
3.1.4. Summary. We have shown
Lemma 3.3.
aφ×sym2f (p) = λφ(p)λf (p2) (3.30)
aφ×sym2f (p2) = (λφ(p2)− 1) · (λf (p4)− λf (p2) + 1). (3.31)
As we shall see below, the single term ‘−1’ in the first factor of (3.31) is respon-
sible for flipping the symmetry from symplectic (for the {sym2f} family of [ILS]
which had asym2f = λf (p4)− λf (p2) + 1) to SO(even) (for the {φ× sym2f} family
we are considering). This behavior is described in more detail in [DM].
Using the results from Kim-Sarnak [K], we have |β±1p | ≤ p
7
64 . Since |αp| ≤ 1,
equation (3.8) yields
aφ×sym2f (pν) = (βνp + β−νp ) · (α2νp + α−2νp + 1) ≪ p
7ν
64 . (3.32)
Therefore
aφ×sym2f (pν)
pν/2 ≪ p
− 25ν64 . (3.33)
This immediately implies
Lemma 3.4. The contribution from terms with ν ≥ 3 in (3.24) can be absorbed
into the error term.
Remark 3.5. We do not need the full strength of |β±1p | ≪ p
7
64 ; any exponent less
than 16 suffices. Without such a bound, we would later need to obtain cancellation
when averaging the Fourier coefficients over the family (a result of this nature is
significantly weaker than proving bounds towards Ramanujan, and follows from the
Petersson formula).
3.2. 1-Level Density. As Fφ×sym2Hk = {φ×sym2f, f ∈ Hk}, we have |Fφ×sym2Hk | =
|Hk|. For each L-function from Fφ×sym2Hk we calculate the 1-level density for its
low lying zeros via the explicit formula; we then average over the family Fφ×sym2Hk .
3.2.1. Preliminaries. Let g be an even Schwartz function with supp(ĝ) ⊂ (−σ, σ).
Following Iwaniec-Luo-Sarnak [ILS] or Royer [Ro], we consider a weighted average
over the family Fφ×sym2Hk of the expressions (3.24). The weight factors that we
use are ωf = ζ(2)/L(1, φ× sym2f). These are positive (by GRH for L(s, sym2f)),
slowly varying, and satisfy
k−ǫ ≪ǫ
1
L(1, sym2f) ≪ǫ k
ǫ (3.34)
and
1
|Hk|
∑
f∈Hk
ζ(2)
L(1, sym2f) = 1 +O
(
1
k
)
. (3.35)
To simplify the application of the Petersson Formula, we have introduced the slowly
varying weights ζ(2)/L(1, sym2f); arguing along the lines of [ILS] allows one to
remove these weights at no cost. We have chosen to leave in the weights in order
to emphasize the features of this GL(6) family.
By GRH, we may denote the non-trivial zeros of L(s, φ×sym2f) by 12+iγ
(j)
φ×sym2f
with γ(j)φ×sym2f ∈ R. Let R = k4. All L(s, φ × sym2f) have the same analytic
conductor, which up to lower order terms is k4. Averaging (3.24) by incorporating

THE LOW LYING ZEROS OF TWO FAMILIES OF L-FUNCTIONS 13
the weights and using Lemma 3.4 to absorb the ν ≥ 3 terms into the error shows
that the 1-level density for the family Fφ×sym2Hk is
D1,Fφ×sym2Hk (g)
=
1
|Hk|
∑
f∈Hk
ζ(2)
L(1, sym2f)
∑
j
g
(
γ(j)φ×sym2f
log cfφ
2π
)
= ĝ (0)− 2|Hk|
∑
f∈Hk
1
L(1, sym2f)
2∑
ν=1
Rσ∑
p=2
aφ×sym2f (pν) log p
pν/2 logR ĝ
(
ν log p
logR
)
+ O
(
1
logR
)
. (3.36)
We are left with analyzing the contribution from the ν = 1, 2 terms, and com-
paring this to (2.6). For small support, we will show there is no contribution from
the ν = 1 term, and the ν = 2 term contributes 12g(0). This proves the symmetry
group is neither unitary nor symplectic. We cannot discard the O and SO(odd)
symmetries; however, we will be able to eliminate them later by studying the 2-level
density.
Remark 3.6. Since Iwaniec-Luo-Sarnak exclusively use 1-level density arguments,
they must use extra averaging to extend their support past [−1, 1]. By studying the
2-level density we provide compelling evidence for the underlying symmetry being
SO(even) without extra averaging.
3.2.2. Contribution from ν = 1.
We must evaluate
T1 =
1
|Hk|
∑
f∈Hk
ζ(2)
L(1, sym2f)
Rσ∑
p=2
aφ×sym2f (p) log p√p logR ĝ
(
log p
logR
)
. (3.37)
By Lemma 3.3, aφ×sym2f (p) = λφ(p)λf (p2). Since λf (1) = 1,
T1 =
Rσ∑
p=2
λφ(p) log p√p logR ĝ
(
log p
logR
) ζ(2)
|Hk|
∑
f∈Hk
λf (1)λf (p2)
L(1, sym2f) . (3.38)
We are led to studying
ζ(2)
|Hk|
∑
f∈Hk
λf (1)λf (p2)
L(1, sym2f) . (3.39)
We have a non-diagonal term since p2 6= 1. By (2.16) of Lemma 2.5, δ(1, p2) = 0;
for p≪ k these terms are≪ p2k . Substituting into the expansion for T1, we see there
is no contribution for Rσ < k. Since R = k4, this implies there is no contribution
for σ < 14 . Note that in executing the prime sum, any polynomial bound on λφ(p)
suffices, since the decay in k is exponential.
For primes p > R 14 , we cannot use (2.16); instead we use (2.15), which gives
≪
√p log p
k5/6 . This yields a p-sum of
1
k5/6
k4σ∑
p
log2 p
logR
λφ(p)
√p
√p . (3.40)

14 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
Let δ be the the best bound towards Ramanujan for λφ(p); namely, λφ(p) ≪ pδ
(the Ramanujan conjecture is δ = 0). We find this sum is ≪ k4σ(1+δ)− 56 ; thus,
σ < 524(1+δ) . Even assuming Ramanujan does not help —this bound is worse than
the previous one. Thus (2.16) is better, and we obtain that there is no contribution
for support up to 14 .
Remark 3.7. The reason there is no contribution for small support is that we have
a non-diagonal term in the the Petersson formula.
3.2.3. ν = 2. We must evaluate
T2 = −
2
|Hk|
∑
f∈Hk
ζ(2)
L(1, sym2f)
Rσ∑
p=2
aφ×sym2f (p2) log p
p logR ĝ
(
2
log p
logR
)
. (3.41)
By Lemma 3.3, aφ×sym2f (p2) = (λφ(p2) − 1) · (λf (p4)− λf (p2) + 1). Almost all of
the terms are non-diagonal. Using λf (1) = 1 we have the following terms: from
λφ(p2), we get
λφ(p2) · λf (1)λf (p4), −λφ(p2) · λf (1)λf (p2), λφ(p2) · λf (1)λf (1).
(3.42)
The first two terms are non-diagonal; the Petersson formula yields no contribu-
tion for small support. The third term is diagonal. For small support, up to lower
order terms it yields +1 by (3.35):
ζ(2)
|Hk|
∑
f∈Hk
λf (1)λf (1)
L(1, sym2f) = 1 +O
(
1
k
)
. (3.43)
This gives
−2
Rσ∑
p=2
λφ(p2) log p
p logR ĝ
(
2
log p
logR
)
. (3.44)
By GRH for L(s, sym2φ), this sum is O( 1logR ) (see Section 4 of [ILS]).
We now handle the three terms from the −1 in the first factor of aφ×sym2f (p2);
these are
−λf (1)λf (p4), λf (1)λf (p2), −λf (1)λf (1). (3.45)
The first two are non-diagonal, and by the Petersson formula do not contribute
for small support. The third term, however, is a diagonal term; up to lower order
corrections, from the Petersson formula its contribution is −1, and we are left with
2
∑
p
log p
p logR ĝ
(
2
log p
logR
)
. (3.46)
By Lemma 2.7, the above sum (up to lower order terms) is 12g(0).
Therefore, for small support, the ν = 2 piece contributes 12g(0) + o(1), with the
main term arising from the sixth term in the expansion of aφ×sym2f (p2). At this
point we have enough evidence to discard the unitary and symplectic symmetries.
Since the functional equations in (3.12) are even, this certainly points to the un-
derlying symmetry being SO(even), but we cannot yet discard the full orthogonal
or SO(odd) symmetries. This will be done in §3.3.
We now determine how large we may take the support. Of the six pieces which do
not contribute to the main term, the worst error term is from λφ(p2) · λf (1)λf (p4).

THE LOW LYING ZEROS OF TWO FAMILIES OF L-FUNCTIONS 15
By (2.16), if 1 · p4 ≪ k2, the sum over f ∈ Hk is ≪ p
2
2k . Again, any polynomial
bound for λφ(p2) yields the sum over primes p ≪ k
1
2 is a lower order term. Since
R = k4, this yields no contribution for σ < 18 . Therefore, up to lower order terms
the contribution is 12g(0) for σ < 18 .
The reason for the sharp decrease in support (relative to the ν = 1 term) is
because we have a λf (1)λf (p4). Another possibility is to use (2.15), which gives for
the λφ(p2) · λf (1)λf (p4) term
1
k5/6
k4σ∑
p
log2 p
logR
λφ(p2) · p
p . (3.47)
From (2.12), λφ(p2) = λφ(p)2 − 1. Substituting into (3.47), the -1 does not con-
tribute for σ < 524 , and we are left with bounding
1
k5/6
k4σ∑
p=2
log2 p
logR λφ(p)
2 ≪ log k
4
k5/6
k4σ∑
n=1
|λφ(n)|2. (3.48)
One could use bounds towards Ramanujan; however, all we need is that Ramanujan
holds on average, namely the sum of |λφ(n)|2 (see [Iw1], equation 8.7) is
X∑
n=1
|λφ(n)|2 ≪φ X. (3.49)
This yields the sum in (3.48) does not contribute for σ < 524 .
A similar argument applied to the other terms show none of them contribute
as well for such support; one must check that the error term for the λf (1)λf (1)
term (which is the diagonal piece responsible for 12g(0)) does not contribute in this
range. This completes the proof of the first part of Theorem 1.2.
As supp(φ̂) ⊂ (−1, 1), while every Λ(s, φ× sym2f) has even functional equation,
we cannot conclude the symmetry is SO(even) and not O or SO(odd). There are
two natural ways to try and increase the support. The first is to average over even
Maass forms φ; unfortunately, we would have to let tj grow to a power of k, which
would change the conductor arguments. Another approach is to average over the
weight k (as in the investigation of sym2f in [ILS]). By averaging over weight, they
triple the support; however, as we start with support less than 13 , such methods
are insufficient to break (−1, 1). We therefore study the 2-level density, which
even for arbitrarily small support can distinguish the three orthogonal groups (see
[Mil1, Mil2]).
3.3. 2-Level Density. We complete the proof of Theorem 1.2. As in §3.2, for
convenience in applying the Petersson formula we study a weighted 2-level density.
Thus (2.3) becomes
D2,Fφ×sym2Hk (g) =
1
|Hk|
∑
f∈Hk
ζ(2)
L(1, sym2f)
∑
j1,j2
j1 6=±j2
g1
(
logR
2π γ
(j1)
f
)
g2
(
logR
2π γ
(j2)
f
)
. (3.50)
The sum is over zeros j1 6= ±j2. As all functional equations are even and we are
assuming GRH, the zeros occur in complex conjugate pairs; this is the first time the
sign of the functional equation enters our arguments. We may rewrite the 2-level

16 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
expression as a sum over all pairs of zeros, minus twice the sum over all zeros,
yielding
D2,Fφ×sym2Hk (g) =
1
|Hk|
∑
f∈Hk
ζ(2)
L(1, sym2f)
∑
j1,j2
g1
(
logR
2π γ
(j1)
f
)
g2
(
logR
2π γ
(j2)
f
)
−2D1,Fφ×sym2Hk (g1g2). (3.51)
In the above, the second term is a 1-level density with test function (g1g2)(x). For
small support, we have shown this term is just ĝ1g2(0) + 12 (g1g2)(0). Crucial in
the above expansion is that each Λ(s, φ× sym2f) has even sign. This allows us to
pair off the zeros, γ(j)f and γ
(−j)
f . Thus summing over distinct zeros is the same
as subtracting off twice a 1-level sum over all zeros. This would be false if the
functional equation were odd. In that case we would have to add back g1(0)g2(0)
for the extra zero at the central point, and in fact the presence or absence of this
additional term is the cause of the differences in the 2-level densities of the three
orthogonal groups.
For the first term in (3.51), since we are summing over all zeros, we may use the
explicit formula for the sum over each ji. Let
bφ×sym2f (p) = aφ×sym2f (p)
bφ×sym2f (p2) = aφ×sym2f (p2) + 1. (3.52)
In the expansions with the explicit formula, we isolate the contribution from the
part of the ν = 2 term which contributes 12gi(0) for small support; this is the +1
term in bφ×sym2f (p2). We have also removed the error terms arising from ν ≥ 3; the
argument is standard (see [Ru2, RS]). We are left with considering the weighted
average of
2∏
i=1
[(
ĝi (0) +
1
2
gi(0)
)
− 2
2∑
νi=1
∑
pi
bφ×sym2f (pνi) log p
pνi/2 logR ĝi
(
νi
log pi
logR
)]
. (3.53)
There are three terms for each i. For small support, we have shown the νi-sums
by themselves do not contribute, though we will see that there are contributions
when a ν1-sum hits a ν2-sum. Thus when a ĝi (0)+ 12gi(0) hits a ν-sum, there is no
contribution. We have [ĝ1 (0)+ 12g1(0)] · [ĝ2 (0)+ 12g2(0)], plus the weighted average
of the mixed sums
4
∑
p1
∑
p2
bφ×sym2f (pν11 )bφ×sym2f (pν22 ) log p1 log p2
pν1/21 p
ν2/2
2 log
2R
ĝ1
(
ν1
log p1
logR
)
ĝ2
(
ν2
log p2
logR
)
(3.54)
for (ν1, ν2) ∈ {(1, 1), (2, 1), (2, 1), (2, 2)}. For each pair there are two cases, when
p1 = p2 and p1 6= p2. Since we are only interested in the 2-level density for
arbitrarily small support as a means to distinguish SO(even) from orthogonal and
SO(odd) symmetry, we do not record how large we may take the support.

THE LOW LYING ZEROS OF TWO FAMILIES OF L-FUNCTIONS 17
3.3.1. (1, 1) Terms. We have
4
|Hk|
∑
f∈Hk
ζ(2)
L(1, sym2f)
∑
p1
∑
p2
aφ×sym2f (p1)aφ×sym2f (p2) log p1 log p2√p1p2 log2R
×ĝ1
(
log p1
logR
)
ĝ2
(
log p2
logR
)
. (3.55)
As aφ×sym2f (p) = λφ(p)λf (p2), we have
4
|Hk|
∑
f∈Hk
ζ(2)
L(1, sym2f)
∑
p1
∑
p2
λφ(p1)λφ(p2)
λf (p21)λf (p22) log p1 log p2√p1p2 log2R
× ĝ1
(
log p1
logR
)
ĝ2
(
log p2
logR
)
. (3.56)
If p1 6= p2, when we use the Petersson formula there is no contribution for small
support, since it is a non-diagonal term. Note p1 = p2 is a diagonal term, giving
1
|Hk|
∑
f
ζ(2)
L(1,sym2f)λf (p2)λf (p2) = 1 + O(
√
p4/2k). For sufficiently small support
the error term is negligible, and thus the diagonal term is
4
∑
p
λφ(p)λφ(p)
log2 p
p log2R
ĝ1
(
log p1
logR
)
ĝ2
(
log p2
logR
)
+ o(1). (3.57)
We use λφ(p)λφ(p) = 1 + λφ(p2); we saw in §3.2.3 that the λφ(p2) term will not
contribute by GRH for L(s, sym2φ). The +1 will contribute, with test function
ĝ1ĝ2. By Lemma 2.7, we have
4
∑
p
log2 p
p log2R
· ĝ1ĝ2
(
log p
logR
)
= 2
∫
|u|ĝ1ĝ2(u)du+O
(
1
logR
)
. (3.58)
3.3.2. (1, 2) and (2, 1) Terms. As these terms are handled identically, we confine
ourselves to (1, 2). We have weighted averages of
4
∑
p1
∑
p2
aφ×sym2f (p1)bφ×sym2f (p22) log p1 log p2
p1/21 p2 log2R
ĝ1
(
log p1
logR
)
ĝ2
(
2
log p2
logR
)
. (3.59)
By Lemma 3.3 and (3.52)
aφ×sym2f (p) = λφ(p)λf (p2) (3.60)
bφ×sym2f (p2) = λφ(p2)(λf (p4)− λf (p2) + 1)− (λf (p4)− λf (p2)). (3.61)
If p1 6= p2, all terms are non-diagonal, and the Petersson formula yields no contri-
bution for small support.
If p1 = p2, only two terms give diagonal terms: λφ(p)λφ(p2) · λf (p2)λf (p2)
and −λφ(p)λf (p2)λf (p2). However, while the Petersson formula will give a ±1
for each of these diagonal terms, this is immaterial since we are dividing by p 32 .
Using the Kim-Sarnak bound of p 764 for Maass forms is enough to show there is
no contribution, for any support, from these terms. We are dividing by p 32 , and
we have at most p 7·364 log p in the numerator. This gives p−1− 1164 log p, which yields
a O( 1logR ) contribution. Arguing as in (3.47) to (3.49), one may replace the Kim-
Sarnak bound with any non-trivial bound towards Ramanujan.

18 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
3.3.3. (2, 2) Term. We now consider the (2, 2) term. We have a weighted average
of
4
∑
p1
∑
p2
bφ×sym2f (p21)bφ×sym2f (p22) log p1 log p2
p1p2 log2R
ĝ1
(
2
log p1
logR
)
ĝ2
(
2
log p2
logR
)
,
(3.62)
where by Lemma 3.3 and (3.52)
bφ×sym2f (p2) = λφ(p2) · (λf (p4)− λf (p2) + 1)
−1 · (λf (p4)− λf (p2)). (3.63)
When p1 = p2, as λφ(p) = O(pδ) for some δ ∈ [0, 12 ],
bφ×sym2f (p
2)2
p2 = O(p4δ−2), and
these terms will not contribute for any δ < 14 . We do not need the full strength of
the Kim-Sarnak bound; the 528 of [BDHI] suffices.
We are left with the case p1 6= p2. The only diagonal term will be λφ(p21)λφ(p22) ·
λf (1)λf (1). For small support, the other terms will not contribute, and by Peters-
son’s formula we have
4
2∏
i=1
∑
pi
λφ(p2i ) log pi
pi logR
ĝi
(
2
log pi
logR
)
. (3.64)
As in §3.2.3, by GRH for L(s, sym2φ) each prime sum is O( 1logR ). Thus there is no
contribution from the (2, 2) terms.
3.3.4. Summary. We have shown
D2,Fφ×sym2Hk (g) =
[
ĝ1 (0) +
1
2
g1(0)
]
·
[
ĝ2 (0) +
1
2
g2(0)
]
+ 2
∫
u
|u|ĝ1ĝ2(u)du − 2ĝ1g2(0)− g1(0)g2(0), (3.65)
and (3.65) agrees only with the 2-level density for SO(even) (see (2.7)), completing
the proof of Theorem 1.2.
4. Fφ×Hk = {φ× f : f ∈ Hk}
In this section we prove Theorem 1.1, namely that the symmetry type of the
family Fφ×Hk = {φ × f : f ∈ Hk} (k → ∞) agrees only with symplectic, where
φ is a fixed even cuspidal Hecke-Maass form with eigenvalue λφ = 14 + t2φ and
f ∈ Hk is a Hecke holomorphic modular form of weight k. These L-functions have
associated Euler products of degree 4 and are indeed associated to automorphic
representations of GL(4) [Ra]. As in §3.1 we first derive the explicit formula and
find the analytic conductors for the family. In order to compute the 1-level density
we analyze the local parameters and find evidence for symplectic symmetry (for
small support). Since the symplectic 1-level density is distinguishable from the
other classical compact groups for arbitrarily small support, there is no need to
investigate the 2-level density. As the arguments are similar to those for φ×sym2f ,
we merely sketch the calculations below. See also Appendix A for details of the
determination of the gamma factors and signs of the functional equation.

THE LOW LYING ZEROS OF TWO FAMILIES OF L-FUNCTIONS 19
4.1. Logarithmic Derivative, Gamma Factors, Functional Equation. In
terms of the Fourier coefficients {λf (n)}, {λφ(n)} and the Maass eigenvalue tφ (14 +
t2φ is the Laplacian eigenvalue), we have
L(s, φ× f) = ζ(2s)
∑
m
λφ(m)λf (m)m−s =
∑
m
λφ×f (m)m−s, (4.1)
where
λφ×f (m) =
∑
m21m2=m
λφ(m2)λf (m2). (4.2)
The logarithmic derivative of L(s, φ× f) is
L′
L (s, φ× f) =
∑
m
Λ(m)aφ×f (m)m−s, (4.3)
with
aφ×f(pν) =
4∑
j=1
τp(j)ν . (4.4)
The archimedean (gamma) factor is
L∞(s, φ× f) := ΓR(s+ itφ + k−12 )ΓR(s− itφ + k−12 )
× ΓR(s+ itφ + k+12 )ΓR(s− itφ + k+12 ), (4.5)
and the completed L-function
Λ(s, φ× f) := L∞(s, φ× f)L(s, φ× f) (4.6)
satisfies the functional equation
Λ(s, φ× f) = Λ(1− s, φ× f). (4.7)
Note the functional equation is even. We define the archimedean parameters µj ,
1 ≤ j ≤ 4, to be the numbers
k ± 1
2
± itφ. (4.8)
4.2. Explicit Formula. As in §3.1.1, let R > 0 be a parameter; later we take
R = k4. Assuming GRH we may write the non-trivial zeros of L(s, φ × f) as
ρj = 12 + iγj, j ∈ Z − {0} (since all signs are even). Then
∑
j
g
( γj
2π logR
)
=
A
logR − 2
∑
p
∞∑
ν=1
gˆ
(ν log p
logR
) aφ×f (pν) log p
pν/2 logR , (4.9)
where
A =
∫ ∞
−∞
4∑
j=1
(
Γ′R
ΓR
(
µj +
1
2
+
2πix
logR
)
+
Γ′R
ΓR
(
µj +
1
2
+
2πix
logR
))
g(x)dx. (4.10)
An analogous calculation as in §3.1.2 gives, up to lower order terms, that the
conductor is
(
k2/4
)2
. As we only care about the logarithm of the conductor, we
take R = k4. The contribution to the 1-level density will be ĝ(0) plus lower order
terms.

20 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
4.3. Relation of aφ×f to λφ and λf . We consider the local parameters α±1p at
any prime p for f , as well as β±1p for φ. The local parameters τp(j) (j = 1, 2, 3, 4)
for the automorphic representation associated to L(s, φ× f) are the four numbers
α±1p β±1p . A calculation similar to but simpler than that in §3.1.3 gives
aφ×f (p) = λφ(p)λf (p) (4.11)
aφ×f (pν) = (λφ(pν)− λφ(pν−2)(λf (pν)− λf (pν−2)), ν ≥ 2. (4.12)
In particular
aφ×f (p2) = (λφ(p2)− 1)(λf (p2)− 1). (4.13)
4.3.1. ν ≥ 3 Terms. We show there is no contribution to the 1-level density from
terms with ν ≥ 3 in (4.9). The Satake parameters are the four numbers α±1p β±1p ,
each of which is bounded by pδ with δ < 16 by Kim-Sarnak [K]. Thus
aφ,f (pν)
p ν2 ≪ p
ν(δ− 12 ), (4.14)
and as δ < 16 , summing over p and ν ≥ 3 is O(1). Dividing by logR = log k4, we
see there is no contribution.
4.3.2. ν = 1 Terms. As aφ×f = λφ(p)λf (p) and logR = log k4, we have
− 2
∑
p
gˆ
(
log p
logR
) λφ(p)λf (p) log p√p logR . (4.15)
As 1 = λf (1), summing over f ∈ Hk yields no contribution for small support, as
λf (p)λf (1) is a non-diagonal term.
4.3.3. ν = 2. Terms
As aφ×f (p2) = (λφ(p2)− 1)(λf (p2)− 1), we have
−2
∑
p
gˆ
(
2 log p
logR
)
(λφ(p2)− 1)(λf (p2)− 1) log p
p logR . (4.16)
There are four types of terms:
λφ(p2) · λf (p2)λf (1), −λφ(p2), −λf (p2)λf (1), (−1)(−1). (4.17)
The first and third are non-diagonal, and by the Petersson formula will not con-
tribute for small support. The second is diagonal; however, by GRH for L(s, sym2φ)
(see (3.44)), this term is O( 1logR ). We are left with the fourth piece,
−(−1)2 · 2
∑
p
gˆ
(
2 log p
logR
)
log p
p logR. (4.18)
By Lemma 2.7, the p-sum is g(0)4 , thus, the ν = 2 terms contribute, for small
support, − 12g(0).

THE LOW LYING ZEROS OF TWO FAMILIES OF L-FUNCTIONS 21
4.3.4. Summary. The previous subsections proved Theorem 1.1, that as k → ∞
the 1-level density of Fφ×Hk agrees only with symplectic. We took two orthogonal
families (when k ≡ 2 mod 4 then Hk has SO(odd) symmetry, and when k ≡ 0 mod 4
then Hk has SO(even) symmetry), and showed that their twists by a fixed even, full
level Maass form give a symplectic family. This should be compared to Theorem
1.2, where we twisted a symplectic family and obtained an SO(even) family.
Remark 4.1. The reason for the symmetry flipping can be found in (4.13). For
the ν = 2 terms, the Maass form introduces an extra factor of −1 in the diagonal
contribution. This changes the sign of the contribution from the ν = 2 terms,
and switches us from symplectic to orthogonal symmetries (if we have orthogonal
symmetries, we need to evaluate the 2-level density to determine which one as our
supports are too small to distinguish SO(even), O and SO(odd)).
5. Conclusion
We investigated the distribution of low lying zeros for two families. The first is
a GL(6) family, {φ × sym2f : f ∈ Hk}; here φ is a fixed Hecke-Maass cusp form
and k → ∞. Though this is a GL(6) family, only four of the six Gamma factors
depend on k, and the analytic conductor is k4. Since all elements of this family have
even functional equation and there is no natural complementary family with odd
sign, a folklore conjecture predicted that the underlying group symmetry should be
symplectic. However, the symmetry type is SO(even), proving that low lying zeros
is more than just a theory of signs of functional equations.
We calculated the 1-level density for test functions g such that supp(ĝ) is small
(supp(ĝ) ⊂ (− 524 , 524 )). For such small support, only the diagonal terms in the
Petersson formulas contribute. Thus we can eliminate two of the five classical
compact groups, namely symplectic and unitary. Unfortunately, since the support
is significantly less than (−1, 1), all three orthogonal candidates are still possible;
however, as all members of the family have even functional equation, we do not
expect the underlying symmetry to be either O or SO(odd). Observe that the
reason for the flipping of symmetry from symplectic to (some flavor of) orthogonal
is that the contribution from the squares of primes (which is what is responsible in
any case for the term ± 12g(0)) changes sign.
More precisely, in [ILS] for the family {sym2f}, there is a contribution of− 12 ĝ (0),
arising from the diagonal term +1 in λf (p4)− λf (p2) + 1. In our family, this term
is multiplied by the factor λφ(p2)− 1 (see (3.29)), and the −1 results in a diagonal
contribution of opposite sign, hence the symmetry flipping.
To discard the O and SO(odd) symmetries, we calculated the 2-level density. It
is shown in [Mil1, Mil2] that for arbitrarily small support the 2-level densities of
the three orthogonal groups are distinguishable. We see that our answer agrees
only with SO(even), further supporting the claim that the symmetry group of this
family is SO(even).
Our second example is a GL(4) family, {φ × f : f ∈ Hk}. Here, for the same
reason as before, twisting flips the symmetry (this time from orthogonal to sym-
plectic). In a subsequent paper, we will describe the interplay between twisting
by a fixed GL(n) form (or family) and the symmetry type (in certain cases). The
arguments of this paper can be generalized to families satisfying certain natural
technical conditions. It can be shown that a natural “family constant” c can be
attached to a family in such a manner that cF×G = cF · cG , where c = 0 (1,−1)

22 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
for unitary (symplectic, orthogonal) symmetry. Here F × G is the family obtained
by Rankin-Selberg convolution of the L-functions in the families F and G. These
results are similar in spirit to the universality found by Rudnick and Sarnak [RS]
in the n-level correlations of high zeros, and will be described in further detail in
[DM]. Specifically, it again seems that the second moment of the Satake parameters
determines the answer. For the GL(6) family, the main term from averaging the
second moment of the Satake parameters (see (3.8)) is −1 and leads to SO(even)
symmetry, while in the GL(4) family the main term (see (4.13)) is +1 and leads to
symplectic symmetry.
Appendix A. Gamma factors and signs of functional equations
In this appendix we derive the precise forms (equations (3.9) and (4.5)) of the
gamma factors for the completed L-functions L(s, φ × sym2f) and L(s, φ × f), as
well as their functional equations (equations (3.12) and (4.7)). In particular, we
show that both functional equations are even.
Being Hecke eigenforms of level 1, f and φ can be identified with automorphic
cuspidal representations F and Φ of GL2(AQ) with trivial central character [Gel].
The latter are isomorphic to restricted tensor products
F ∼=
∏′
v
Fv Φ ∼=
∏′
v
Φv (A.1)
of representations of GL2(Qv) for each place v of Q (here v = p for p prime, or
v = ∞, in which case Q∞ = R). In the case at hand, as every (finite) prime p
is unramified, the corresponding principal series representations of GL2(Qp) have
associated SL2(C)-conjugacy classes1
Fp ↔
(
αp 0
0 α−1p
)♮
, Φp ↔
(
βp 0
0 β−1p
)♮
. (A.2)
Denote by M(αp),M(βp) the matrices in (A.2).
Taking the symmetric square of the standard representation of GL2(R), one
obtains the conjugacy class sym2M(αp)♮ = diag(α2p, 1, α−2p )♮, whence the Satake
parameters of sym2f . Similarly, the conjugacy classes ofM(αp)⊗M(βp) in GL4(R)
and sym2M(αp)⊗M(βp) in GL6(R) define the Satake parameters of φ× f and φ×
sym2f . The local L-factors are defined in terms of these Satake parameters in
the usual manner, and the product of all local factors defines the (incomplete)
L-functions (3.5) and (4.1).
The representations F∞ and Φ∞ are the discrete series representation of weight
k, and the representation I(| · |it, | · |−it) of GL2(R), respectively (recall that 14 + t2
is the Laplacian eigenvalue of φ.)2 Selberg proved that t is real for Maass forms
of level 1 (and conjectured that t is still real for forms of any weight). The first
published proof is due to Roelcke [Roe]; see also [Iw1].
Proofs of equations (3.9) and (4.5) involve parametrizing the representations F∞
and Φ∞, through the Langlands correspondence, by semisimple representations of
the Weil group WR (see [Kn] and [CM]). We number the discrete series as in [Kn],
1A♮ denotes the conjugacy class of A.
2I(| · |it, | · |−it) is the unitary induction from the group Q of upper-triangular matrices to
GL2(R) of the representation
(
a ∗
d
)
7→ |a|it|d|−it.

THE LOW LYING ZEROS OF TWO FAMILIES OF L-FUNCTIONS 23
(see the note at the top of page 1588 of [CM]), so replacing k by ℓ + 1 in what
follows would make our notation agree with that of [CM]. Then (cf., equations
(3.2) and (3.3) of [Kn])
F∞ ↔ ρ(k−1,0), (A.3)
Φ∞ ↔ ρ(+,it) ⊕ ρ(+,−it). (A.4)
We have denoted by ρ(a,b) the semisimple representation of WR with parameters
(a, b). The known cases of functoriality [GeJa, Ra, KiSh] imply the existence of
automorphic (cuspidal) representations sym2F , Φ× sym2F , and Φ× F such that,
by the archimedean Langlands correspondence,
(sym2F )∞ ↔ sym2ρ(k−1,0) (A.5)
(Φ× sym2F )∞ ↔
(
ρ(+,it) ⊕ ρ(+,−it)
)
⊗ sym2ρ(k−1,0) (A.6)
(Φ× F )∞ ↔
(
ρ(+,it) ⊕ ρ(+,−it)
)
⊗ ρ(k−1,0). (A.7)
By Proposition 3.1 of [CM],3
sym2ρ(k−1,0) ∼= ρ(−,0) ⊕ ρ(2k,0). (A.8)
Moreover, it is easily checked that
ρ(+,±it) ⊗ ρ(−,0) ∼= ρ(−,±it) (A.9)
ρ(+,±it) ⊗ ρ(ℓ,0) ∼= ρ(ℓ,±it). (A.10)
Hence,
(Φ× sym2F )∞ ↔ ρ(−,it) ⊕ ρ(−,−it) ⊕ ρ(2k−2,it) ⊕ ρ(2k−2,−it) (A.11)
(Φ× F )∞ ↔ ρ(k−1,it) ⊕ ρ(k−1,−it). (A.12)
The archimedean (gamma) factors can be found using these decompositions and the
local Langlands correspondence. In terms of irreducible semisimple representations
of WR, we have
L(s, ρ) =



ΓR(s± it) ρ = ρ(+,±it),
ΓR(s± it+ 1) ρ = ρ(−,±it),
ΓR(s± it+ ℓ2 )ΓR(s± it+ ℓ2 + 1) ρ = ρ(ℓ,±it).
(A.13)
These local factors are multiplicative under direct sums of representations of WR,
so definitions (3.9) and (4.5) are consistent with (A.11) and (A.12) via (A.13).
Since all automorphic representations under discussion are self-contragredient,
the functional equations relate each L-function to itself as s 7→ 1 − s. In general,
the root number ε(s, π) associated to an automorphic cuspidal representation π is
a product
ε(s, π) =
∏
v
ε(s, πv)
of local root numbers.4 For self-contragredient representations π = π˜ the ε-factor
agrees with the sign of the functional equation (up to a factor Qs which is not
present in the level-1 case that concerns us). Moreover, local root numbers can be
computed via the local Langlands correspondence as root numbers associated to
3Note that the weight k of our f is always even. Thus, the appearance of (−, 0) in (A.8) is due
to the fact that (−1)k−1 = −1.
4We have omitted the dependence of the local root numbers on the choice of additive character
ψ of AQ.

24 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
Weil group representations. Additionally, ε(s, πp) = 1 at any prime p such that πp
is unramified.5 The local root number of an irreducible representation ρ of WR is
ε(s, ρ) =



1 ρ = ρ(+,±it),
i ρ = ρ(−,±it),
iℓ+1 ρ = ρ(ℓ,±it).
(A.14)
From (A.14) and (A.11), (A.12) we obtain
ε((Φ× sym2F )∞) = i · i · i2k−1 · i2k−1 = +1, (A.15)
and, since k is even,
ε((Φ× F )∞) = ik · ik = +1. (A.16)
Since all (finite) primes p are unramified for Φ×F and Φ× sym2F , we conclude
ε(s,Φ× sym2F ) = ε(s,Φ× F ) = +1, (A.17)
so the global functional equations have even sign, proving (3.12) and (4.7).
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26 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
Department of Mathematics, The University of Texas at San Antonio, San Antonio,
TX 78249
E-mail address: eduenez@math.utsa.edu
Department of Mathematics, Brown University, 151 Thayer Street, Providence, RI
02912
E-mail address: sjmiller@math.brown.edu