Taylor expansion of an Eisenstein series

Abstract

In this paper, we give an explicit formula for the first two terms of the Taylor expansion of a classical Eisenstein series of weight 2 k + 1 2k+1 for Γ 0 ( q ) \Gamma _{0}(q) . Both the first term and the second term have interesting arithmetic interpretations. We apply the result to compute the central derivative of some Hecke L L -functions.