Taylor Expansion of an Eisenstein Series

In this paper, we give an explicit formula for the first two terms of the Taylor expansion of a classical Eisenstein series of weight 2k + 1 for Γ0(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-functions. 0. Introduction. Consider the classical Eisenstein series ∑ γ∈Γ∞\ SL2(Z) Im(γτ) which has a simple pole at s = 1. The well-known Kronecker limit formula gives a closed formula for the next term (the constant term) in terms of the Dedekind η-function and has a lot of applications in number theory. It seems natural and worthwhile to study the same question for more general Eisenstein series. For example, consider the Eisenstein series (0.1) E(τ, s) = ∑ γ∈Γ∞\Γ0(q) 2(d)(cτ + d)−2k−1 Im(γτ) s 2−k Here γ = ( a b c d ) , −q is a fundamental discriminant of an imaginary quadratic field, and 2 = (−q ). This Eisenstein series was used in the celebrated work of Gross and Zagier ([GZ, Chapter IV]) to compute the central derivative of cuspidal modular forms of weight 2k + 2. The Eisenstein series is holomorphic 1991 Mathematics Subject Classification. 11G05 11M20 14H52.