Determining cusp forms by central values of Rankin–Selberg L-functions

Central Values of Degree Six L-functions

Let κ′, κ ≥ 3 be two odd integers. Let f (resp. g) be a normalized holomorphic modular form of weight 2κ (resp. κ′ + 1) and level one on the upper half plane h. Assume that they are Hecke eigenforms. Let L(s, Sym g×f) be the completed degree six L-function and we normalize so that s = 1 2 is the center of symmetry. Let 〈−,−〉 be the Petersson inner product, defined using the usual measure on h s...

Computing Central Values of L-functions

How fast can we compute the value of an L-function at the center of the critical strip? We will divide this question into two separate questions while also making it more precise. Fix an elliptic curve E defined over Q and let L(E, s) be its L-series. For each fundamental discriminant D let L(E,D, s) be the L-series of the twist ED of E by the corresponding quadratic character; note that L(E, 1...

Special Values of Koecher-maass Series of Siegel Cusp Forms

The purpose of this paper is to give a generalization of the above result to the case of a Siegel cusp form f , where now L(f, χ, s) is replaced by an appropriate χ-twist of the Koecher-Maass series attached to f . More precisely, let f be a cusp form of even integral weight k ≥ g + 1 w.r.t. the Siegel modular group Γg := Spg(Z) of genus g and write a(T ) (T a positive definite half-integral ma...

Central Values of Hecke L-functions of Cm Number Fields

0. Introduction. It is well known that the zeta function of CM (complex multiplication) abelian varieties can be given in terms of L-functions of associated Hecke characters. In this paper, we prove a formula expressing the central special value of the L-function of certain Hecke characters in terms of theta functions. The formula easily implies that the central value is nonnegative and yields ...

The Cubic Moment of Central Values of Automorphic L-functions