A formula for the central value of certain Hecke L-functions

Abstract

Let N = 1 mod 4 be the negative of a prime, K=Q(sqrt{N}) and O_K its ring of integers. Let D be a prime ideal in O_K of prime norm congruent to 3 modulo 4. Under these assumptions, there exists Hecke characters $\psi_{\D}$ of K with conductor $(\D)$ and infinite type $(1,0)$. Their L-series L(\psi_\D,s)$ are associated to a CM elliptic curve E(N,\D) defined over the Hilbert class field of $K$. We will prove a Waldspurger-type formula for L(\psi_\D,s) of the form L(\psi_\D,1) = \Omega \sum_{[\A],I} r(\D,[\A],I) m_{[\A],I}([\D]) where the sum is over class ideal representatives I of a maximal order in the quaternion algebra ramified at |N| and infinity and [\A] are class group representatives of $K$. An application of this formula for the case N=-7 will allow us to prove the non-vanishing of a family of L-series of level $7|D|$ over $K$.