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 ( N ) and O K its ring of integers. Let D be a prime ideal in O K of prime norm congruent to 3 mod 4 . Under these assumptions, there exists Hecke characters ψ D of K with conductor ( D ) and infinite type ( 1 , 0 ) . Their L-series L ( ψ D , s ) are associated to a CM elliptic curve A ( N , D ) defined over the Hilbert class field of K. We will prove a Waldspurger-type formula for L ( ψ D , s ) of the form L ( ψ D , 1 ) = Ω ∑ [ 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.