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

AbstractLet N≡1mod4 be the negative of a prime, K=Q(N) and OK its ring of integers. Let D be a prime ideal in OK of prime norm congruent to 3mod4. Under these assumptions, there exists Hecke characters ψD of K with conductor (D) and infinite type (1,0). Their L-series L(ψD,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(ψD,s) of the form L(ψD,1)=Ω∑[A],Ir(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