A Formula for the Central Value of Certain Hecke L-functions

Let N ≡ 1 mod 4 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 3 modulo 4. 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],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.