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

الأصل · EN

Let N = 1 mod 4 be the negative of a prime, K=Q(sqrtN) 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 ψ of K with conductor () and infinite type (1,0). Their L-series L(ψ,s) are associated to a CM elliptic curve E(N,) defined over the Hilbert class field of K. We will prove a Waldspurger-type formula for L(ψ,s) of the form L(ψ,1) = Ω∑[],ᵢ r(,[],I) m[],ᵢ([]) where the sum is over class ideal representatives I of a maximal order in the quaternion algebra ramified at |N| and infinity and [] 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.