A formula for the central value of certain Hecke L-functions - Pacetti, A. (2005)
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 mod 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],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. © 2005 Elsevier Inc. All rights reserved.