AN EXTENSION OF HECKE’S CONVERSE THEOREM

J. B. Conrey and D. W. Farmer

Abstract. Associated to a newform f(z) is a Dirichlet series L_{f} (s) with functional equation and Euler product. Hecke showed that if the Dirichlet series F (s) has a functional equation of the appropriate form, then F (s) = L_{f} (s) for some holomorphic newform f(z) on Γ(1). Weil extended this result to Γ_{0}(N) under an assumption on the twists of F (s) by Dirichlet characters. We show that, at least for small N, the assumption on twists can be replaced by an assumption on the local factors of the Euler product of F (s).

Typeset by AMS-TEX