An extension of Hecke's converse theorem

1995/02/22 by Conrey, J. Brian, Farmer, David W.
#Complex Variables (math.CV) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.math/9502209

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 Γ₀(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).

Related

• Converse theorems assuming a partial Euler product
• A converse theorem for Γ₀(13)
• L-functions and higher order modular forms
• Identities for the product of Two Dirichlet Series Satisfying Hecke's Functional Equation
• An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials