A converse theorem for $Γ_0(13)$

J. B. Conrey, David W. Farmer, B. E. Odgers, N. C. Snaith

Published 2006-01-23Version 1

We prove that a Dirichlet series with a functional equation and Euler product of a particular form can only arise from a holomorphic cusp form on the Hecke congruence group $\Gamma_0(13)$. The proof does not assume a functional equation for the twists of the Dirichlet series. The main new ingredient is a generalization of the familiar Weil's lemma that played a prominent role in previous converse theorems.