A converse theorem for Γ0(13)

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 Γ0(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.