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A converse theorem for $Γ_0(13)$

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

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 $Γ_0(13)$. 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.

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J. B. Conrey, David W. Farmer, B. E. Odgers, N. C. Snaith. 2006-01-23. A converse theorem for $Γ_0(13)$. https://arxiv.org/abs/math/0601549

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