$L$-functions of degree $2$ and conductor $1$: underlying ideas and a generalisation

TL;DR

The paper classifies normalized degree-2, conductor-1 $L$-functions in the extended Selberg class by linking nonlinear twists to a universal invariant $\chi_F$, proving that such $F$ arise from level-1 Hecke or Maass $L$-functions. The approach proceeds through four steps: (i) extracting structural invariants via nonlinear twists, (ii) formalizing virtual $\gamma$-factors and aligning $F$ with a Hecke/Maass-type equation, (iii) analyzing period functions to restrict the equation's form, and (iv) concluding that a constant ratio forces a genuine Hecke/Maass functional equation. A corrigendum fixes a slip in a previous section without affecting the final result. Theorem 2 extends the framework to functional equations involving two Dirichlet series. Together the results support automorphic descriptions for these low-degree cases and sharpen the connection between the invariants $\chi_F$ and the underlying automorphic data.

Abstract

We present a streamlined account of a recent theorem on the classification of the $L$-functions of degree 2 and conductor 1 from the extended Selberg class. We also present a more general new result dealing with functional equations involving two Dirichlet series. Further, we correct a slip in the original proof of the above theorem, which however does not affect the final result.