Beyond the extended Selberg class: $1<d_F< 2$

TL;DR

This paper proves that a broad class of Dirichlet series, including many automorphic L-functions, contains no elements with degrees strictly between 1 and 2, extending previous results with a simpler proof.

Contribution

It establishes the non-existence of elements with degrees between 1 and 2 in a large class of Dirichlet series, generalizing prior work and providing a shorter proof.

Findings

No elements of degrees between 1 and 2 in the class ${rakA}^{ ext{ extasteriskcentered}}$

Includes standard, tensor, exterior, and symmetric square L-functions of automorphic forms

Simpler proof compared to previous methods

Abstract

We show that a class of Dirichlet series ${\mathfrak{A}}^{\#}$ that is much larger than the extended Selberg class ${\mathscr{S}}^{\#}$, and also contains the standard as well as the tensor product, exterior square and symmetric square $L$-functions of automorphic $L$-functions of $GL_n$ over number fields, does not have any elements of degrees between $1$ and $2$. The proof of our more general theorem is very different from the proof of Kaczorowski and Perelli for the class ${\mathscr{S}}^{\#}$, and is much shorter and simpler even in that case.