Twists, Euler products and a converse theorem for $L$-functions of degree 2 in the Selberg class

TL;DR

This paper establishes a link between the Euler product structure and the analytic properties of linear twists of degree 2 $L$-functions, culminating in a uniqueness result for the Riemann zeta function squared within the Selberg class.

Contribution

It introduces a new criterion connecting Euler products to linear twists and proves a converse theorem characterizing $\

Findings

The only degree 2 $L$-function in the Selberg class with a pole at 1 is $\

The paper provides a method to verify analytic properties of $L$-functions via linear twists

It demonstrates that $\

Abstract

We prove a general result relating the shape of the Euler product of an $L$-function to the analytic properties of certain linear twists of the $L$-function itself. Then, by a sharp form of the transformation formula for linear twists, we check the required analytic properties in the case of $L$-functions of degree 2 and conductor 1 in the Selberg class. Finally we prove a converse theorem, showing that $\zeta(s)^2$ is the only member of the Selberg class satisfying the above conditions and, moreover, having a pole at $s=1$.