# On the linear twist of degree 1 functions in the extended Selberg class

## TL;DR

This paper investigates the properties of linear twists of degree 1 functions in the extended Selberg class, establishing a functional equation and analyzing zero distribution to deepen understanding of their analytic behavior.

## Contribution

It introduces a functional equation for linear twists of degree 1 functions and studies their zero distribution, extending the theory of Selberg class functions.

## Key findings

- Linear twists satisfy a Hurwitz-Lerch type functional equation.
- Distribution of zeros of the linear twist is characterized.
- New insights into the symmetry properties of degree 1 functions in the Selberg class.

## Abstract

Given a degree 1 function $F\in\mathcal{S}^{\sharp}$ and a real number $\alpha$, we consider the linear twist $F(s,\alpha)$, proving that it satisfies a functional equation reflecting $s$ into $1-s$, which can be seen as a Hurwitz-Lerch type of functional equation. We also derive some results on the distribution of the zeros of the linear twist.