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

Abstract

Given a degree 1 function $F\in {\mathcal{S}}^{♯}$ 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.

12 pages