On the invariants of L-functions of degree 2, I: twisted degree and internal shift

TL;DR

This paper investigates how certain invariants of degree 2 L-functions, specifically degree and internal shift, behave under twisting by Dirichlet characters, aiming to support a generalized Weil converse theorem.

Contribution

It demonstrates that degree and internal shift invariants remain unchanged under twist for degree 2 L-functions under certain conditions, advancing the understanding of their structural properties.

Findings

Degree and internal shift are invariant under twist.

Conditions identified for invariance to hold.

Lays groundwork for a generalized Weil converse theorem.

Abstract

This is the first part of a series of papers where the behaviour of the invariants under twist by Dirichlet characters is studied for $L$-functions of degree 2. Here we show, under suitable conditions, that degree and internal shift remain unchanged under twist. The ultimate goal of the series is to prove a general version of Weil converse theorem with minimal assumptions on the shape of the functional equation of the twists.