On a multiplicative non-Hecke twist of motivic L-functions

TL;DR

This paper introduces a novel multiplicative twist of motivic L-functions using a family of characters, revealing new analytic properties and applications to Dirichlet and modular form L-functions, as well as p-adic series.

Contribution

It defines a multiplicative non-Hecke twist of motivic L-functions and explores its analytic properties and applications, including p-adic series construction.

Findings

Enhanced half-plane of absolute convergence

Preservation of Euler product structure

Construction of convergent p-adic Dirichlet series

Abstract

We investigate the twisting of motivic $L$-functions by a family of multiplicative characters $\psi$, defined on prime ideals $\mathfrak{p}$ via $\psi(\mathfrak{p})=\alpha^{N(\mathfrak{p})}$ for a fixed $\alpha \in \mathbb{C}$. One can extend $\psi$ to a continuous non-Hecke character on the idele group of a number field. For $|\alpha|<1$, the resulting $\psi$-twisted $L$-function has interesting analytic properties: an enhanced half-plane of absolute convergence, preservation of the Euler product structure, and meromorphic continuation to the complex plane. We give applications to Dirichlet $L$-functions and $L$-functions associated to modular forms. Furthermore, we show that $\psi$-twisting allows the construction of convergent $p$-adic Dirichlet series and $p$-adic Euler products which have some similarities with their complex counterparts.