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

TL;DR

This work introduces a novel multiplicative non-Hecke twist $\psi$ defined by $\psi(\mathfrak{p}) = \alpha^{N(\mathfrak{p})}$, extended to finite ideles, and demonstrates that twisting motivic L-functions $L(M,s)$ by $\psi$ yields $L(M,s,\psi)$ with an expanded domain of absolute convergence, preserved Euler product, and meromorphic continuation to $\mathbb{C}$ for $|\alpha|<1$. The authors specialize to Dirichlet and modular L-functions, establishing explicit abscissas of convergence, pole structures, and meromorphic continuations, including how poles arise from equations of the form $\alpha^{p} c_{p,i} p^{-s} = 1$. On the $p$-adic side, they construct convergent $p$-adic Dirichlet series and Euler products via the $\psi$-twist and derive a Mahler expansion for these twisted $p$-adic objects, yielding a coherent $p$-adic analogue to the complex theory. Collectively, the results provide a robust framework for non-Hecke twists of motivic and automorphic L-functions and open new avenues for $p$-adic analytic number theory in the context of arithmetic twists.

Abstract

We investigate the twisting of motivic $L$-functions by a family of multiplicative characters $ψ$, defined on prime ideals $\mathfrak{p}$ via $ψ(\mathfrak{p})=α^{N(\mathfrak{p})}$ for a fixed $α\in \mathbb{C}$. One can extend $ψ$ to a continuous non-Hecke character on the idele group of a number field. For $|α|<1$, the resulting $ψ$-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 $ψ$-twisting allows the construction of convergent $p$-adic Dirichlet series and $p$-adic Euler products which have some similarities with their complex counterparts.

Figures (3)

• Figure 1: Graph of the integer log function $S$. The diagonal dotted lines are given by the values of $S$ at $p$, $2p$, $3p$, $\dots$, where $p$ is a prime number. The graphic also shows the lower bound $\frac{3}{\log(3)} \log(n)$.
• Figure 2: Poles of the $\psi$-twisted Riemann zeta function for $\alpha=\frac{1}{2}$. The largest real parts of poles are at $\frac{-3\log(2)}{\log(3)} \approx -1.89$.
• Figure 3: Real part $\sigma$ of the largest $30$ poles of the $\psi$-twisted Riemann zeta function for each $0<\alpha<1$. The largest $\sigma$ (i.e., the upper curve) is associated to $p=3$, the next to $p=2$, then $p=5$ etc. The upper curve also gives the abscissa of absolute convergence $\sigma_a$.

Theorems & Definitions (51)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Remark 2.5
• Definition 2.6
• Definition 2.7
• Proposition 2.8
• proof