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

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.