The standard $L$-function attached to a vector valued modular form

TL;DR

This paper defines and relates two $L$-functions associated with vector valued modular forms, demonstrating their meromorphic continuation across the entire complex plane, thus advancing understanding of their analytic properties.

Contribution

It introduces two new $L$-functions for vector valued eigenforms and establishes their meromorphic continuation, linking algebraic and analytic perspectives.

Findings

Both $L$-functions are related and can be continued meromorphically.

The first $L$-function is defined via eigenvalues of the form.

The second $L$-function is expressed as an Euler product with rational $p$-factors.

Abstract

We define two $L$-functions associated to a common vector valued eigenform $f$ transforming with the ``finite'' Weil representation. The first one can be seen as a standard zeta function defined by the eigenvalues of $f$. The second one can be interpreted as standard $L$-function defined as an Euler product where each $p$-factor is a rational function in terms of two unramified characters of the $p$-adic field $\Q_p$. We show that both $L$-functions are related and prove further that they both can be continued meromorphically to the whole complex $s$-plane.