A note on the invariants of the $L$-functions

TL;DR

This paper clarifies the concept of invariants of $L$-functions by demonstrating that invariants are characterized by stability under specific transformations and establishing the existence of rational extensions with these invariance properties.

Contribution

It introduces the construction of rational extensions of invariants of $L$-functions, providing a formal framework for understanding their invariance properties.

Findings

Every invariant admits a rational extension with desired invariance properties

The existence of such an extension is established through a novel construction

Clarifies the relationship between invariants and their stability under transformations

Abstract

We explain the exact meaning of a statement we made in a previous paper on invariants, namely that a complex-valued function of the data of the functional equation of an $L$-function is an invariant if and only if it is stable under the multiplication and factorial formulae. To this end, we show that every invariant has a so-called rational extension, having some desired invariance properties. The existence of such an extension, which enables to express formally the above heuristic concept, is not apparent and its construction is the main novelty of the paper.