Forbidden conductors of L-functions and continued fractions of particular form

TL;DR

This paper investigates forbidden conductors of degree 2 L-functions in the extended Selberg class by connecting the problem to continued fractions and their weights, establishing uniqueness results and deriving theoretical and computational conclusions.

Contribution

It introduces a novel technique linking L-function conductors to continued fractions, proving the uniqueness of the associated weight and deriving new theoretical and computational results.

Findings

Uniqueness of the weight $w_q$ for existing L-functions with conductor $q$

Identification of forbidden conductors based on continued fraction properties

Theoretical and computational results supporting the main theorem

Abstract

In this paper we study the forbidden values of the conductor $q$ of the $L$-functions of degree 2 in the extended Selberg class by a novel technique, linking the problem to certain continued fractions and to their weight $w_q$. Our basic result states that if an $L$ function with conductor $q$ exists, then the weight $w_q$ is unique in a suitable sense. From this we deduce several results, both of theoretical and computational nature.