Forbidden conductors of $L$-functions and continued fractions of particular form
Volume 210 / 2023
Acta Arithmetica 210 (2023), 1-21
MSC: Primary 11M41; Secondary 11A55.
DOI: 10.4064/aa220721-30-9
Published online: 26 October 2023
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. Published in Open Access (under CC-BY license).
