Forbidden conductors of $L$-functions and continued fractions of particular form
Acta Arithmetica MSC: Primary 11M41; Secondary 11A55. DOI: 10.4064/aa220721-30-9 Opublikowany online: 26 October 2022
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.