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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, thenthe weight $w_q$ is unique in a suitable sense. From this we deduce severalresults, both of theoretical and computational nature.

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