  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.
