In this paper we study the forbidden values of conductor $q$ $L$-functions degree 2 in extended Selberg class by a novel technique, linking problem to certain continued fractions and their weight $w_q$. Our basic result states tha

We prove a generalization of the q-Selberg integral evaluation formula. The integrand is that multiplied by factor same form with respect to part variables. proof relies on quadratic norm formula Koornwinder polynomials. also derive generalizations Mehta's as limit cases our integral.