On the reciprocity law for the twisted second moment of Dirichlet $L$-functions

We investigate the reciprocity law, studied by Conrey~\cite{Con07} and Young~\cite{You11a}, for the second moment of Dirichlet L-functions twisted by $\chi(a)$ modulo a prime $q$. We show that the error term in this reciprocity law can be extended to a continuous function of $a/q$ with respect to the real topology. Furthermore, we extend this reciprocity result, proving an exact formula involving also shifted moments. We also give an expression for the twisted second moment involving the coefficients of the continued fraction expansion of $a/q$, and, consequently, we improve upon a classical result of Selberg on the second moment of Dirichlet L-functions with two twists. Finally, we obtain a formula connecting the shifted second moment of the Dirichlet $L$-functions with the Estermann function. In particular cases, this result can be used to obtain some simple explicit exact formulae for the moments.