Multiplicity results for the functional equation of the Dirichlet $L$-functions

Given an integer $q$ and a primitive character $\chi$ modulo $q$, the functional equation of the Dirichlet $L$-function $L(s,\chi)$ is determined by the \emph{signature} of $\chi$, i.e. by $\chi(-1)$ (the parity) and $\tau(\chi)$ (the Gauss sum). In this paper we prove several results about the cardinalities of the sets $T(\chi):=\{\psi:\,\tau(\psi)=\tau(\chi)\}$ and $W(\chi):=\{\psi:\,\tau(\psi) = \tau(\chi),\ \psi(-1)=\chi(-1)\}$, mainly an algorithm for their computation and optimal upper and lower bounds for their values, when $q$ is either an odd prime power or a composite number of special form. For the same $q$ we compute also the number of distinct Gauss sums and of distinct signatures: the latter number deserves a special attention because it coincides with the number of non-trivial functional equations of degree $1$ and conductor $q$ in the Selberg class.