Twisted quadratic moments for Dirichlet $L$-functions at $s=2$

Let $c$, $n$ be given positive integers. Let $q>2$ be coprime with $c$. Let $X_q$ be the multiplicative group of order $\phi(q)$ of the Dirichlet characters modulo $q$. Set $$ M(q,c,n):={2\over\phi(q)}\sum_{\chi\in X_q\atop\chi(-1)=(-1)^n}\chi(c)\vert L(n,\chi)\vert^2. $$ The goal of this paper is to explain how one can compute explicit formulas for $M(q,c,n)$ for given small integers $n$ and $c$. As an example, we give explicit formulas for $M(q,c,2)$ for $c\in\{1,2,3,4,6\}$, and for $M(p,5,2)$ for $p$ a prime integer. As a consequence, we show that a previously published formula for $M(p,3,2)$ is false.