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 φ(q) of the Dirichlet characters modulo q. Set M (q, c, n) := 2 φ(q) χ∈Xq χ(−1)=(−1) n χ(c)|L(n, χ)| 2. The goal of this paper is to explain how one can compute explicit formulas for M (q, c, n) for a given small integers n and c. As an example, we give explicit formulas for M (q, c, 2) for c ∈ {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. 0 2010 Mathematics Subject Classification. Primary. 11M06.