Let k(S) be the product of the ϕ(k) Dirichlet L-functions formed with characters modulo k. We prove the existence of explicit numerical zero-free regions for k(S). The first result is that k(S) has at most a single zero in the region {, where R = 9.645908801 and M = max {k, k |t|, 10}. The only possible zero in this region is a simple real zero arising from an L-function formed with a real non-principal character. The second result is that if χ1 and χ2 are distinct real primitive characters modulo k1 and k2, respectively, and if β1 is a zero of L(s, χi), i = 1, 2, then min , where , and .