Alexandra Florea, a number theorist at UC Irvine, presents a research seminar talk for the joint IAS/Princeton Number Theory seminar on when Dirichlet L-functions fail to vanish at the central point 1/2. She walks through non-vanishing results, conditional on the Generalized Riemann Hypothesis, for pairs of L-functions attached to Dirichlet characters sitting in the same Galois orbit, then extends the argument to simultaneous non-vanishing of four L-functions as the character chi varies modulo q, this time assuming both GRH and the Ramanujan-Petersson conjecture. The talk draws on joint work with Bui-Ngo and with Bui-Milinovich, laying out the analytic machinery, moment computations, and conjectural inputs behind each result. Delivered at the IAS's Simonyi Hall, it is a technical specialist talk aimed at researchers in analytic number theory rather than a general audience introduction, assuming familiarity with L-functions and moment methods.