Zero free regions for Dirichlet series

International audienceIn this paper, we are interested in explicit zero-free discs for some Dirichlet series and we also study a general Beurling-Nyman criterion for $L$-functions. Our results generalize and improve previous results obtained by N.~Nikolski and by A.~de Roton. As a concrete application, we get, for example, a Beurling-Nyman type criterion for the Siegel zero problem