In this paper we introduce and investigate a new type of series which we call a "Dirichlet-power" series. This series includes both Dirichlet series and power series as a special case. It a rose during the investigation of series representations of the second order hypergeometric zeta function. The Dirichlet-power series has many properties analogous to Dirichlet series. For a Dirichlet-power series we establish Domain of convergence and show the existence of zero free regions to the right half plane.