TURAN INEQUALITIES AND ZEROS OF DIRICHLET SERIES ASSOCIATED WITH CERTAIN CUSP FORMS

The ''Turan inequalities'' are a countably infinite set of conditions about the power series coefficients of certain entire functions which are necessary in order for the function to have only real zeros. We gi ve a one-parameter family of generalized Dirichlet series, each with f unctional equation, for which the Turan inequalities hold for the asso ciated xi-function (normalized so that the critical line is the real a xis). For a discrete set of values of the parameter the Dirichlet seri es has an Euler product and is the L-series associated to a modular fo rm. For these we expect the analogue of the Riemann Hypothesis to hold . For the rest of the values of the parameter we do not expect an anal ogue of the Riemann Hypothesis. We show for one particular value of th e parameter that the Dirichlet series in fact has zeros within the reg ion of absolute convergence.