On the Linnik-Sprindzuk theorem about the zeros of L-functions

The classical Linnik-Sprindzuk phenomenon shows that the Riemann Hypothesis for Dirichlet L-functions is equivalent to the Riemann Hypothesis plus a suitable property of the vertical distribution of the zeros of the Riemann zeta function. We prove a different form of such a phenomenon (in the general framework of the Selberg class), relating certain properties of the zeros of the twists of an L-function to other properties of the zeros of the L-function itself.