On zeros of Mellin transforms of $\textrm {SL}_{2}(\textbf {Z})$ cusp forms

Abstract:

We compute zeros of Mellin transforms of modular cusp forms for $S{L_2}({\mathbf {Z}})$. Such Mellin transforms are eigenforms of Hecke operators. We recall that, for all weights k and all dimensions of cusp forms, the Mellin transforms of cusp forms have infinitely many zeros of the form $k/2 + t\sqrt { - 1}$, i.e., infinitely many zeros on the critical line. A new basis theorem for the space of cusp forms is given which, together with the Selberg trace formula, renders practicable the explicit computations of the algebraic Fourier coefficients of cusp eigenforms required for the computations of the zeros. The first forty of these Mellin transforms corresponding to cusp eigenforms of weight $k \leqslant 50$ and dimension $\leqslant 4$ are computed for the sections of the critical strips, $\sigma + t\sqrt { - 1}$, $k - 1 < 2\sigma < k + 1$, $- 40 \leqslant t \leqslant 40$. The first few zeros lie on the respective critical lines $k/2 + t\sqrt { - 1}$ and are simple. A measure argument, depending upon the Riemann hypothesis for finite fields, is given which shows that Hasse-Weil L-functions (including the above) lie among Dirichlet series which do satisfy Riemann hypotheses (but which need not have functional equations nor analytic continuations).