math.NT — Number Theory
Unitary character moments and a zero-free half-plane for unitary Weil L-functions
We extend Liu's moment estimate for ordinary ideal-character sums to two independent fixed unitary twists, preserving the full length range of the finite-order theorem. The proof keeps the two character coefficients separate through the Gauss transformations and controls exceptional terms by the poles of their Mellin transforms. The case of two equal twists strengthens the fourth moment used in zero detection. Using the analytic results of Hu and Liu specified in the paper, we obtain the zero-free half-plane for every unitary Hecke -function over every number field, including characters of infinite order. This extends the known finite-order bound to the full unitary family. Unitary induction gives the same half-plane for finite-dimensional continuous unitary Weil representations, with the norm-character poles counted exactly.