A Riemann Hypothesis Condition for Metaplectic Eisenstein Series

Let A be the adeles of the function field F_q(T). If E{g, s) is the Eisenstein series associated with the spherical vector in the principal series representation of the double cover of GL₂{A), then the Fourier coefficients (Whittaker functions) of E[g, s) have an Euler product and satisfy a 'Riemann Hypothesis' condition. Namely, the zeros occur only on the line K(s) = 0. These Whittaker functions are essentially quadratic L-functions associated with hyperelliptic curves.

This result adds to the list of known examples for which the Fourier coefficients of Eisenstein series coming from the spherical vector in the principal series representation satisfy a Riemann Hypothesis condition.