Real zeros of Eisenstein series and Rankin-Selberg L-functions

Abstract

We prove that the Eisenstein series E(z, s) have no real zeroes for s ∈ (0, 1) when the value of the imaginary part of z is in the range $$\tfrac{1}{5}$$ < Im z < 4.94. For very large and very small values of the imaginary part of z, E(z, s) have real zeros in (½, 1), i.e. GRH does not hold for the Eisenstein series. Using these properties, we prove that the Rankin-Selberg L-function attached with the Ramanujan τ-function has no real zeros in the critical strip, except at the central point s = ½.