The highest lowest zero of general L-functions

Abstract

Stephen D. Miller showed that, assuming the generalized Riemann Hypothesis, every entire L-function of real archimedian type has a zero in the interval 1 2 + it with −t 0 < t < t 0 , where t 0 ≈ 14.13 corresponds to the first zero of the Riemann zeta function. We give a numerical example of a self-dual degree-4 L-function whose first positive imaginary zero is at t 1 ≈ 14.496. In particular, Miller's result does not hold for general L-functions. We show that all L-functions satisfying some additional (conjecturally true) conditions have a zero in the interval (−t 2 , t 2 ) with t 2 ≈ 22.661.

A note on the zeros of zeta and L-functions

“…where 2) and the sum runs over the nontrivial zeros ρ = 1 2 + iγ of ζ(s) counted with multiplicity. Drawing upon the work of Carneiro and Littmann [7], upper and lower bounds for S(t) are established by replacing F(x) in the sum over zeros by (optimally chosen) real entire majorants and minorants of F(x) and then applying the explicit formula.…”

“…2 Note the difference between the definition of N (t, π) and the definition for N (t) given in the Sect. 1.…”

“…In connection to our Theorem 7, the interesting papers of S. D. Miller [17] and Bober et al [2] consider questions related to estimating the height of the lowest zero and the size of the largest gap between zeros for L-functions in certain families.…”

On the argument of L-functions

“…The examples include the Dirichlet L-functions L(·, χ) for primitive characters χ. 1 Our notation is motivated by L-functions arising from cuspidal automorphic representations π of GL(m) over a number field.…”

Maass Forms on GL(3) and GL(4)

“…The first Sp(4, Z) L−function, which has (λ 1 , λ 2 ) = (12.46875, 4.72095) has the curious property that its lowest zero on the critical line is at 1 2 ± 14.496i, which is higher than the lowest zero of the Riemann zeta-function. This is discussed further by Bober, et al [3]. As a check on these calculations, we also compute the standard (degree 5) L−function using the data from the spin (degree 4) L−function (see section 6).…”