# Zeros of the extended Selberg class zeta-functions and of their derivatives

## TL;DR

This paper investigates the zeros of extended Selberg class zeta-functions and their derivatives, showing they have similar zero distributions near the critical line and exploring zero trajectories of these functions.

## Contribution

The paper extends known results about the zeros of the Riemann zeta-function and its derivative to a broader class of zeta-functions within the extended Selberg class.

## Key findings

- Zeros of zeta-functions and their derivatives are similarly distributed near the critical line.
- In small regions to the left of the critical line, these functions have the same number of zeros.
- Analysis of zero trajectories for a family of zeta-functions from the extended Selberg class.

## Abstract

Levinson and Montgomery proved that the Riemann zeta-function $\zeta(s)$ and its derivative have approximately the same number of non-real zeros left of the critical line. R. Spira showed that $\zeta'(1/2+it)=0$ implies $\zeta(1/2+it)=0$. Here we obtain that in small areas located to the left of the critical line and near it the functions $\zeta(s)$ and $\zeta'(s)$ have the same number of zeros. We prove our result for more general zeta-functions from the extended Selberg class $S$. We also consider zero trajectories of a certain family of zeta-functions from $S$.