# Eichler cohomology and zeros of polynomials associated to derivatives of $L$-functions

## TL;DR

This paper investigates the roots of cohomological analogues of period polynomials related to higher derivatives of $L$-functions, proposing conjectures and providing numerical and partial theoretical evidence.

## Contribution

It introduces a new class of higher derivative period polynomials, formulates conjectures on their root locations, and proves a special case for Eisenstein series.

## Key findings

- Numerical evidence supports conjectures on root locations.
- Proved a special case for Eisenstein series.
- Proposed general conjectures for higher derivative polynomials.

## Abstract

In recent years, a number of papers have been devoted to the study of roots of period polynomials of modular forms. Here, we study cohomological analogues of the Eichler-Shimura period polynomials corresponding to higher $L$-derivatives. We state general conjectures about the locations of the roots of the full and odd parts of the polynomials, in analogy with the existing literature on period polynomials, and we also give numerical evidence that similar results hold for our higher derivative "period polynomials" in the case of cusp forms. We prove a special case of this conjecture in the case of Eisenstein series.