Non-Vanishing of Derivatives of L-Functions of Hilbert Modular Forms in The Critical Strip

TL;DR

This paper proves that, on average, derivatives of L-functions of cuspidal Hilbert modular forms with large weight do not vanish in certain regions of the critical strip, extending classical modular form results.

Contribution

It establishes non-vanishing results for derivatives of Hilbert modular L-functions in the critical strip, generalizing known results from classical modular forms.

Findings

Derivatives of L-functions do not vanish on average in specified regions

Results are valid for forms with sufficiently large weight

Extends classical modular form non-vanishing results to Hilbert modular forms

Abstract

In this paper, we show that, on average, the derivatives of $L$-functions of cuspidal Hilbert modular forms with sufficiently large weight $k$ do not vanish on the line segments $\Im(s)=t_{0}$, $\Re(s)\in(\frac{k-1}{2},\frac{k}{2}-\epsilon)\cup(\frac{k}{2}+\epsilon,\frac{k+1}{2})$. This is analogous to the case of classical modular forms.