On quotients of derivatives of $L$-functions inside the critical strip

TL;DR

This paper extends the study of the non-vanishing and transcendental properties of derivatives of $L$-functions from the central point to arbitrary points within the critical strip, broadening the scope of previous research.

Contribution

It introduces a new analysis of $L$-function derivatives at arbitrary critical points, expanding beyond the central point focus of prior work.

Findings

Extended non-vanishing results to arbitrary critical points.

Analyzed transcendental nature of derivatives at these points.

Provided new insights into the behavior of $L$-functions within the critical strip.

Abstract

In 2011, Gun, Murty and Rath studied non-vanishing and transcendental nature of special values of a varying class of $L$-functions and their derivatives. This led to a number of works by several authors in different set-ups including studying higher derivatives. However, all these works were focused around the central point of the critical strip. In this article, we extend the study to arbitrary points in the critical strip.