# A converse theorem without root numbers

## TL;DR

This paper proves a version of Weil's converse theorem that relaxes the requirement of fixed root numbers, allowing for more general functional equations with varying root numbers.

## Contribution

It introduces a converse theorem that assumes functional equations for character twists without fixing their root numbers, expanding the scope of previous results.

## Key findings

- Proves a generalized Weil's converse theorem
- Allows root numbers to vary arbitrarily in functional equations
- Extends the applicability of converse theorems in number theory

## Abstract

We answer a challenge posed in (Math. Ann. 363 (2015), no. 1-2, 423-454) by proving a version of Weil's converse theorem that assumes a functional equation for character twists but allows their root numbers to vary arbitrarily.