---
abstract: 'We show that the finite part of the adjoint $L$-function (including contributions from all non-archimedean places, including ramified places) is holomorphic in ${{\mathrm{Re}}}(s) \ge 1/2$ for a cuspidal automorphic representation of ${{\mathrm{GL}}}_3$ over a number field. This improves the main result of [@H18]. We obtain more general results for twisted adjoint $L$-functions of both ${{\mathrm{GL}}}_3$ and quasisplit unitary groups. For unitary groups, we explicate the relationship between poles of twisted adjoint $L$-functions, endoscopy, and the structure of the stable base change lifting.'
address:
- 'Department of Mathematics, State University of New York at Buffalo, Buffalo, NY, USA'
- 'Department of Mathematics and Statistics, University of Calgary, Canada'
author:
- Joseph Hundley
- Qing Zhang
title: 'Adjoint $L$-functions for ${{\mathrm{GL}}}(3)$ and ${{\mathrm {U}}}_{2,1}$'
---

Introduction
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Motivation and Background
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Let $F$ be a number field and ${{\mathbb {A}}}$ be its ring of adeles. Let $H$ be a connected, reductive $F$-group, isomorphic to ${{\mathrm{GL}}}_3$ over the algebraic closure ${\overline}F$ of $F.$ The $L$-group $^LH$ of $H$ is then the semi-direct product of ${{\mathrm{GL}}}_3({\mathbb{C}})$ with a Galois or Weil group, depending on which form of the $L$-group we consider (cf.[@BorelCorvallis]). Differentiating the action of $^LH$ on its own identity component 