#### Vol. 304, No. 1, 2020

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On the archimedean local gamma factors for adjoint representation of GL$_3$, part I

### Fangyang Tian

Vol. 304 (2020), No. 1, 303–345
##### Abstract

Studying the analytic properties of the partial Langlands $L$-function via Rankin–Selberg method has proved to be successful in various cases. Yet in few cases is the local theory studied at the archimedean places, which causes a tremendous gap in the completion of the analytic theory of the complete $L$-function. We establish the meromorphic continuation and the functional equation of the archimedean local integrals associated with D. Ginzburg’s global integral (1991) for the adjoint representation of ${GL}_{3}$. Via the local functional equation, the local gamma factor $\Gamma \left(s,\pi ,Ad,\psi \right)$ can be defined. In a forthcoming paper, we will compute the local gamma factor $\Gamma \left(s,\pi ,Ad,\psi \right)$ explicitly, which fill in some blanks in the archimedean local theory of Ginzburg’s global integral.

##### Keywords
archimedean local integral, Rankin–Selberg integral, adjoint L-function for general linear group
##### Mathematical Subject Classification 2010
Primary: 22E45, 22E50
Secondary: 11F70
##### Milestones
Revised: 29 November 2018
Accepted: 13 March 2019
Published: 18 January 2020
##### Authors
 Fangyang Tian Department of Mathematics National University of Singapore Singapore