Artin L-function on PGL3

Abstract

We study Artin $L$-functions on a finite $2$-dimensional complex $X_{\Gamma }$ arising from PGL${}_3$ attached to finite-dimensional representations $\rho$ of its fundamental group. Some key properties, such as rationality, functional equation, and invariance under induction, of these functions are proved. Moreover, using a cohomological argument, we establish a connection between the Artin $L$- functions and the $L$-function of $\rho$, extending the identity on zeta functions of $X_{\Gamma }$ obtained in [KL,KLW].