Automorphic L-function

introduction

• an automorphic L-function is a function $L(s,\pi,r)$
  • complex variable $s$,
  • associated to an automorphic form $\pi$ of a reductive group $G$ over a global field
  • a finite-dimensional complex representation $r$ of the Langlands dual group $L_G$ of $G$,
• They were introduced by Langlands (1967, 1970, 1971)
• generalizing the Dirichlet $L$-series of a Dirichlet character and the Mellin transform of a modular form