On the degree of certain local L-functions

Let π be an irreducible supercuspidal representation of GL_n(F), where F is a p-adic field. By a result of Bushnell and Kutzko, the group of unramified self-twists of π has cardinality n/e, where e is the o_F-period of the principal o_F-order in M_n(F) attached to π. This is the degree of the local Rankin-Selberg L-function L(s, π × π^∨). In this paper, we compute the degree of the Asai, symmetric square, and exterior square L-functions associated to π. As an application, assuming p is odd, we compute the conductor of the Asai lift of a supercuspidal representation, where we also make use of the conductor formula for pairs of supercuspidal representations due to Bushnell, Henniart, and Kutzko (1998).