Root Numbers of Asai L-Functions

Let E/F be a quadratic extension of p-adic fields. We compute the value of is an element of(1/2, pi, r,psi) for a square integrable representation pi of GL(n)(E), which is (Galois) conjugate self-dual, where r denotes the Asai representation. This is the twisted version of a well-known result due to Bushnell and Henniart. The proof makes use of a result on the corresponding global root number, which is proved by a method conceived by Lapid and Rallis.