Computation of epsilon factors of pairs for GLn on a local field, Part One

Let F be a non-Archimedean local field and phi a non-trivial additive character of F. Let sigma be a representation of the Weil-Deligne group of F and sigma its contragredient representation. We compute epsilon(sigma x sigma, phi, 1/2). Analogously, we compute epsilon(pi x pi, phi, 1/2) for all irreducible admissible representations pi of GL(n)(F). Consequently, if F has characteristic zero, and sigma, pi correspond via the Langlands correspondence established by M. Harris or the correspondence constructed by the authors, then we have epsilon(sigma x sigma, phi, s) = epsilon(pi x pi,phi,s) for all s is an element of C.