ON THE GLOBAL ROOT NUMBERS OF GL(n)× GL(m)

which converges absolutely in R(s) > 1. One knows by Jacquet, Piatetski-Shapiro and Shalika ([JPSS]), and Shahidi ([Sh1]), that this L-function extends to a meromorphic function on all of C and admits a functional equation L(s, π × π) = W (π × π)(N(π × π)dF ) 1 2 L(1− s, π × π ∨ ). Here π (resp. π ∨ ) denotes the contragredient of π (resp. π), dF the discriminant of F , and N(π × π) a positive rational number (“the conductor”). The arithmetically important global root numberW (π×π) is non-zero and satisfies W (π × π)W (π × π ∨ ) = 1. We will be particularly interested in the self-dual situation, when W (π × π) = ±1. It is of importance to determine the sign. Our object here is to make some modest progress on this question.