132 J.W. COGDELL, £-FUNCTIONS FOR GLnand can be found in [15,23]. In essence, in the family of integrals w(s; W, W', ), if
(O) = O then the integral will again reduce to a finite sum and hence be entire. If
(O) =/= 0 and if s 0 is a pole of w(s; W, W', ) then the residue of the pole at s =so
will be of the form
c<I>(O) ( W(g)W'(g)I det(g)ls^0 dg
Jzn(k) Nn(k)\ GLn(k)which is the Whittaker form of an invariant pairing between 7r and 7r^1 ® I det Isa.
Thus we must have s 0 is pure imaginary and 7r ~ 71"^1 ®I det 180 for the residue to be
nonzero. This condition is also sufficient.
Theorem 3.4. If 7r and 7r^1 are both (unitary) supercuspidal, then L(s, 7r x 7r^1 ) = 1
if m < n and if m = n we have
L(s, 7r x 7r') =IT (1-aq-s)-1with the product over all a= q^80 with 7r ~ 71"^1 ® I det 180 •
3.1.5. Remarks on the general calculation
In the other cases, we must rely on the Bernstein-Zelevinsky classification of generic
representations of GLn(k) [97]. All generic representations can be realized as pre-
scribed constituents of representations parabolically induced from supercuspidals.
One can proceed by analyzing the Whittaker functions of induced representations
in terms of Whittaker functions of the inducing data as in [42] or by analyzing
the poles of the local integrals in terms of quasi invariant pairings of derivatives
of 7r and 7r^1 as in [15] to compute L(s, 7r x 7r^1 ) in terms of L-functions of pairs of
supercuspidal representations. We refer you to those papers or [58] for the explicit
formulas.
3.1.6. Multiplicativity and stability of ')'-factors
To conclude this section, let us mention two results on the ')'-factors. One is used
in the computations of L-factors in the general case. This is the multiplicativity of
')'-factors [42]. The second is the stability of ')'-factors [46]. Both of these results
are necessary in applications of the Converse Theorem to liftings, which we discuss
in Lecture 6.
Proposition (Multiplicativity of ')'-factors). If 7r = Ind(7r 1 ® 7r 2 ), with 'Tri and
irreducible admissible representation of GL,., (k), then
'Y(s, 7r x 71"^1 , 'If;)= ')'(s, 7r1 x 71"^1 , 'lf;)'Y(s, 7r2 x 71"^1 , 'If;)and similarly for 71"^1 • Moreover L( s, 7r x 7r^1 )-^1 divides [L( s, 7r 1 x 7r^1 )L( s, 7r 2 x 7r^1 )]-^1.
Proposition (Stability of ')'-factors). If 7r 1 and 7r 2 are two irreducible admissible
generic representations of G Ln ( k), having the same central character, then for every
sufficiently highly ramified character T/ of G L 1 ( k) we have
')'(S,7r1 X TJ,'l/J) = ')'(S,7r2 X TJ,'l/J)
and
L(s, 7r1 x TJ) = L(s, 7r2 x TJ) = 1.