LECTURE 1. ANALYTIC PROPERTIES OF INDMDUAL £-FUNCTIONS 191Lemma 1.1.1. For n ~ 1, ftr®ir ( n) ~ O; in particular, for any n ~ 1,>-7r 0 .;r(n) ~ 0.This lemma is easily proved for n coprime with q7r by (1.2); that it holds for
every n follows from the structure of the admissible representations of G Ld and the
expression of their local factors of pairs (see [JPSS2, HR, RS2]).Remark 1.3. This positivity property extends to isobaric sums of unitary cuspidal
representations
As explained in [Co2] 4.4, this property together with the non-vanishing of the
local factors Lv ( 7r ® 7r^1 , s) and the fact that L 00 ( 7r ® 1T, s) L ( 7r ® 1T, s) has no pole for
~es > 1, implies by Landau's Lemma that L(q,.)(7r ® 1T, s) is absolutely convergent
for ~es> 1, and non-vanishing in this domain; moreover, by Cauchy/ Schwarz, the
same is true of the L-series L < q,.) ( 7r, s) and L ( q,. q,.,) ( 7r ® 7r^1 , s) for all pairs ( 7r, 7r').
One can deduce the following bounds for the local parameters:
Proposition 1.1. For 7r E A~(Q) and all i E {l, ... , d} one has(1.6)1 1.
~eμtr ,i ~
2
and logP I a7r ,i (p) I ~
2
, for all pnmes p.Moreover, for an unramified place, one has
(1.7)Proof. For each place v, the local factor Lv ( 7r ® 1T, s) has no pole for ~es > 1,
since this would contradict the non-vanishing of L(7r ® 1T, s ) and the holomorphy of
L 00 (7r ® 1T, s)L(7r ® 1T, s ) in this domain. In particular one can deduce that
logP latr®ir,i(P)I, ~eμtr®ii",i ~ 1, i = l ... dd'.Now the expression of the local factor at an unramified place (1.2) implies (1. 7).
The bounds (1.6) for the remaining (ramified) places follow from the structure of
the admissible representations of GLd(Qv ) and the expression of local factors of
pairs; for more detail we refer to the Appendix of [RS2]. D
In fact, the bounds (1.6) and (1. 7) can be obtained with the stronger inequality
< 1/ 2 by purely local arguments, using the fact that the local components of 7r are
generic (see [JS2, HR]). However, as we shall see, improving beyond 1/2 requires
global arguments. We refer to the bounds given in Proposition 1.1 as the "trivial"
bounds for the local parameters of 7r. More generally, given 0 ~ () ~ 1/2, one can
consider the following:
Hypothesis Hd(()). For all d' ~ d, all 7r E A~(Q) and all i E {l, ... , d'}, one has
(1.8) ~eμtr,i ~()and logP latr,i(P)I ~()for all primes p.Moreover, for an unramified place, one has for all i E { 1, ... , d}
(1.9)Remark 1.4. Usually these bounds are presented for the unramified places only
in the form of (1. 9); however, for several technical purposes the corresponding
bounds (1.8) for the ramified places are useful.