190 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS
£-functions of pairs are defined more generally for automorphic representa-tions. In particular, for isobaric representations 7r = 7r 1 EE · · · EE 7rr and 7r^1 =
7r~ EE · · · EE 7r~,, one has
L(7r 0 7r^1 , s) = II II L(7ri 0 7rj, s).
i=l. .. r j=l. .. r'Remark 1.2. In particular for 7r unitary cuspidal and II an isobaric sum of unitary
representations, the multiplicity of 7r as a component of II is given by -ords=1L(s,
7r®7T).
In fact, £ -functions of pairs are expected to be automorphic. More precisely a
special case of the Langlands Functoriality Conjecture predicts the following
Functoriality Conjecture for Pairs. Given 7r = 7r 1 EE · · · EE 7r r and 7r^1 = 7r~ EE · · ·EE 7r~,
two isobaric sums of unitary cuspidal representations of GLd(Aq) and GLd'(Aq)
respectively, there exists an isobaric sum of unitary cuspidal representations 7r ~ 7r^1 of
G Ldd' ( Aq) such that
L(7r@7r',s) = L(7r~7r^1 ,s).
Moreover, one has the distributive formula
7r ~ 7r
1
= EEi=l...r EEj=l.. .r' 'Tri ~ 7rj ·
Consequently, L( 7r 0 7r^1 , s) should factor as a product of principal cuspidal £ -
functions,
i=l... r^11where 7rr E A~'.' ( Q), i = 1 ... r" with d{ + · · · + d~,, = dd', and the analytic
properties that v-le will discuss below regarding principal £-functions should extend
to £-functions of pairs. See [Ral] for further discussions about this conjecture.
By now this conjecture is known for a few cases, namely in the trivial but im-
portant case of d = 1 (twists by characters), and ford = 2, d' = 2, 3 by the works
of (among others) Weil, Shimura, Gelbart/ Jacquet, Cogdell/Piatetski-Shapiro, Ra-
makrishnan, Kim and Shahidi (see [Co2] Leet. 6).
1.1.3. The Ramanujan/ Petersson Conjecture
It is of fundamental importance for the study of automorphic forms (and for many
applications of an arithmetic nature) to have good bounds for the local parameters
atr,i(p), μtr,i, i = l ... d. By the general theory it is known that L(7r, s) is absolutely
convergent for ~es > B for some B ~ 1. Since L( 7r, s ) is an Euler product, it does
not vanish in this domain and since L( 7r 00 , s) L( 7r, s) is holomorphic, this implies
that
logp latr,i(P)I, ~eμtr,i ~ e, i = l ... d
(clearly we have assumed that 7r is not equal to I· lit for t E R). As was discovered
by Rankin and Selberg, the analytic properties of L( 7r 0 if, s) are very usefull in pro-
viding stronger bounds. A fundamental property of L(7r®7T, s) is the non-negativity
of the Dirichlet coefficients >-1r1 81 ;r ( n): set
log L(7r 0 7T, s) := """'Ctr®ii-(n) ;
~ ns
n~l