1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. ANALYTIC PROPERTIES OF INDIVIDUAL £-FUNCTIONS 189

absolutely convergent for ~es large enough. This is an Euler product of degree dd'

with local factors of the form

at the finite places, and at the infinite one,

dd'

Loo(7r 0 1f

1

, s) = L(noo 0 7f~, s) = II rR(s - μ1'@1'' ,i )·

i=l

Moreover, at places v for which n v is unramified, Lv ( n © n' , s) has the explicit

expression

(1.2)

at v = p a finite place, and at the infinite place,

d d'

(1.3) Loo(7r 0 n' , s) =II II rR(s - μ1',i - μ1'',i' ).

i=l i'=l

These completed L-functions have the following analytic properties, which are

proved in the above cited papers (again see [Co2]):

• L 00 ( n © n', s) L ( n © n', s) has a meromorphic continuation to C with at

most two simple poles; the latter occur if and only if n' '.::::'.ii"© I det lit for

some t E R and are located at s = -it, 1 - it.

• L 00 ( n © n', s )L( n © n', s) satisfies a functional equation of the form

(1.4) q;~1',L 00 (n 0 n', s)L(n 0 n', s) =

w(n 0 n')q~1;:)^12 L 00 (ii" 0 ii-' , 1 - s)L(ii" 0 ii-' , 1 - s).

where q7r®7r' ~ 1 is an integer and w(n © n') has modulus one.

• The "completed" L-function L 00 (n © n', s)L(n © n', s) is bounded in ver-
  tical strips as l~msl ---t +oo (with, in fact, exponential decay) and is of
  finite order away from its poles (if any).
  The integer q7r®7r' is by definition the conductor of (the pair) n © n'; it is supported
  on the primes dividing q1'q1'', and in fact one has the following upper bound [BH]

q7r@7r' ~ q1' d' q1'd ,/ ( q1', q1'1),

which now is an easy consequence of the local Langlands correspondance for GLd.
We denote, for t E R, by

d d '
Q1'Q91'^1 (t) = q1'@1'' ITU+ lit-μ7r®7r',il),
i=l
the analytic conductor of n © n'. An archimedean analog of the bound given above
for the conductors shows (and follow easily from (1.3) if n 00 is unramified) that
one has
(1.5)