1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. ANALYTIC PROPERTIES OF INDMDUAL £-FUNCTIONS 205

the following integral: let G(u) be an even function, normalized by G(O) = 1,

holomorphic and bounded in the vertical strip l~eul :( 4. For X > 0, we set

Irr(s,X) = -

2

1

. jq~s+u)f^2 L(7r 00 ,s+u)L(7r,s+u)XuG(u)du
ITT U
(2)

= q'Tr s/2 L.., "°' >.'Tr(n) s 2. _l J L( 'lroo, s + u )(-n-)-uG(. lnX u ) du.

n 'Tri v q'Tr u

n~l ( 2 )

Because of the exponential decay of q~s+u)/^2 L(7r 00 , s+u)L(7r, s+u) in vertical strips

we may shift the line of integration to ~eu = -2, passing a pole at u = 0 (here

we are assuming that L(7r, s) -/= ((s +it) for any t ER, otherwise there is an extra

harmless contribution), with residue q!^12 L(7r 00 , s )L(7r, s). On the resulting integral,

we apply the functional equation (1.1) and make the change of variable u f-+ -u to

obtain

q!/^2 L(7r 00 , s)L(7r, s) = I'Tr(s, X) + w(7r)J7i-(1-s, x-^1 ).

Now, taking the Dirichlet series expression of L( 7r, s) and dividing by q!/^2 L( 7r 00 , s ),

one finds

(1.30)

L(7r s) = "°' >.'Tr(n) V. ( n ) w(7r s) "°' A7i-(n) V _ ( nX )

> ~ ns S,'Troo x JQ'Tr(t) + > ~ nl-s l-S,'Troo JQ'Tr(t)

where

(1.31) V. S,'Troo ( ) Y = _1 2 · J L(7r£(^00 , s + ) u) Q 'll"oo (t)-u/2 Y -uc(u) du.
'Tri 7r 00 , S U
(2)
and
i-2• L(ir 00 , 1 - s)
w(7r, s) = w(7r)q'Tr^2 L( ).
'lroo,S
Note that w(7r, s) is well defined when 1 - sis not a pole of L(ir 00 , s) (in particular
when <J < 1/2+1/(d^2 +1)). Note also that w(7r, s) has modulus one on ~es= 1/2
(which justifies our notation). To be more specific, we assume that ~es= 1 /2 and
we take
G(u) = (cos(~))-Ad,

for some parameter A > 4. In (1.31) we shift the line of integration to ~eu = B
with either B = A, if y ~ 1, or with B satisfying 0 > B > supi=l...d ~eμ'Tr,i - 1/2
if y < 1; in the latter case, we hit a pole at u = 0 with residue 1. Differentiating j
times, we see that

v<j) ( )«·6 + -B-jl IL(7roo,s+u)Q-B/211G(u)lluljldul_

S,'Troo y J B<O J=O y (B) L(7r oo, s) 'll"oo (t) lul

By Stirling's formula one sees that

I

L(7roo,s+u) -B/2I ('Tr I I)

L(7roo, s) Q'Troo(t) «d,B exp 4d u '

hence we infer that for any j ~ 0 and any 0 < 'f/ < 1/2 - supi=l...d ~eμ'Tr,i>

(j) - y-J + T/

(1.32) VS,'Troo (y) - 6y<l + Q( (l )A)'

j=O + y