1549380232-Automorphic_Forms_and_Applications__Sarnak_

218 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF L-FUNCTIONS

type theorem for L( 7r f ®ir f, s) (which is known, see Section 1.2.3), and from (1.24)
(applied to 7r = sym^2 7r f 0 XJ ), one has for any t E R,
(2.16) (Q11"J(l +!ti))-"«,, ress=1L(7rJ ® irJ,s) «,, (Q71" 1 (1 +!ti))".

The same bounds holds for its classical counterpart L(f x 7, 1 +it). In particular,
one has the following form of RPC on average:

(2.17) L l>-t(n)l^2 «,, (QJN)" N.

n~N

Moreover, the residue at s = 1 of L(f x 7, s) is related to the inner product (!, !) =
IPJ(l)l-^2 (see [DFI8] Sect. 19):
ress=1L(f x 7, s)

{

(^32) 71"^3 (JJ) if f is a Maass form of weight k=0,1
_ vol(Xo(q))!r(~+itt+~)l2'

-^4 11"
2
<^4 11")k(f ,f) if f is holomorphic.
vol(Xo(q))r(k)'

It follows that
1 k 1 k
(2.18) (QJ )-"qlr(2 + itf + 2 )1^2 «,, (!, !) «,, (QJ )"qlr(2 + itf + 2 )12,

if f is a Maass form of weight k = 0 , 1, or

(2.19)

if f is holomorphic.

2.3.1. Voronoi's summation formula

From the automorphy relations (2.1), one can deduce the following Voronoi-type
summation formula; we display it for holomorphic forms in a simple case (see
[KMV2] for more general formulae).

Lemma 2.3.1. Let W : R*+ _, C be a smooth function with compact support. Let
c = O(q) and a be an integer coprime to c. For g E Sk(q, x) we have:

c L vfnp 9 (n)e(n~)W(n)

n;;,l c

·k ~ a 1

00

= 27ri x(a) Lt vfnp 47ry'nX

9 (n)e(-n-) W(x)Jk_^1 ( )dx.

n;;,l C 0 C

Proof. The proof follows from the automorphic relation

az + b k

g(--d) = (cz + d) x(a)g(z )

cz+

applied to Zt = -d/ c + i / (ct) for t E R >o. Then taking the Mellin transform of this

equality, one deduces that the L-function

L(g, ~' s) = L Pg(n)n-1/ 2-se(an)

c n;;,l c

satisfies a functional equation with the same factors at infinity, conductor and root
number, but related to L(g, ~a, 1 - s). The formula follows then from the inverse