22 2. PRELIMINARIES

2.8. Adjoint square L-functions

For π as in 2.6.1, its adjoint square L-function L(s, π; Ad) is defined to be the

Euler product

L(s, π;Ad) =

v∈Σfin∪Σ∞

Lv(s, πv;Ad)

convergent on a right half-plane, where the local factors are given as follows. If πv

is Kv-spherical, then

Lv(s, πv;Ad) =

(1 − qv −s+2νv )−1 (1 − qv −s−2νv )−1 (1 − qv −s)−1, v ∈ Σfin,

ΓR (s + 2νv) ΓR (s − 2νv) ΓR(s), v ∈ Σ∞

with νv ∈ C such that πv

∼

=

Iv(||vv ν ). If v ∈ Σfin with c(πv) = 1, then Lv(s, πv;Ad) =

(1 − qv

−(s+1)

)−1.

It is known that L(s, π; Ad) is continued to an entire function on C ([11]). The

product ζF (s) L(s, π; Ad) is the convolution L-function for the pair (π, ¯), π which is

studied by the Rankin-Selberg integral involving the Eisenstein series ([22]); in our

case, it is explicated as follows.

Lemma 2.14. Let ϕπ

new

∈ Vπ be the newform of π. Then, for Re(s) 0,

GF ZA\GA

E1,oF (2s − 1; g) ϕπ

new(g)

ϕπ new(g) dg (2.29)

= [Kfin :

K0(fπ)]−1 N(fπ)s

DF−3/2 s

ζF

(2s)−1

ζF (s) L(s, π;Ad).

Proof. By the standard unfolding procedure, the integral is decomposed to

the product of the local integrals

Zv(s) =

Kv

dk

Fv×

φ0,v ([ t 0

0 1

] k) φ0,v ([ t 0

0 1

] k)

|t|v−1 s d×t.

If v ∈ Σfin with c(πv) = 0, then we obtain

Zv(s) = vol(ov

×) qvv(s−1) d ζF,v(2s)−1

ζF,v(s) Lv(s, πv;Ad) (2.30)

using (2.11) as in [5, Proposition 3.8.1]. If v ∈ Σ∞, then by

φ0,v ([ t 0

0 1

]) = 2

t1/2

Kνv (2πt), t 0,

we have

Zv(s) =

ΓR(2s)−1

ΓR(s + 2ν) ΓR(s − 2ν)

ΓR(s)2,

Re(s) 2|Re(νv)|

using the formula in [36, p.101]. Finally, let v ∈ Σfin with c(πv) = 1. According to

the disjoint dicomposition Kv = K0(pv) ∪ {

ξ∈ov /pv

1 ξ

0 1

w0K0(pv)}, the integral

Zv(s) breaks up to the sum of the following integrals.

Zv

(1)(s)

= vol(K0(pv))

Fv×

φ0,v ([ t 0

0 1

]) φ0,v ([ t 0

0 1

])

|t|s−1

v

d×t,

Zv

(2)(ξ;

s) = vol(K0(pv))

Fv×

φ0,v

(

[ t 0

0 1

]

1 ξ

0 1

w0

)

φ0,v

(

[ t 0

0 1

]

1 ξ

0 1

w0

)

|t|s−1

v

d×t

with ξ ∈ ov/pv. If ϕ0,v denotes the Kv-spherical Whittaker function of Iv(sgnvv |

|v/2),1

which is given by (2.11) with

qνv

= (−1)

v

qv/2,1

φ0,v ([ t 0

0 1

]) = ϕ0,v ([ t 0

0 1

]) − (−1)

v

ϕ0,v tv

−1

0

0 1

.