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
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
.
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