20 2. PRELIMINARIES
where
Aχ,c(ν) =
˜2(c)N(c)−ν
χ
v∈S(c)
L(1 + ν, χv) 2
L(1 ν, χv
−2)
.
Proof. By a well-known procedure found in the proof of [5, Theorem 3.7.1],
the proof is reduced to a computation of local intertwining operators on the vectors
f0,χv
(ν)
and
˜(ν)
f
1,χv
. Thus, we are done by Lemma 2.8.
Though ϕ = Eχ,c(ν; −) Eχ,c(ν;

−) is not left GF -invariant, it is still BF -
invariant; thus the integral
Z∗(s,
η; ϕ) makes sense.
Lemma 2.11. Let η be a unitary idele-class character satisfying (2.18), (2.19)
and (2.20). Then,
Z∗(s,
η; Eχ,c(ν) Eχ,c(ν))

(Re(s) 0) equals
(DF
N(c))−ν/2
˜(dF/Qc) χ G(η) Bχ,c(s,
η
ν)
L(s + ν/2,χη) L(s ν/2,χ−1η)
L(ν + 1,χ2)
with
Bχ,c(s,
η
ν) =
DF−1/2 s
v∈S(c)
{(qv
−1
+ 1) L(1 + ν, χv)
2
ηv(
v)qv/2−s 1
Q(Iv(χv||v
ν/2))
}.
(2.27)
Proof. Similar to Lemma 2.5.
Lemma 2.12. Let ν iR and η as in Lemma 2.11. The meromorphic function
Bχ,c
η
(
−z +
1
2
, ν
) L
(
−z +
ν+1
2
, χη
)
L
(
−z +
1−ν
2
, χ−1η
)
L(ν + 1,χ2)
in z C is holomorphic except possible simple poles at z = ± 1)/2, (−ν ± 1)/2.
The relevant residues satisfy the relations
Resz=(ν+1)/2 =
−(−1)S(c)
N(c) Resz=(−ν−1)/2
= δχ,η vol(F
×\A1)
˜(c) χ (DF
N(c))(ν+1)/2,
Resz=(1−ν)/2 =
−(−1)S(c)
N(c) Resz=(−1+ν)/2
= δχ,η vol(F
× \A1)
˜(c) χ (DF
N(c))(ν+1)/2
DF
−1/2
Aχ,c(ν)
ζF (ν)
ζF + 1)
.
Proof. The first statement is evident. The formulas of residues are obtained
by a direct computation with the aid of the easily confirmed relations
Bη,c
η
(
−ν
2
, ν
)
= (DF
−1N(c))(ν+1)/2
˜(c), η
Bη,c
η
(
1
ν
2
, ν
)
= (DF
−1N(c))(ν−1)/2
˜(c) η
(−1)S(c)
Aη,c(ν),
Bη,c
η
(
ν
2
, ν
)
= (DF
N(c))(ν−1)/2
N(c) ˜(c) η Aη,c(ν),
Bη,c
η
(
1 +
ν
2
, ν
)
= (DF
N(c))(ν+1)/2 (−1)S(c) N(c)−1
˜(c), η
combined with the functional equation of L(ν, 1) = ζF (ν).
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