Affiliation:
1. School of Mathematics and Statistics, Shandong Normal University , Jinan , 250014 Shandong , P. R. China
Abstract
Abstract
Let
f
f
be a self-dual Hecke-Maass eigenform for the group
S
L
3
(
Z
)
S{L}_{3}\left({\mathbb{Z}})
. For
1
2
<
σ
<
1
\frac{1}{2}\lt \sigma \lt 1
fixed we define
m
(
σ
)
m\left(\sigma )
(
≥
2
\ge 2
) as the supremum of all numbers
m
m
such that
∫
1
T
∣
L
(
s
,
f
)
∣
m
d
t
≪
f
,
ε
T
1
+
ε
,
\underset{1}{\overset{T}{\int }}| L\left(s,f){| }^{m}{\rm{d}}t{\ll }_{f,\varepsilon }{T}^{1+\varepsilon },
where
L
(
s
,
f
)
L\left(s,f)
is the Godement-Jacquet L-function related to
f
f
. In this paper, we first show the lower bound of
m
(
σ
)
m\left(\sigma )
for
2
3
<
σ
<
1
\frac{2}{3}\lt \sigma \lt 1
. Then we establish asymptotic formulas for the second, fourth and sixth powers of
L
(
s
,
f
)
L\left(s,f)
as applications.
Reference29 articles.
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Cited by
2 articles.
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