Properties

Label 4-2800e2-1.1-c1e2-0-23
Degree $4$
Conductor $7840000$
Sign $1$
Analytic cond. $499.885$
Root an. cond. $4.72843$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·9-s + 4·19-s + 12·29-s + 8·31-s + 12·41-s − 49-s − 12·59-s + 16·61-s + 16·79-s − 5·81-s + 12·89-s − 4·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 8·171-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 2/3·9-s + 0.917·19-s + 2.22·29-s + 1.43·31-s + 1.87·41-s − 1/7·49-s − 1.56·59-s + 2.04·61-s + 1.80·79-s − 5/9·81-s + 1.27·89-s − 0.383·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.611·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7840000\)    =    \(2^{8} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(499.885\)
Root analytic conductor: \(4.72843\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7840000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.533144352\)
\(L(\frac12)\) \(\approx\) \(3.533144352\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.077133290737055348619846408104, −8.492645480827432431338036796154, −8.208353102112304477596525584497, −7.934015786652707917340161408485, −7.41671646408840425288303624923, −7.22934325202366282567774005548, −6.54322420019086101678713562990, −6.41826934755362089389463452296, −6.11361252642616746112082231875, −5.38399087255994091411054806936, −5.10658766928732920691806759752, −4.72666445225850708592257068957, −4.13545814548122198967972245572, −4.05658807070316465844734957914, −3.24117678965586432282899807578, −2.85963383086949779789187079139, −2.48868163363041978162932746505, −1.79375369235433350047610789685, −1.03569233487136146757510540359, −0.75469463338118639725849482399, 0.75469463338118639725849482399, 1.03569233487136146757510540359, 1.79375369235433350047610789685, 2.48868163363041978162932746505, 2.85963383086949779789187079139, 3.24117678965586432282899807578, 4.05658807070316465844734957914, 4.13545814548122198967972245572, 4.72666445225850708592257068957, 5.10658766928732920691806759752, 5.38399087255994091411054806936, 6.11361252642616746112082231875, 6.41826934755362089389463452296, 6.54322420019086101678713562990, 7.22934325202366282567774005548, 7.41671646408840425288303624923, 7.934015786652707917340161408485, 8.208353102112304477596525584497, 8.492645480827432431338036796154, 9.077133290737055348619846408104

Graph of the $Z$-function along the critical line