Properties

Label 4-4600e2-1.1-c1e2-0-8
Degree $4$
Conductor $21160000$
Sign $1$
Analytic cond. $1349.17$
Root an. cond. $6.06062$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·9-s + 4·11-s + 4·19-s + 6·29-s + 14·31-s − 18·41-s + 14·49-s + 4·61-s − 2·71-s + 28·79-s + 16·81-s − 32·89-s + 20·99-s + 20·101-s + 8·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 20·171-s + ⋯
L(s)  = 1  + 5/3·9-s + 1.20·11-s + 0.917·19-s + 1.11·29-s + 2.51·31-s − 2.81·41-s + 2·49-s + 0.512·61-s − 0.237·71-s + 3.15·79-s + 16/9·81-s − 3.39·89-s + 2.01·99-s + 1.99·101-s + 0.766·109-s − 0.909·121-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 + 1/13·169-s + 1.52·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21160000\)    =    \(2^{6} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1349.17\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21160000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.014426029\)
\(L(\frac12)\) \(\approx\) \(5.014426029\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 145 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 178 T^{2} + p^{2} T^{4} \)
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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

−8.659274547511491376797276220149, −8.179601580147223344006452475914, −7.63038858744467132932930325995, −7.49242049364059786024122014112, −6.90025614703684659844922500466, −6.71432553154183620787476990508, −6.47325099482051362561726648766, −6.16947130669112473693139586873, −5.32107833308116842769210031163, −5.30802100754912077238355864120, −4.62941543333005518116565524367, −4.43446087846467611734654205892, −4.05583855104512051400844996245, −3.59259555539317274659978741662, −3.20122168115195979571617579194, −2.70129889536784819637221992176, −2.07597613751231664573489242707, −1.59101803780633745754875348953, −1.04584270128031457834865558683, −0.76274912844162435656476452823, 0.76274912844162435656476452823, 1.04584270128031457834865558683, 1.59101803780633745754875348953, 2.07597613751231664573489242707, 2.70129889536784819637221992176, 3.20122168115195979571617579194, 3.59259555539317274659978741662, 4.05583855104512051400844996245, 4.43446087846467611734654205892, 4.62941543333005518116565524367, 5.30802100754912077238355864120, 5.32107833308116842769210031163, 6.16947130669112473693139586873, 6.47325099482051362561726648766, 6.71432553154183620787476990508, 6.90025614703684659844922500466, 7.49242049364059786024122014112, 7.63038858744467132932930325995, 8.179601580147223344006452475914, 8.659274547511491376797276220149

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