Properties

Label 4-6300e2-1.1-c1e2-0-14
Degree $4$
Conductor $39690000$
Sign $1$
Analytic cond. $2530.66$
Root an. cond. $7.09265$
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  + 4·11-s − 4·19-s + 12·29-s − 4·31-s + 20·41-s − 49-s − 16·59-s − 4·61-s + 20·71-s + 4·89-s + 12·101-s + 28·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 1.20·11-s − 0.917·19-s + 2.22·29-s − 0.718·31-s + 3.12·41-s − 1/7·49-s − 2.08·59-s − 0.512·61-s + 2.37·71-s + 0.423·89-s + 1.19·101-s + 2.68·109-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 + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(39690000\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2530.66\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 39690000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.829230816\)
\(L(\frac12)\) \(\approx\) \(3.829230816\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.091172535600355663489655844032, −7.937671080429113715282872728935, −7.39064780084228608815784087944, −7.35056163543842104933475572535, −6.55453481382828141098344411268, −6.46028655286727064212423854705, −6.23077888690775140778736351497, −5.93598055542009928829687955892, −5.17571917677945202904205026067, −5.11542120668297631948622964551, −4.44478499298663224056762754507, −4.20668315968916506154938187359, −4.00647991642999781317710018220, −3.42244125722796490654879978995, −2.81508504838901768470439952619, −2.75739453643418687511031109826, −1.89907005178994095225488619318, −1.74391184899707157466825444562, −0.858282277204015813740215577933, −0.64164850107965483703340264522, 0.64164850107965483703340264522, 0.858282277204015813740215577933, 1.74391184899707157466825444562, 1.89907005178994095225488619318, 2.75739453643418687511031109826, 2.81508504838901768470439952619, 3.42244125722796490654879978995, 4.00647991642999781317710018220, 4.20668315968916506154938187359, 4.44478499298663224056762754507, 5.11542120668297631948622964551, 5.17571917677945202904205026067, 5.93598055542009928829687955892, 6.23077888690775140778736351497, 6.46028655286727064212423854705, 6.55453481382828141098344411268, 7.35056163543842104933475572535, 7.39064780084228608815784087944, 7.937671080429113715282872728935, 8.091172535600355663489655844032

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