Properties

Label 4-8470e2-1.1-c1e2-0-19
Degree $4$
Conductor $71740900$
Sign $1$
Analytic cond. $4574.26$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s − 2·5-s + 2·7-s + 4·8-s − 3·9-s − 4·10-s + 4·14-s + 5·16-s − 6·17-s − 6·18-s − 6·20-s − 2·23-s + 3·25-s + 6·28-s − 12·29-s − 10·31-s + 6·32-s − 12·34-s − 4·35-s − 9·36-s − 8·37-s − 8·40-s + 10·41-s − 2·43-s + 6·45-s − 4·46-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.755·7-s + 1.41·8-s − 9-s − 1.26·10-s + 1.06·14-s + 5/4·16-s − 1.45·17-s − 1.41·18-s − 1.34·20-s − 0.417·23-s + 3/5·25-s + 1.13·28-s − 2.22·29-s − 1.79·31-s + 1.06·32-s − 2.05·34-s − 0.676·35-s − 3/2·36-s − 1.31·37-s − 1.26·40-s + 1.56·41-s − 0.304·43-s + 0.894·45-s − 0.589·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(71740900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(4574.26\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 71740900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T + 35 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 10 T + 104 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + 92 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 6 T + 112 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - 18 T + 188 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 2 T + 72 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 14 T + 159 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 10 T + 143 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 184 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T + 186 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−7.62411222907060406709138730199, −7.22592761046942875241385547494, −6.76070870126114461801546249776, −6.72520954808886430070276833741, −6.03144340701380284026464953274, −5.83608022482588086986869238466, −5.37494713506306487768934594134, −5.29746791446997564804252526116, −4.75110246875056894776769642472, −4.42510479571197996299813113217, −4.07537452381307099611362023362, −3.78020148965902975545436800739, −3.29792979557349095136928455981, −3.18917601452885925376823720353, −2.32844727071305867102170425799, −2.22295808728723007715856854247, −1.77740871338094001696749851093, −1.18274868067522233243619076641, 0, 0, 1.18274868067522233243619076641, 1.77740871338094001696749851093, 2.22295808728723007715856854247, 2.32844727071305867102170425799, 3.18917601452885925376823720353, 3.29792979557349095136928455981, 3.78020148965902975545436800739, 4.07537452381307099611362023362, 4.42510479571197996299813113217, 4.75110246875056894776769642472, 5.29746791446997564804252526116, 5.37494713506306487768934594134, 5.83608022482588086986869238466, 6.03144340701380284026464953274, 6.72520954808886430070276833741, 6.76070870126114461801546249776, 7.22592761046942875241385547494, 7.62411222907060406709138730199

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