Properties

Label 8-21e8-1.1-c5e4-0-4
Degree $8$
Conductor $37822859361$
Sign $1$
Analytic cond. $2.50262\times 10^{7}$
Root an. cond. $8.41006$
Motivic weight $5$
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  + 10·2-s − 9·4-s − 440·8-s + 1.95e3·11-s − 1.43e3·16-s + 1.95e4·22-s + 7.13e3·23-s − 4.86e3·25-s + 3.35e3·29-s − 3.45e3·32-s − 9.20e3·37-s + 2.04e4·43-s − 1.75e4·44-s + 7.13e4·46-s − 4.86e4·50-s + 1.02e5·53-s + 3.35e4·58-s − 3.02e4·64-s − 2.28e4·67-s + 1.53e5·71-s − 9.20e4·74-s − 9.06e4·79-s + 2.04e5·86-s − 8.58e5·88-s − 6.42e4·92-s + 4.38e4·100-s + 1.02e6·106-s + ⋯
L(s)  = 1  + 1.76·2-s − 0.281·4-s − 2.43·8-s + 4.86·11-s − 1.39·16-s + 8.59·22-s + 2.81·23-s − 1.55·25-s + 0.740·29-s − 0.595·32-s − 1.10·37-s + 1.68·43-s − 1.36·44-s + 4.97·46-s − 2.75·50-s + 5.03·53-s + 1.30·58-s − 0.922·64-s − 0.623·67-s + 3.62·71-s − 1.95·74-s − 1.63·79-s + 2.98·86-s − 11.8·88-s − 0.791·92-s + 0.438·100-s + 8.89·106-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(2.50262\times 10^{7}\)
Root analytic conductor: \(8.41006\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 7^{8} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(24.07283228\)
\(L(\frac12)\) \(\approx\) \(24.07283228\)
\(L(\frac{7}{2})\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$D_{4}$ \( ( 1 - 5 T + 21 p T^{2} - 5 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
5$D_4\times C_2$ \( 1 + 4868 T^{2} + 22562806 T^{4} + 4868 p^{10} T^{6} + p^{20} T^{8} \)
11$D_{4}$ \( ( 1 - 976 T + 556178 T^{2} - 976 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + 224500 T^{2} - 32848089674 T^{4} + 224500 p^{10} T^{6} + p^{20} T^{8} \)
17$D_4\times C_2$ \( 1 + 3065760 T^{2} + 6374095942466 T^{4} + 3065760 p^{10} T^{6} + p^{20} T^{8} \)
19$D_4\times C_2$ \( 1 + 8544504 T^{2} + 30052000046306 T^{4} + 8544504 p^{10} T^{6} + p^{20} T^{8} \)
23$D_{4}$ \( ( 1 - 3568 T + 12712350 T^{2} - 3568 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 - 1676 T + 24360510 T^{2} - 1676 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 29395756 T^{2} + 658228795954534 T^{4} + 29395756 p^{10} T^{6} + p^{20} T^{8} \)
37$D_{4}$ \( ( 1 + 4604 T + 143922030 T^{2} + 4604 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 + 212331360 T^{2} + 36724946155085474 T^{4} + 212331360 p^{10} T^{6} + p^{20} T^{8} \)
43$D_{4}$ \( ( 1 - 10224 T + 293018130 T^{2} - 10224 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 568199404 T^{2} + 172542701189040294 T^{4} + 568199404 p^{10} T^{6} + p^{20} T^{8} \)
53$D_{4}$ \( ( 1 - 51460 T + 1308627278 T^{2} - 51460 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 1609944152 T^{2} + 1391734325057519170 T^{4} + 1609944152 p^{10} T^{6} + p^{20} T^{8} \)
61$D_4\times C_2$ \( 1 + 1094138468 T^{2} + 1609346124964468758 T^{4} + 1094138468 p^{10} T^{6} + p^{20} T^{8} \)
67$D_{4}$ \( ( 1 + 11448 T - 1089307338 T^{2} + 11448 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 - 76912 T + 4562060670 T^{2} - 76912 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 2170565248 T^{2} + 425171557315203874 T^{4} + 2170565248 p^{10} T^{6} + p^{20} T^{8} \)
79$D_{4}$ \( ( 1 + 45344 T + 6154499470 T^{2} + 45344 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 6721792472 T^{2} + 21939991980485194594 T^{4} + 6721792472 p^{10} T^{6} + p^{20} T^{8} \)
89$D_4\times C_2$ \( 1 + 14718300768 T^{2} + \)\(11\!\cdots\!58\)\( T^{4} + 14718300768 p^{10} T^{6} + p^{20} T^{8} \)
97$D_4\times C_2$ \( 1 + 22489794400 T^{2} + 25388976518139394 p^{2} T^{4} + 22489794400 p^{10} T^{6} + p^{20} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.02664568108799308881009367171, −7.01095288841720600085506661374, −6.53620672013784157337492297482, −6.40438565033029593167913416804, −6.29710337840667688276505477340, −5.79032646207954608974514372715, −5.45643236240654797602436610653, −5.38496528986627373950642193570, −5.31960083440640789517428680599, −4.68112288608020926837452705748, −4.39643888911104012084316368324, −4.39467154116960977911908823002, −4.21662650325741952844136628595, −3.82129815268477657546611273202, −3.60585309500521017549745403225, −3.51484784521431073877568077678, −3.43911455345558934617982142119, −2.63659823885883562474776045585, −2.44961379065162586697594535564, −1.86845692562583794574536132190, −1.66387198162066838971252768010, −1.16085660965213121240531385096, −0.899448298931730680019440020296, −0.77639318743304759421234666333, −0.41282025781609969284278534725, 0.41282025781609969284278534725, 0.77639318743304759421234666333, 0.899448298931730680019440020296, 1.16085660965213121240531385096, 1.66387198162066838971252768010, 1.86845692562583794574536132190, 2.44961379065162586697594535564, 2.63659823885883562474776045585, 3.43911455345558934617982142119, 3.51484784521431073877568077678, 3.60585309500521017549745403225, 3.82129815268477657546611273202, 4.21662650325741952844136628595, 4.39467154116960977911908823002, 4.39643888911104012084316368324, 4.68112288608020926837452705748, 5.31960083440640789517428680599, 5.38496528986627373950642193570, 5.45643236240654797602436610653, 5.79032646207954608974514372715, 6.29710337840667688276505477340, 6.40438565033029593167913416804, 6.53620672013784157337492297482, 7.01095288841720600085506661374, 7.02664568108799308881009367171

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