Properties

Label 8-63e8-1.1-c1e4-0-4
Degree $8$
Conductor $2.482\times 10^{14}$
Sign $1$
Analytic cond. $1.00886\times 10^{6}$
Root an. cond. $5.62962$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·4-s − 10·8-s + 4·11-s + 8·22-s − 20·23-s − 14·25-s − 4·29-s + 18·32-s − 16·37-s − 12·43-s − 12·44-s − 40·46-s − 28·50-s + 16·53-s − 8·58-s + 11·64-s − 12·67-s + 4·71-s − 32·74-s − 40·79-s − 24·86-s − 40·88-s + 60·92-s + 42·100-s + 32·106-s + 12·107-s + ⋯
L(s)  = 1  + 1.41·2-s − 3/2·4-s − 3.53·8-s + 1.20·11-s + 1.70·22-s − 4.17·23-s − 2.79·25-s − 0.742·29-s + 3.18·32-s − 2.63·37-s − 1.82·43-s − 1.80·44-s − 5.89·46-s − 3.95·50-s + 2.19·53-s − 1.05·58-s + 11/8·64-s − 1.46·67-s + 0.474·71-s − 3.71·74-s − 4.50·79-s − 2.58·86-s − 4.26·88-s + 6.25·92-s + 21/5·100-s + 3.10·106-s + 1.16·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.00886\times 10^{6}\)
Root analytic conductor: \(5.62962\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3969} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{16} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad3 \( 1 \)
7 \( 1 \)
good2$D_{4}$ \( ( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
5$D_4\times C_2$ \( 1 + 14 T^{2} + 94 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
13$D_4\times C_2$ \( 1 + 28 T^{2} + 454 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 + 30 T^{2} + 902 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 88 T^{2} + 3538 T^{4} + 88 p^{2} T^{6} + p^{4} T^{8} \)
37$D_{4}$ \( ( 1 + 8 T + 45 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 + 30 T^{2} + 2462 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 + 6 T + 15 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 22 T^{2} - 906 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 - 8 T + 77 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 40 T^{2} + 38 p T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 10 T^{2} + 5262 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 + 6 T + 123 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 - 2 T + 123 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 76 T^{2} + 8182 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 20 T + 253 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 206 T^{2} + 24262 T^{4} + 206 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 + 138 T^{2} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + 262 T^{2} + 35374 T^{4} + 262 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.29871613104541327664975348221, −5.92471521385459265558799499114, −5.79139918791828671142766929488, −5.76971433271049052774411885271, −5.69142955341112128936953969223, −5.18057375771130797176391201197, −5.09419432305918916752547724142, −5.04015321128888718628311227855, −4.95826973695171130909339385862, −4.34078675424844224403559736965, −4.17246377236454117374214944959, −4.13787546287448282545836021119, −4.11744463512663093955185739729, −3.78452404676519489992536788313, −3.67858528210579546848526599244, −3.62685269407757783154481104067, −3.46295126040655819178667189851, −2.94398126864783889964151272690, −2.60440045520731440966278171180, −2.47215739719307441325815145112, −2.12401430888258278763190651210, −1.78116947301212301719726055390, −1.59805000374395681385710129544, −1.47269916421327283812331377086, −1.06432943688538997592263563449, 0, 0, 0, 0, 1.06432943688538997592263563449, 1.47269916421327283812331377086, 1.59805000374395681385710129544, 1.78116947301212301719726055390, 2.12401430888258278763190651210, 2.47215739719307441325815145112, 2.60440045520731440966278171180, 2.94398126864783889964151272690, 3.46295126040655819178667189851, 3.62685269407757783154481104067, 3.67858528210579546848526599244, 3.78452404676519489992536788313, 4.11744463512663093955185739729, 4.13787546287448282545836021119, 4.17246377236454117374214944959, 4.34078675424844224403559736965, 4.95826973695171130909339385862, 5.04015321128888718628311227855, 5.09419432305918916752547724142, 5.18057375771130797176391201197, 5.69142955341112128936953969223, 5.76971433271049052774411885271, 5.79139918791828671142766929488, 5.92471521385459265558799499114, 6.29871613104541327664975348221

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