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Properties

Label 8-4536e4-1.1-c1e4-0-1
Degree $8$
Conductor $4.233\times 10^{14}$
Sign $1$
Analytic cond. $1.72107\times 10^{6}$
Root an. cond. $6.01831$
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  + 4·5-s + 4·7-s − 6·11-s + 3·13-s + 8·17-s − 2·19-s − 5·23-s + 5·25-s + 29-s − 11·31-s + 16·35-s + 27·37-s + 2·41-s + 11·43-s + 7·47-s + 10·49-s + 4·53-s − 24·55-s + 9·59-s + 7·61-s + 12·65-s + 12·67-s + 12·71-s + 13·73-s − 24·77-s + 22·79-s − 6·83-s + ⋯
L(s)  = 1  + 1.78·5-s + 1.51·7-s − 1.80·11-s + 0.832·13-s + 1.94·17-s − 0.458·19-s − 1.04·23-s + 25-s + 0.185·29-s − 1.97·31-s + 2.70·35-s + 4.43·37-s + 0.312·41-s + 1.67·43-s + 1.02·47-s + 10/7·49-s + 0.549·53-s − 3.23·55-s + 1.17·59-s + 0.896·61-s + 1.48·65-s + 1.46·67-s + 1.42·71-s + 1.52·73-s − 2.73·77-s + 2.47·79-s − 0.658·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{16} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.72107\times 10^{6}\)
Root analytic conductor: \(6.01831\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{16} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(18.81069666\)
\(L(\frac12)\) \(\approx\) \(18.81069666\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{4} \)
good5$C_2 \wr S_4$ \( 1 - 4 T + 11 T^{2} - 31 T^{3} + 91 T^{4} - 31 p T^{5} + 11 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 6 T + 35 T^{2} + 117 T^{3} + 474 T^{4} + 117 p T^{5} + 35 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 3 T + 25 T^{2} - 18 T^{3} + 18 p T^{4} - 18 p T^{5} + 25 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 8 T + 35 T^{2} - 167 T^{3} + 886 T^{4} - 167 p T^{5} + 35 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 2 T + 55 T^{2} + 41 T^{3} + 1309 T^{4} + 41 p T^{5} + 55 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 5 T + 53 T^{2} + 221 T^{3} + 1729 T^{4} + 221 p T^{5} + 53 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - T + 50 T^{2} - 172 T^{3} + 1192 T^{4} - 172 p T^{5} + 50 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 11 T + 160 T^{2} + 1058 T^{3} + 8008 T^{4} + 1058 p T^{5} + 160 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 27 T + 412 T^{2} - 4104 T^{3} + 29436 T^{4} - 4104 p T^{5} + 412 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 2 T + p T^{2} - 257 T^{3} + 2008 T^{4} - 257 p T^{5} + p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 11 T + 148 T^{2} - 1112 T^{3} + 9532 T^{4} - 1112 p T^{5} + 148 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 7 T + 110 T^{2} - 298 T^{3} + 4906 T^{4} - 298 p T^{5} + 110 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 4 T + 83 T^{2} + 197 T^{3} + 2722 T^{4} + 197 p T^{5} + 83 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 9 T + 200 T^{2} - 1458 T^{3} + 16542 T^{4} - 1458 p T^{5} + 200 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 7 T + 193 T^{2} - 1333 T^{3} + 16135 T^{4} - 1333 p T^{5} + 193 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 12 T + 283 T^{2} - 2367 T^{3} + 28956 T^{4} - 2367 p T^{5} + 283 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 12 T + 167 T^{2} - 591 T^{3} + 7587 T^{4} - 591 p T^{5} + 167 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 13 T + 232 T^{2} - 1504 T^{3} + 18820 T^{4} - 1504 p T^{5} + 232 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 22 T + 247 T^{2} - 1909 T^{3} + 16129 T^{4} - 1909 p T^{5} + 247 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 6 T + 149 T^{2} + 1299 T^{3} + 11256 T^{4} + 1299 p T^{5} + 149 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 14 T + 323 T^{2} - 2963 T^{3} + 41152 T^{4} - 2963 p T^{5} + 323 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - T + 124 T^{2} - 490 T^{3} + 19024 T^{4} - 490 p T^{5} + 124 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.91512274541796578652451983202, −5.50373357685083137968474935132, −5.49893620776071665237511191232, −5.43558455913534634114763395750, −5.26653600551740875743986194600, −4.95861154004653689483549481712, −4.81540004578282422193976387279, −4.54713231719433730717316007497, −4.39358394348841078223399013987, −3.88147295657897978553189170768, −3.85345137435667092993327319210, −3.78029576294876543235646964754, −3.73228775798425060329698857257, −3.08281791869806940019618624354, −2.81144476722588910566486572743, −2.69831879478649147507145627836, −2.59569176031453020597128847403, −2.06689414400170080115379847026, −2.05952079497576312461526052922, −1.91843635477693520873584456425, −1.86974631145368837265314873002, −1.06643081097423492147783894136, −0.980046573994672445750962033329, −0.68789737083656265845589614489, −0.64389156369771366813049623263, 0.64389156369771366813049623263, 0.68789737083656265845589614489, 0.980046573994672445750962033329, 1.06643081097423492147783894136, 1.86974631145368837265314873002, 1.91843635477693520873584456425, 2.05952079497576312461526052922, 2.06689414400170080115379847026, 2.59569176031453020597128847403, 2.69831879478649147507145627836, 2.81144476722588910566486572743, 3.08281791869806940019618624354, 3.73228775798425060329698857257, 3.78029576294876543235646964754, 3.85345137435667092993327319210, 3.88147295657897978553189170768, 4.39358394348841078223399013987, 4.54713231719433730717316007497, 4.81540004578282422193976387279, 4.95861154004653689483549481712, 5.26653600551740875743986194600, 5.43558455913534634114763395750, 5.49893620776071665237511191232, 5.50373357685083137968474935132, 5.91512274541796578652451983202

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