Properties

Label 8-3e8-1.1-c55e4-0-0
Degree $8$
Conductor $6561$
Sign $1$
Analytic cond. $8.83861\times 10^{8}$
Root an. cond. $13.1310$
Motivic weight $55$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.08e8·2-s − 3.09e16·4-s − 1.43e19·5-s − 2.05e23·7-s + 1.04e25·8-s + 3.00e27·10-s − 1.94e28·11-s − 4.44e30·13-s + 4.28e31·14-s + 8.55e30·16-s − 8.59e33·17-s + 3.39e35·19-s + 4.45e35·20-s + 4.06e36·22-s − 4.98e37·23-s − 4.33e38·25-s + 9.28e38·26-s + 6.35e39·28-s + 1.83e40·29-s + 2.28e41·31-s − 3.15e41·32-s + 1.79e42·34-s + 2.95e42·35-s − 3.21e43·37-s − 7.08e43·38-s − 1.50e44·40-s − 2.46e44·41-s + ⋯
L(s)  = 1  − 1.09·2-s − 0.858·4-s − 0.864·5-s − 1.18·7-s + 1.53·8-s + 0.949·10-s − 0.448·11-s − 1.03·13-s + 1.29·14-s + 0.00659·16-s − 1.24·17-s + 2.31·19-s + 0.741·20-s + 0.492·22-s − 1.77·23-s − 1.56·25-s + 1.13·26-s + 1.01·28-s + 1.11·29-s + 2.21·31-s − 1.28·32-s + 1.37·34-s + 1.02·35-s − 2.40·37-s − 2.54·38-s − 1.32·40-s − 1.09·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(56-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+55/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(6561\)    =    \(3^{8}\)
Sign: $1$
Analytic conductor: \(8.83861\times 10^{8}\)
Root analytic conductor: \(13.1310\)
Motivic weight: \(55\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 6561,\ (\ :55/2, 55/2, 55/2, 55/2),\ 1)\)

Particular Values

\(L(28)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{57}{2})\) 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 \( 1 \)
good2$C_2 \wr S_4$ \( 1 + 26077815 p^{3} T + 72710344310675 p^{10} T^{2} + 5491911371338614105 p^{21} T^{3} + \)\(18\!\cdots\!59\)\( p^{37} T^{4} + 5491911371338614105 p^{76} T^{5} + 72710344310675 p^{120} T^{6} + 26077815 p^{168} T^{7} + p^{220} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 2879387592677433432 p T + \)\(20\!\cdots\!56\)\( p^{5} T^{2} + \)\(10\!\cdots\!16\)\( p^{10} T^{3} + \)\(31\!\cdots\!66\)\( p^{17} T^{4} + \)\(10\!\cdots\!16\)\( p^{65} T^{5} + \)\(20\!\cdots\!56\)\( p^{115} T^{6} + 2879387592677433432 p^{166} T^{7} + p^{220} T^{8} \)
7$C_2 \wr S_4$ \( 1 + \)\(59\!\cdots\!00\)\( p^{3} T + \)\(67\!\cdots\!00\)\( p^{7} T^{2} + \)\(15\!\cdots\!00\)\( p^{12} T^{3} + \)\(38\!\cdots\!02\)\( p^{18} T^{4} + \)\(15\!\cdots\!00\)\( p^{67} T^{5} + \)\(67\!\cdots\!00\)\( p^{117} T^{6} + \)\(59\!\cdots\!00\)\( p^{168} T^{7} + p^{220} T^{8} \)
11$C_2 \wr S_4$ \( 1 + \)\(17\!\cdots\!28\)\( p T + \)\(18\!\cdots\!88\)\( p^{3} T^{2} - \)\(56\!\cdots\!44\)\( p^{5} T^{3} + \)\(45\!\cdots\!70\)\( p^{8} T^{4} - \)\(56\!\cdots\!44\)\( p^{60} T^{5} + \)\(18\!\cdots\!88\)\( p^{113} T^{6} + \)\(17\!\cdots\!28\)\( p^{166} T^{7} + p^{220} T^{8} \)
13$C_2 \wr S_4$ \( 1 + \)\(34\!\cdots\!20\)\( p T + \)\(25\!\cdots\!00\)\( p^{3} T^{2} + \)\(40\!\cdots\!80\)\( p^{6} T^{3} + \)\(13\!\cdots\!26\)\( p^{9} T^{4} + \)\(40\!\cdots\!80\)\( p^{61} T^{5} + \)\(25\!\cdots\!00\)\( p^{113} T^{6} + \)\(34\!\cdots\!20\)\( p^{166} T^{7} + p^{220} T^{8} \)
17$C_2 \wr S_4$ \( 1 + \)\(50\!\cdots\!80\)\( p T + \)\(65\!\cdots\!00\)\( p^{2} T^{2} + \)\(13\!\cdots\!80\)\( p^{4} T^{3} + \)\(32\!\cdots\!26\)\( p^{7} T^{4} + \)\(13\!\cdots\!80\)\( p^{59} T^{5} + \)\(65\!\cdots\!00\)\( p^{112} T^{6} + \)\(50\!\cdots\!80\)\( p^{166} T^{7} + p^{220} T^{8} \)
19$C_2 \wr S_4$ \( 1 - \)\(33\!\cdots\!00\)\( T + \)\(64\!\cdots\!84\)\( p T^{2} - \)\(33\!\cdots\!00\)\( p^{3} T^{3} + \)\(17\!\cdots\!94\)\( p^{5} T^{4} - \)\(33\!\cdots\!00\)\( p^{58} T^{5} + \)\(64\!\cdots\!84\)\( p^{111} T^{6} - \)\(33\!\cdots\!00\)\( p^{165} T^{7} + p^{220} T^{8} \)
23$C_2 \wr S_4$ \( 1 + \)\(49\!\cdots\!60\)\( T + \)\(10\!\cdots\!00\)\( p T^{2} + \)\(73\!\cdots\!60\)\( p^{3} T^{3} + \)\(42\!\cdots\!86\)\( p^{5} T^{4} + \)\(73\!\cdots\!60\)\( p^{58} T^{5} + \)\(10\!\cdots\!00\)\( p^{111} T^{6} + \)\(49\!\cdots\!60\)\( p^{165} T^{7} + p^{220} T^{8} \)
29$C_2 \wr S_4$ \( 1 - \)\(18\!\cdots\!00\)\( T + \)\(63\!\cdots\!96\)\( T^{2} - \)\(32\!\cdots\!00\)\( p T^{3} + \)\(30\!\cdots\!66\)\( p^{2} T^{4} - \)\(32\!\cdots\!00\)\( p^{56} T^{5} + \)\(63\!\cdots\!96\)\( p^{110} T^{6} - \)\(18\!\cdots\!00\)\( p^{165} T^{7} + p^{220} T^{8} \)
31$C_2 \wr S_4$ \( 1 - \)\(22\!\cdots\!08\)\( T + \)\(18\!\cdots\!88\)\( p T^{2} - \)\(75\!\cdots\!96\)\( p^{2} T^{3} + \)\(32\!\cdots\!70\)\( p^{3} T^{4} - \)\(75\!\cdots\!96\)\( p^{57} T^{5} + \)\(18\!\cdots\!88\)\( p^{111} T^{6} - \)\(22\!\cdots\!08\)\( p^{165} T^{7} + p^{220} T^{8} \)
37$C_2 \wr S_4$ \( 1 + \)\(32\!\cdots\!20\)\( T + \)\(22\!\cdots\!00\)\( p T^{2} + \)\(98\!\cdots\!40\)\( p^{2} T^{3} + \)\(41\!\cdots\!66\)\( p^{3} T^{4} + \)\(98\!\cdots\!40\)\( p^{57} T^{5} + \)\(22\!\cdots\!00\)\( p^{111} T^{6} + \)\(32\!\cdots\!20\)\( p^{165} T^{7} + p^{220} T^{8} \)
41$C_2 \wr S_4$ \( 1 + \)\(24\!\cdots\!08\)\( T + \)\(46\!\cdots\!88\)\( p^{2} T^{2} + \)\(12\!\cdots\!76\)\( p^{2} T^{3} + \)\(90\!\cdots\!70\)\( p^{3} T^{4} + \)\(12\!\cdots\!76\)\( p^{57} T^{5} + \)\(46\!\cdots\!88\)\( p^{112} T^{6} + \)\(24\!\cdots\!08\)\( p^{165} T^{7} + p^{220} T^{8} \)
43$C_2 \wr S_4$ \( 1 + \)\(20\!\cdots\!00\)\( p T + \)\(14\!\cdots\!00\)\( p^{2} T^{2} + \)\(20\!\cdots\!00\)\( p^{3} T^{3} + \)\(80\!\cdots\!98\)\( p^{4} T^{4} + \)\(20\!\cdots\!00\)\( p^{58} T^{5} + \)\(14\!\cdots\!00\)\( p^{112} T^{6} + \)\(20\!\cdots\!00\)\( p^{166} T^{7} + p^{220} T^{8} \)
47$C_2 \wr S_4$ \( 1 + \)\(90\!\cdots\!20\)\( p T + \)\(27\!\cdots\!00\)\( p^{2} T^{2} - \)\(79\!\cdots\!60\)\( p^{3} T^{3} - \)\(14\!\cdots\!42\)\( p^{4} T^{4} - \)\(79\!\cdots\!60\)\( p^{58} T^{5} + \)\(27\!\cdots\!00\)\( p^{112} T^{6} + \)\(90\!\cdots\!20\)\( p^{166} T^{7} + p^{220} T^{8} \)
53$C_2 \wr S_4$ \( 1 - \)\(92\!\cdots\!60\)\( p T + \)\(11\!\cdots\!00\)\( p^{2} T^{2} - \)\(66\!\cdots\!80\)\( p^{3} T^{3} + \)\(41\!\cdots\!58\)\( p^{4} T^{4} - \)\(66\!\cdots\!80\)\( p^{58} T^{5} + \)\(11\!\cdots\!00\)\( p^{112} T^{6} - \)\(92\!\cdots\!60\)\( p^{166} T^{7} + p^{220} T^{8} \)
59$C_2 \wr S_4$ \( 1 + \)\(81\!\cdots\!00\)\( T + \)\(88\!\cdots\!96\)\( T^{2} + \)\(42\!\cdots\!00\)\( T^{3} + \)\(28\!\cdots\!06\)\( T^{4} + \)\(42\!\cdots\!00\)\( p^{55} T^{5} + \)\(88\!\cdots\!96\)\( p^{110} T^{6} + \)\(81\!\cdots\!00\)\( p^{165} T^{7} + p^{220} T^{8} \)
61$C_2 \wr S_4$ \( 1 - \)\(70\!\cdots\!08\)\( T + \)\(37\!\cdots\!28\)\( T^{2} - \)\(39\!\cdots\!56\)\( T^{3} + \)\(67\!\cdots\!70\)\( T^{4} - \)\(39\!\cdots\!56\)\( p^{55} T^{5} + \)\(37\!\cdots\!28\)\( p^{110} T^{6} - \)\(70\!\cdots\!08\)\( p^{165} T^{7} + p^{220} T^{8} \)
67$C_2 \wr S_4$ \( 1 - \)\(41\!\cdots\!60\)\( T + \)\(16\!\cdots\!00\)\( T^{2} - \)\(36\!\cdots\!80\)\( T^{3} + \)\(74\!\cdots\!98\)\( T^{4} - \)\(36\!\cdots\!80\)\( p^{55} T^{5} + \)\(16\!\cdots\!00\)\( p^{110} T^{6} - \)\(41\!\cdots\!60\)\( p^{165} T^{7} + p^{220} T^{8} \)
71$C_2 \wr S_4$ \( 1 + \)\(99\!\cdots\!08\)\( T + \)\(16\!\cdots\!28\)\( T^{2} + \)\(51\!\cdots\!56\)\( T^{3} + \)\(83\!\cdots\!70\)\( T^{4} + \)\(51\!\cdots\!56\)\( p^{55} T^{5} + \)\(16\!\cdots\!28\)\( p^{110} T^{6} + \)\(99\!\cdots\!08\)\( p^{165} T^{7} + p^{220} T^{8} \)
73$C_2 \wr S_4$ \( 1 + \)\(89\!\cdots\!40\)\( T + \)\(85\!\cdots\!00\)\( T^{2} + \)\(99\!\cdots\!80\)\( T^{3} + \)\(33\!\cdots\!98\)\( T^{4} + \)\(99\!\cdots\!80\)\( p^{55} T^{5} + \)\(85\!\cdots\!00\)\( p^{110} T^{6} + \)\(89\!\cdots\!40\)\( p^{165} T^{7} + p^{220} T^{8} \)
79$C_2 \wr S_4$ \( 1 - \)\(48\!\cdots\!00\)\( T + \)\(16\!\cdots\!96\)\( T^{2} - \)\(35\!\cdots\!00\)\( T^{3} + \)\(62\!\cdots\!06\)\( T^{4} - \)\(35\!\cdots\!00\)\( p^{55} T^{5} + \)\(16\!\cdots\!96\)\( p^{110} T^{6} - \)\(48\!\cdots\!00\)\( p^{165} T^{7} + p^{220} T^{8} \)
83$C_2 \wr S_4$ \( 1 - \)\(71\!\cdots\!20\)\( T + \)\(88\!\cdots\!00\)\( T^{2} - \)\(64\!\cdots\!40\)\( T^{3} + \)\(43\!\cdots\!98\)\( T^{4} - \)\(64\!\cdots\!40\)\( p^{55} T^{5} + \)\(88\!\cdots\!00\)\( p^{110} T^{6} - \)\(71\!\cdots\!20\)\( p^{165} T^{7} + p^{220} T^{8} \)
89$C_2 \wr S_4$ \( 1 + \)\(15\!\cdots\!00\)\( T + \)\(13\!\cdots\!96\)\( T^{2} + \)\(81\!\cdots\!00\)\( T^{3} + \)\(37\!\cdots\!06\)\( T^{4} + \)\(81\!\cdots\!00\)\( p^{55} T^{5} + \)\(13\!\cdots\!96\)\( p^{110} T^{6} + \)\(15\!\cdots\!00\)\( p^{165} T^{7} + p^{220} T^{8} \)
97$C_2 \wr S_4$ \( 1 - \)\(51\!\cdots\!40\)\( T + \)\(69\!\cdots\!00\)\( T^{2} - \)\(28\!\cdots\!20\)\( T^{3} + \)\(18\!\cdots\!98\)\( T^{4} - \)\(28\!\cdots\!20\)\( p^{55} T^{5} + \)\(69\!\cdots\!00\)\( p^{110} T^{6} - \)\(51\!\cdots\!40\)\( p^{165} T^{7} + p^{220} T^{8} \)
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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

−8.404979695142599400575701457744, −7.68172885843353565353759191432, −7.53973424838829997205998366037, −7.35492532713815641883723513768, −6.88025826627412658925917066793, −6.70194706490111214339299410271, −6.13894132845242408825065149438, −6.11265392173474851101458029967, −5.65887097353171215909074671348, −5.04910070359119344842080262117, −4.94551343141869547782874500495, −4.84453036315678142821534509959, −4.54972641963636612905862758339, −3.85056731806724830099430778438, −3.82135100597928262135843940006, −3.56698611911906322055082297179, −3.27404969543616702351733529701, −3.00830486003073860468701497168, −2.38538298098842907165328815819, −2.23558749944029451362940790241, −2.15377363437121195115562945977, −1.37963246714910314446420294243, −1.29441710772998478389793447301, −0.930575110950750158110845405209, −0.57706698260158879411810262740, 0, 0, 0, 0, 0.57706698260158879411810262740, 0.930575110950750158110845405209, 1.29441710772998478389793447301, 1.37963246714910314446420294243, 2.15377363437121195115562945977, 2.23558749944029451362940790241, 2.38538298098842907165328815819, 3.00830486003073860468701497168, 3.27404969543616702351733529701, 3.56698611911906322055082297179, 3.82135100597928262135843940006, 3.85056731806724830099430778438, 4.54972641963636612905862758339, 4.84453036315678142821534509959, 4.94551343141869547782874500495, 5.04910070359119344842080262117, 5.65887097353171215909074671348, 6.11265392173474851101458029967, 6.13894132845242408825065149438, 6.70194706490111214339299410271, 6.88025826627412658925917066793, 7.35492532713815641883723513768, 7.53973424838829997205998366037, 7.68172885843353565353759191432, 8.404979695142599400575701457744

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