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Properties

Label 8-1176e4-1.1-c3e4-0-4
Degree $8$
Conductor $1.913\times 10^{12}$
Sign $1$
Analytic cond. $2.31789\times 10^{7}$
Root an. cond. $8.32984$
Motivic weight $3$
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  − 12·3-s − 8·5-s + 90·9-s + 40·11-s − 48·13-s + 96·15-s − 72·17-s − 32·19-s + 8·23-s − 136·25-s − 540·27-s + 144·29-s − 48·31-s − 480·33-s + 48·37-s + 576·39-s − 72·41-s + 512·43-s − 720·45-s − 160·47-s + 864·51-s + 536·53-s − 320·55-s + 384·57-s − 240·59-s − 896·61-s + 384·65-s + ⋯
L(s)  = 1  − 2.30·3-s − 0.715·5-s + 10/3·9-s + 1.09·11-s − 1.02·13-s + 1.65·15-s − 1.02·17-s − 0.386·19-s + 0.0725·23-s − 1.08·25-s − 3.84·27-s + 0.922·29-s − 0.278·31-s − 2.53·33-s + 0.213·37-s + 2.36·39-s − 0.274·41-s + 1.81·43-s − 2.38·45-s − 0.496·47-s + 2.37·51-s + 1.38·53-s − 0.784·55-s + 0.892·57-s − 0.529·59-s − 1.88·61-s + 0.732·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(2.31789\times 10^{7}\)
Root analytic conductor: \(8.32984\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{4} \)
7 \( 1 \)
good5$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 8 p^{2} T^{2} + 1544 T^{3} + 40898 T^{4} + 1544 p^{3} T^{5} + 8 p^{8} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 40 T + 5092 T^{2} - 155016 T^{3} + 10018838 T^{4} - 155016 p^{3} T^{5} + 5092 p^{6} T^{6} - 40 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 48 T + 8912 T^{2} + 307440 T^{3} + 29511218 T^{4} + 307440 p^{3} T^{5} + 8912 p^{6} T^{6} + 48 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 72 T + 18328 T^{2} + 1017000 T^{3} + 132511218 T^{4} + 1017000 p^{3} T^{5} + 18328 p^{6} T^{6} + 72 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 32 T + 12444 T^{2} + 444960 T^{3} + 127487926 T^{4} + 444960 p^{3} T^{5} + 12444 p^{6} T^{6} + 32 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 44724 T^{2} - 187080 T^{3} + 792000774 T^{4} - 187080 p^{3} T^{5} + 44724 p^{6} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 144 T + 52836 T^{2} - 9502704 T^{3} + 1581400822 T^{4} - 9502704 p^{3} T^{5} + 52836 p^{6} T^{6} - 144 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 48 T + 61260 T^{2} + 6694896 T^{3} + 2155682342 T^{4} + 6694896 p^{3} T^{5} + 61260 p^{6} T^{6} + 48 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 48 T + 107220 T^{2} + 3877040 T^{3} + 5733855606 T^{4} + 3877040 p^{3} T^{5} + 107220 p^{6} T^{6} - 48 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 72 T + 132408 T^{2} + 25735656 T^{3} + 8640420370 T^{4} + 25735656 p^{3} T^{5} + 132408 p^{6} T^{6} + 72 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 512 T + 336396 T^{2} - 105348608 T^{3} + 39800861942 T^{4} - 105348608 p^{3} T^{5} + 336396 p^{6} T^{6} - 512 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 160 T + 326572 T^{2} + 48645664 T^{3} + 47161260198 T^{4} + 48645664 p^{3} T^{5} + 326572 p^{6} T^{6} + 160 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 536 T + 507404 T^{2} - 234466568 T^{3} + 107272884982 T^{4} - 234466568 p^{3} T^{5} + 507404 p^{6} T^{6} - 536 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 240 T + 373212 T^{2} - 28701456 T^{3} + 58936761430 T^{4} - 28701456 p^{3} T^{5} + 373212 p^{6} T^{6} + 240 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 896 T + 993360 T^{2} + 559778304 T^{3} + 345837093106 T^{4} + 559778304 p^{3} T^{5} + 993360 p^{6} T^{6} + 896 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 1088 T + 1156908 T^{2} - 790291520 T^{3} + 491979608726 T^{4} - 790291520 p^{3} T^{5} + 1156908 p^{6} T^{6} - 1088 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 1288 T + 1012404 T^{2} - 492746568 T^{3} + 280404439750 T^{4} - 492746568 p^{3} T^{5} + 1012404 p^{6} T^{6} - 1288 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 1488 T + 1753472 T^{2} + 1533924432 T^{3} + 1040082721634 T^{4} + 1533924432 p^{3} T^{5} + 1753472 p^{6} T^{6} + 1488 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 416 T + 1109756 T^{2} - 40685856 T^{3} + 519699459142 T^{4} - 40685856 p^{3} T^{5} + 1109756 p^{6} T^{6} - 416 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 112 T + 1678732 T^{2} - 251602288 T^{3} + 1266920594454 T^{4} - 251602288 p^{3} T^{5} + 1678732 p^{6} T^{6} - 112 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 3160 T + 4955832 T^{2} + 5316788664 T^{3} + 4738991693650 T^{4} + 5316788664 p^{3} T^{5} + 4955832 p^{6} T^{6} + 3160 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 2384 T + 4995488 T^{2} + 6771588432 T^{3} + 7465851671042 T^{4} + 6771588432 p^{3} T^{5} + 4995488 p^{6} T^{6} + 2384 p^{9} T^{7} + p^{12} T^{8} \)
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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

−7.07311395084747386650666967213, −6.57400593537917437694237385984, −6.56908464541230493977556745870, −6.50675497314069263681047861932, −6.22197646837641412912503636679, −5.94017583673857056892758776546, −5.58755090346612643593921234088, −5.46110886442174881266709207314, −5.37233903081176296172297453510, −5.11029871405416478283511505132, −4.68086414501444989275640126367, −4.55555332543288288000929668643, −4.47146115771470077234563565350, −3.95275373496728053460901064445, −3.93956394909069476051115920787, −3.77379123344185623429667452278, −3.68690951252763864924660131651, −2.76445278587140312267906967610, −2.70561639700481847680246692637, −2.39456186353322551859158932999, −2.36183336160174760825939835424, −1.47462031569902553606617484185, −1.44329890736239276802597013967, −1.09364473732739488342367524078, −1.07318458867128794535154598641, 0, 0, 0, 0, 1.07318458867128794535154598641, 1.09364473732739488342367524078, 1.44329890736239276802597013967, 1.47462031569902553606617484185, 2.36183336160174760825939835424, 2.39456186353322551859158932999, 2.70561639700481847680246692637, 2.76445278587140312267906967610, 3.68690951252763864924660131651, 3.77379123344185623429667452278, 3.93956394909069476051115920787, 3.95275373496728053460901064445, 4.47146115771470077234563565350, 4.55555332543288288000929668643, 4.68086414501444989275640126367, 5.11029871405416478283511505132, 5.37233903081176296172297453510, 5.46110886442174881266709207314, 5.58755090346612643593921234088, 5.94017583673857056892758776546, 6.22197646837641412912503636679, 6.50675497314069263681047861932, 6.56908464541230493977556745870, 6.57400593537917437694237385984, 7.07311395084747386650666967213

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