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Properties

Label 8-2352e4-1.1-c3e4-0-3
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $3.70863\times 10^{8}$
Root an. cond. $11.7801$
Motivic weight $3$
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  − 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

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

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(3.70863\times 10^{8}\)
Root analytic conductor: \(11.7801\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(3.852968945\)
\(L(\frac12)\) \(\approx\) \(3.852968945\)
\(L(\frac{5}{2})\) 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

−6.20470425151722741982677864992, −5.59240629846379069101675569230, −5.56740401076203401322752706376, −5.49439480836448642090401600645, −5.39442777695429088296655400439, −4.89103495386709809954805240351, −4.87554662771421914713538876669, −4.77236397189150343050974935587, −4.54967582228770980878466344380, −3.86720714161151397618292153121, −3.83563345623149812238579616321, −3.81501351750869924358437545032, −3.81340249208850988286401426088, −3.05898467975307927227807335375, −2.89473863210373399408862699394, −2.67330448953823047186293382137, −2.54767705438025116152259258321, −1.84374970390552561806724164713, −1.70224764335885133634673116433, −1.67312514306597313840634644695, −1.52816769061046797491386467484, −0.800498491916692887339731902263, −0.66504521738311640956374860662, −0.51319214816375689524401751450, −0.34413226614665664911416399725, 0.34413226614665664911416399725, 0.51319214816375689524401751450, 0.66504521738311640956374860662, 0.800498491916692887339731902263, 1.52816769061046797491386467484, 1.67312514306597313840634644695, 1.70224764335885133634673116433, 1.84374970390552561806724164713, 2.54767705438025116152259258321, 2.67330448953823047186293382137, 2.89473863210373399408862699394, 3.05898467975307927227807335375, 3.81340249208850988286401426088, 3.81501351750869924358437545032, 3.83563345623149812238579616321, 3.86720714161151397618292153121, 4.54967582228770980878466344380, 4.77236397189150343050974935587, 4.87554662771421914713538876669, 4.89103495386709809954805240351, 5.39442777695429088296655400439, 5.49439480836448642090401600645, 5.56740401076203401322752706376, 5.59240629846379069101675569230, 6.20470425151722741982677864992

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