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Properties

Label 8-124e4-1.1-c3e4-0-2
Degree $8$
Conductor $236421376$
Sign $1$
Analytic cond. $2865.17$
Root an. cond. $2.70485$
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  − 4·3-s − 14·5-s − 12·7-s − 58·9-s − 98·11-s − 128·13-s + 56·15-s − 86·17-s − 116·19-s + 48·21-s − 214·23-s − 101·25-s + 208·27-s − 168·29-s − 124·31-s + 392·33-s + 168·35-s + 598·37-s + 512·39-s + 218·41-s − 192·43-s + 812·45-s + 32·47-s − 59·49-s + 344·51-s + 290·53-s + 1.37e3·55-s + ⋯
L(s)  = 1  − 0.769·3-s − 1.25·5-s − 0.647·7-s − 2.14·9-s − 2.68·11-s − 2.73·13-s + 0.963·15-s − 1.22·17-s − 1.40·19-s + 0.498·21-s − 1.94·23-s − 0.807·25-s + 1.48·27-s − 1.07·29-s − 0.718·31-s + 2.06·33-s + 0.811·35-s + 2.65·37-s + 2.10·39-s + 0.830·41-s − 0.680·43-s + 2.68·45-s + 0.0993·47-s − 0.172·49-s + 0.944·51-s + 0.751·53-s + 3.36·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 31^{4}\)
Sign: $1$
Analytic conductor: \(2865.17\)
Root analytic conductor: \(2.70485\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{8} \cdot 31^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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 \( 1 \)
31$C_1$ \( ( 1 + p T )^{4} \)
good3$C_2 \wr S_4$ \( 1 + 4 T + 74 T^{2} + 320 T^{3} + 854 p T^{4} + 320 p^{3} T^{5} + 74 p^{6} T^{6} + 4 p^{9} T^{7} + p^{12} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 14 T + 297 T^{2} + 3242 T^{3} + 49228 T^{4} + 3242 p^{3} T^{5} + 297 p^{6} T^{6} + 14 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 12 T + 29 p T^{2} - 3104 T^{3} + 956 p T^{4} - 3104 p^{3} T^{5} + 29 p^{7} T^{6} + 12 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 98 T + 8006 T^{2} + 402386 T^{3} + 17644554 T^{4} + 402386 p^{3} T^{5} + 8006 p^{6} T^{6} + 98 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 128 T + 9810 T^{2} + 37532 p T^{3} + 23368730 T^{4} + 37532 p^{4} T^{5} + 9810 p^{6} T^{6} + 128 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 86 T + 4228 T^{2} + 227866 T^{3} + 26423606 T^{4} + 227866 p^{3} T^{5} + 4228 p^{6} T^{6} + 86 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 116 T + 15443 T^{2} + 1363164 T^{3} + 154721440 T^{4} + 1363164 p^{3} T^{5} + 15443 p^{6} T^{6} + 116 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 214 T + 62312 T^{2} + 8012366 T^{3} + 1217316686 T^{4} + 8012366 p^{3} T^{5} + 62312 p^{6} T^{6} + 214 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 168 T + 65882 T^{2} + 6081948 T^{3} + 1778541866 T^{4} + 6081948 p^{3} T^{5} + 65882 p^{6} T^{6} + 168 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 598 T + 235190 T^{2} - 68455498 T^{3} + 17416236482 T^{4} - 68455498 p^{3} T^{5} + 235190 p^{6} T^{6} - 598 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 218 T + 45317 T^{2} - 24691466 T^{3} + 6307384176 T^{4} - 24691466 p^{3} T^{5} + 45317 p^{6} T^{6} - 218 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 192 T + 95538 T^{2} + 14690108 T^{3} + 6487166754 T^{4} + 14690108 p^{3} T^{5} + 95538 p^{6} T^{6} + 192 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 32 T + 231008 T^{2} - 6835312 T^{3} + 34843926590 T^{4} - 6835312 p^{3} T^{5} + 231008 p^{6} T^{6} - 32 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 290 T + 412506 T^{2} - 127252374 T^{3} + 81640432690 T^{4} - 127252374 p^{3} T^{5} + 412506 p^{6} T^{6} - 290 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 376 T + 303315 T^{2} - 38573252 T^{3} + 12054241288 T^{4} - 38573252 p^{3} T^{5} + 303315 p^{6} T^{6} + 376 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 196 T + 5762 p T^{2} - 24638344 T^{3} + 105287146250 T^{4} - 24638344 p^{3} T^{5} + 5762 p^{7} T^{6} - 196 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 40 T + 680108 T^{2} - 3258024 T^{3} + 263174977430 T^{4} - 3258024 p^{3} T^{5} + 680108 p^{6} T^{6} - 40 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 536 T + 844055 T^{2} + 344174168 T^{3} + 329632059072 T^{4} + 344174168 p^{3} T^{5} + 844055 p^{6} T^{6} + 536 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 1436 T + 2034484 T^{2} - 1602491140 T^{3} + 1240608984182 T^{4} - 1602491140 p^{3} T^{5} + 2034484 p^{6} T^{6} - 1436 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 1010 T + 2068264 T^{2} + 1419955562 T^{3} + 1552028756494 T^{4} + 1419955562 p^{3} T^{5} + 2068264 p^{6} T^{6} + 1010 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 1174 T + 2253054 T^{2} + 1869444326 T^{3} + 1902723538394 T^{4} + 1869444326 p^{3} T^{5} + 2253054 p^{6} T^{6} + 1174 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 518 T + 1123420 T^{2} - 494841510 T^{3} + 252371923334 T^{4} - 494841510 p^{3} T^{5} + 1123420 p^{6} T^{6} + 518 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 466 T + 748589 T^{2} + 398198962 T^{3} + 1179475690352 T^{4} + 398198962 p^{3} T^{5} + 748589 p^{6} T^{6} + 466 p^{9} T^{7} + p^{12} T^{8} \)
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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

−9.908331355948405768547495236472, −9.743827948335188964933685389570, −9.425497500696559711852702988877, −9.156533760745363151738941361763, −8.801162942726218870080667427497, −8.291493259112238885101107706350, −8.021632163016695952072814846213, −7.935216206166005237899942995115, −7.88564207398541850522352582885, −7.38802626178143531206966629080, −7.22827018453745253974585434252, −6.52793379395203378227019374381, −6.51363104739737957840474561475, −5.91136463572116904130048066593, −5.61105462460997553249297717092, −5.57274254717569287539520771240, −5.22213216139620570804233634503, −4.76679552886280092595893286456, −4.32764076068778194840367208608, −4.08780551203771438749953993296, −3.71187779542170186382938953812, −2.88383176420668570512195588458, −2.52198549012378428953338608114, −2.51807916125295442194846844509, −2.12575121962869064035623742167, 0, 0, 0, 0, 2.12575121962869064035623742167, 2.51807916125295442194846844509, 2.52198549012378428953338608114, 2.88383176420668570512195588458, 3.71187779542170186382938953812, 4.08780551203771438749953993296, 4.32764076068778194840367208608, 4.76679552886280092595893286456, 5.22213216139620570804233634503, 5.57274254717569287539520771240, 5.61105462460997553249297717092, 5.91136463572116904130048066593, 6.51363104739737957840474561475, 6.52793379395203378227019374381, 7.22827018453745253974585434252, 7.38802626178143531206966629080, 7.88564207398541850522352582885, 7.935216206166005237899942995115, 8.021632163016695952072814846213, 8.291493259112238885101107706350, 8.801162942726218870080667427497, 9.156533760745363151738941361763, 9.425497500696559711852702988877, 9.743827948335188964933685389570, 9.908331355948405768547495236472

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