Properties

Label 8-12e8-1.1-c2e4-0-2
Degree $8$
Conductor $429981696$
Sign $1$
Analytic cond. $237.022$
Root an. cond. $1.98083$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 12·3-s + 6·5-s + 6·7-s + 90·9-s + 18·11-s − 14·13-s + 72·15-s − 8·19-s + 72·21-s − 30·23-s + 3·25-s + 540·27-s + 6·29-s − 74·31-s + 216·33-s + 36·35-s − 120·37-s − 168·39-s − 138·41-s − 10·43-s + 540·45-s − 174·47-s + 11·49-s + 108·55-s − 96·57-s + 18·59-s − 62·61-s + ⋯
L(s)  = 1  + 4·3-s + 6/5·5-s + 6/7·7-s + 10·9-s + 1.63·11-s − 1.07·13-s + 24/5·15-s − 0.421·19-s + 24/7·21-s − 1.30·23-s + 3/25·25-s + 20·27-s + 6/29·29-s − 2.38·31-s + 6.54·33-s + 1.02·35-s − 3.24·37-s − 4.30·39-s − 3.36·41-s − 0.232·43-s + 12·45-s − 3.70·47-s + 0.224·49-s + 1.96·55-s − 1.68·57-s + 0.305·59-s − 1.01·61-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(237.022\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(18.37052676\)
\(L(\frac12)\) \(\approx\) \(18.37052676\)
\(L(2)\) 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$C_1$ \( ( 1 - p T )^{4} \)
good5$D_4\times C_2$ \( 1 - 6 T + 33 T^{2} - 126 T^{3} + 116 T^{4} - 126 p^{2} T^{5} + 33 p^{4} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} \)
7$D_4\times C_2$ \( 1 - 6 T + 25 T^{2} + 522 T^{3} - 4044 T^{4} + 522 p^{2} T^{5} + 25 p^{4} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} \)
11$D_4\times C_2$ \( 1 - 18 T + 249 T^{2} - 2538 T^{3} + 18308 T^{4} - 2538 p^{2} T^{5} + 249 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \)
13$D_4\times C_2$ \( 1 + 14 T - 95 T^{2} - 658 T^{3} + 22996 T^{4} - 658 p^{2} T^{5} - 95 p^{4} T^{6} + 14 p^{6} T^{7} + p^{8} T^{8} \)
17$D_4\times C_2$ \( 1 - 516 T^{2} + 135302 T^{4} - 516 p^{4} T^{6} + p^{8} T^{8} \)
19$D_{4}$ \( ( 1 + 4 T + 18 p T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 30 T + 1401 T^{2} + 33030 T^{3} + 1091060 T^{4} + 33030 p^{2} T^{5} + 1401 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} \)
29$D_4\times C_2$ \( 1 - 6 T + 1409 T^{2} - 8382 T^{3} + 1254420 T^{4} - 8382 p^{2} T^{5} + 1409 p^{4} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} \)
31$D_4\times C_2$ \( 1 + 74 T + 2281 T^{2} + 94202 T^{3} + 4022068 T^{4} + 94202 p^{2} T^{5} + 2281 p^{4} T^{6} + 74 p^{6} T^{7} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 + 60 T + 3254 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 69 T + 3268 T^{2} + 69 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 + 10 T - 2087 T^{2} - 15110 T^{3} + 1179268 T^{4} - 15110 p^{2} T^{5} - 2087 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} \)
47$D_4\times C_2$ \( 1 + 174 T + 16745 T^{2} + 1157622 T^{3} + 61675956 T^{4} + 1157622 p^{2} T^{5} + 16745 p^{4} T^{6} + 174 p^{6} T^{7} + p^{8} T^{8} \)
53$D_4\times C_2$ \( 1 - 996 T^{2} - 9136858 T^{4} - 996 p^{4} T^{6} + p^{8} T^{8} \)
59$D_4\times C_2$ \( 1 - 18 T + 6969 T^{2} - 123498 T^{3} + 35331908 T^{4} - 123498 p^{2} T^{5} + 6969 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 + 62 T - 4463 T^{2} + 53630 T^{3} + 40856884 T^{4} + 53630 p^{2} T^{5} - 4463 p^{4} T^{6} + 62 p^{6} T^{7} + p^{8} T^{8} \)
67$D_4\times C_2$ \( 1 - 22 T + 985 T^{2} + 208538 T^{3} - 22072796 T^{4} + 208538 p^{2} T^{5} + 985 p^{4} T^{6} - 22 p^{6} T^{7} + p^{8} T^{8} \)
71$D_4\times C_2$ \( 1 - 16452 T^{2} + 117605702 T^{4} - 16452 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 - 20 T + 7302 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 86 T - 2231 T^{2} + 245530 T^{3} + 7570612 T^{4} + 245530 p^{2} T^{5} - 2231 p^{4} T^{6} - 86 p^{6} T^{7} + p^{8} T^{8} \)
83$D_4\times C_2$ \( 1 - 66 T + 9321 T^{2} - 519354 T^{3} + 24465668 T^{4} - 519354 p^{2} T^{5} + 9321 p^{4} T^{6} - 66 p^{6} T^{7} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 25924 T^{2} + 285535302 T^{4} - 25924 p^{4} T^{6} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 - 242 T + 25489 T^{2} - 3450194 T^{3} + 454397668 T^{4} - 3450194 p^{2} T^{5} + 25489 p^{4} T^{6} - 242 p^{6} T^{7} + p^{8} T^{8} \)
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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

−9.358406811728764692358541390362, −8.991470931175957153758386576243, −8.758332285372489003070324274337, −8.583380174774319210251472481930, −8.495040396668544743415759203984, −8.098417251547476018637129560909, −7.83622947317915995758596225837, −7.49624161054073155619483475816, −7.26492404496764337175678030287, −6.85414827213649745326058833936, −6.81030742549333747140828365773, −6.42642807949954484534789560736, −6.09833730928179908222855567453, −5.16054385765402168947639542302, −4.97924532786057456527304325791, −4.86997477683667786308924838923, −4.31617857100484774484524471940, −3.83211646103514031939738262936, −3.53377120617939780271939976928, −3.27858143209061309093459445123, −3.18968364246582292948899385194, −2.05922533163694006560593327019, −1.88074874169919798642099209355, −1.87096195090316916790817694638, −1.64972381781560691134461549315, 1.64972381781560691134461549315, 1.87096195090316916790817694638, 1.88074874169919798642099209355, 2.05922533163694006560593327019, 3.18968364246582292948899385194, 3.27858143209061309093459445123, 3.53377120617939780271939976928, 3.83211646103514031939738262936, 4.31617857100484774484524471940, 4.86997477683667786308924838923, 4.97924532786057456527304325791, 5.16054385765402168947639542302, 6.09833730928179908222855567453, 6.42642807949954484534789560736, 6.81030742549333747140828365773, 6.85414827213649745326058833936, 7.26492404496764337175678030287, 7.49624161054073155619483475816, 7.83622947317915995758596225837, 8.098417251547476018637129560909, 8.495040396668544743415759203984, 8.583380174774319210251472481930, 8.758332285372489003070324274337, 8.991470931175957153758386576243, 9.358406811728764692358541390362

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