Properties

Label 8-2352e4-1.1-c1e4-0-12
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $124410.$
Root an. cond. $4.33368$
Motivic weight $1$
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  − 2·3-s + 4·5-s + 9-s + 4·11-s − 8·15-s − 4·17-s − 8·19-s + 4·23-s + 12·25-s + 2·27-s − 16·31-s − 8·33-s + 8·37-s + 8·41-s + 32·43-s + 4·45-s − 8·47-s + 8·51-s + 4·53-s + 16·55-s + 16·57-s − 16·59-s + 8·61-s − 16·67-s − 8·69-s − 8·71-s − 8·73-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s + 1/3·9-s + 1.20·11-s − 2.06·15-s − 0.970·17-s − 1.83·19-s + 0.834·23-s + 12/5·25-s + 0.384·27-s − 2.87·31-s − 1.39·33-s + 1.31·37-s + 1.24·41-s + 4.87·43-s + 0.596·45-s − 1.16·47-s + 1.12·51-s + 0.549·53-s + 2.15·55-s + 2.11·57-s − 2.08·59-s + 1.02·61-s − 1.95·67-s − 0.963·69-s − 0.949·71-s − 0.936·73-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(124410.\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.063409690\)
\(L(\frac12)\) \(\approx\) \(4.063409690\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T + T^{2} )^{2} \)
7 \( 1 \)
good5$D_4\times C_2$ \( 1 - 4 T + 4 T^{2} - 8 T^{3} + 39 T^{4} - 8 p T^{5} + 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 4 T - 2 T^{2} + 16 T^{3} + 27 T^{4} + 16 p T^{5} - 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 4 T - 4 T^{2} - 56 T^{3} - 161 T^{4} - 56 p T^{5} - 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 + 8 T + 18 T^{2} + 64 T^{3} + 539 T^{4} + 64 p T^{5} + 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
23$C_4\times C_2$ \( 1 - 4 T - 26 T^{2} + 16 T^{3} + 867 T^{4} + 16 p T^{5} - 26 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 16 T + 138 T^{2} + 896 T^{3} + 5027 T^{4} + 896 p T^{5} + 138 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 8 T + 6 T^{2} + 128 T^{3} - 373 T^{4} + 128 p T^{5} + 6 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 4 T + 84 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
47$D_4\times C_2$ \( 1 + 8 T - 38 T^{2} + 64 T^{3} + 5187 T^{4} + 64 p T^{5} - 38 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 4 T + 34 T^{2} + 496 T^{3} - 3333 T^{4} + 496 p T^{5} + 34 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 16 T + 82 T^{2} + 896 T^{3} + 11691 T^{4} + 896 p T^{5} + 82 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 8 T + 24 T^{2} + 656 T^{3} - 6025 T^{4} + 656 p T^{5} + 24 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2^2$ \( ( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 8 T - 48 T^{2} - 272 T^{3} + 2543 T^{4} - 272 p T^{5} - 48 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 16 T + 66 T^{2} + 512 T^{3} + 9635 T^{4} + 512 p T^{5} + 66 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 4 T - 4 T^{2} + 632 T^{3} - 8945 T^{4} + 632 p T^{5} - 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 - 24 T + 320 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.18213052107709928568859641325, −6.12281696972679269027015061820, −6.03803854465560626795340341940, −5.83653026533453844007947924983, −5.71046951547934159512374800317, −5.53071345208376988329340464057, −5.05226268992738086202951479910, −5.00424547992461019113852662489, −4.72415859287628441558232217898, −4.42843307425872158627013801696, −4.32401482583756490396672930322, −4.15298965405557485398773736912, −4.06784138651864694687367080439, −3.52451004385857063902016839595, −3.23815941043690269312109284592, −3.09826541249015753483816918029, −2.78405286191202133044887024297, −2.37710621827355710012074247639, −2.29171940968757990587790912383, −1.95374667719756834656581477402, −1.73503280561763342349569219973, −1.58544750883032014644645527774, −0.889822657573869278369897261474, −0.808663742352786696135307766470, −0.41350071560858887148526125659, 0.41350071560858887148526125659, 0.808663742352786696135307766470, 0.889822657573869278369897261474, 1.58544750883032014644645527774, 1.73503280561763342349569219973, 1.95374667719756834656581477402, 2.29171940968757990587790912383, 2.37710621827355710012074247639, 2.78405286191202133044887024297, 3.09826541249015753483816918029, 3.23815941043690269312109284592, 3.52451004385857063902016839595, 4.06784138651864694687367080439, 4.15298965405557485398773736912, 4.32401482583756490396672930322, 4.42843307425872158627013801696, 4.72415859287628441558232217898, 5.00424547992461019113852662489, 5.05226268992738086202951479910, 5.53071345208376988329340464057, 5.71046951547934159512374800317, 5.83653026533453844007947924983, 6.03803854465560626795340341940, 6.12281696972679269027015061820, 6.18213052107709928568859641325

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