Properties

Label 8-78e4-1.1-c1e4-0-0
Degree $8$
Conductor $37015056$
Sign $1$
Analytic cond. $0.150482$
Root an. cond. $0.789197$
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-s + 6·7-s + 9-s − 6·11-s − 2·12-s − 8·17-s + 6·19-s − 12·21-s − 2·23-s + 6·25-s + 2·27-s + 6·28-s + 2·29-s + 12·33-s + 36-s − 12·37-s + 36·41-s − 2·43-s − 6·44-s + 8·49-s + 16·51-s − 12·53-s − 12·57-s + 8·61-s + 6·63-s − 64-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s + 2.26·7-s + 1/3·9-s − 1.80·11-s − 0.577·12-s − 1.94·17-s + 1.37·19-s − 2.61·21-s − 0.417·23-s + 6/5·25-s + 0.384·27-s + 1.13·28-s + 0.371·29-s + 2.08·33-s + 1/6·36-s − 1.97·37-s + 5.62·41-s − 0.304·43-s − 0.904·44-s + 8/7·49-s + 2.24·51-s − 1.64·53-s − 1.58·57-s + 1.02·61-s + 0.755·63-s − 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(37015056\)    =    \(2^{4} \cdot 3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(0.150482\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{78} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 37015056,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6619664070\)
\(L(\frac12)\) \(\approx\) \(0.6619664070\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
3$C_2$ \( ( 1 + T + T^{2} )^{2} \)
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
good5$C_2^2$$\times$$C_2^2$ \( ( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} ) \)
7$D_4\times C_2$ \( 1 - 6 T + 4 p T^{2} - 96 T^{3} + 291 T^{4} - 96 p T^{5} + 4 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 + 6 T + 28 T^{2} + 96 T^{3} + 267 T^{4} + 96 p T^{5} + 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2$$\times$$C_2^2$ \( ( 1 + 8 T + p T^{2} )^{2}( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} ) \)
19$D_4\times C_2$ \( 1 - 6 T + 44 T^{2} - 192 T^{3} + 891 T^{4} - 192 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 2 T - 16 T^{2} - 52 T^{3} - 221 T^{4} - 52 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 - 2 T - 43 T^{2} + 22 T^{3} + 1252 T^{4} + 22 p T^{5} - 43 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + 48 p T^{4} + 12 p^{2} T^{5} + 85 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 - 36 T + 621 T^{2} - 6804 T^{3} + 51752 T^{4} - 6804 p T^{5} + 621 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 2 T - 8 T^{2} - 148 T^{3} - 1877 T^{4} - 148 p T^{5} - 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 116 T^{2} + 6810 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 + 54 T^{2} - 565 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 8 T - 47 T^{2} + 88 T^{3} + 4696 T^{4} + 88 p T^{5} - 47 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 42 T + 868 T^{2} + 11760 T^{3} + 113307 T^{4} + 11760 p T^{5} + 868 p^{2} T^{6} + 42 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 6 T + 148 T^{2} - 816 T^{3} + 14307 T^{4} - 816 p T^{5} + 148 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 158 T^{2} + 16131 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 228 T^{2} + 24074 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 + 12 T + 202 T^{2} + 1848 T^{3} + 20067 T^{4} + 1848 p T^{5} + 202 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2^3$ \( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \)
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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

−11.14920526402552747920383678637, −10.46952276357695368955839763573, −10.30144854453387930859462299316, −10.27621947446913151759042079185, −9.513381129697581069675827652353, −9.292661793350265110313610425119, −8.788196249468763156294424808134, −8.549207448099631054207377829777, −8.485413756317002429117953726751, −7.72992778196790332747013034511, −7.60494649373673797247094018464, −7.51424907229592473310621805340, −7.11253044695813313555695908315, −6.70868088619909623586300094746, −5.94132277755358982544269229002, −5.84584059943671257987957986023, −5.84002869441533288882442948255, −4.94944937197601151130105246391, −4.87262416371336868153476395314, −4.53422574343959597122805498660, −4.41295548574965369783604137822, −3.29459443722306480203783628955, −2.72857001623338863597294091313, −2.28101002280442087926870223077, −1.45567106767391687444528325286, 1.45567106767391687444528325286, 2.28101002280442087926870223077, 2.72857001623338863597294091313, 3.29459443722306480203783628955, 4.41295548574965369783604137822, 4.53422574343959597122805498660, 4.87262416371336868153476395314, 4.94944937197601151130105246391, 5.84002869441533288882442948255, 5.84584059943671257987957986023, 5.94132277755358982544269229002, 6.70868088619909623586300094746, 7.11253044695813313555695908315, 7.51424907229592473310621805340, 7.60494649373673797247094018464, 7.72992778196790332747013034511, 8.485413756317002429117953726751, 8.549207448099631054207377829777, 8.788196249468763156294424808134, 9.292661793350265110313610425119, 9.513381129697581069675827652353, 10.27621947446913151759042079185, 10.30144854453387930859462299316, 10.46952276357695368955839763573, 11.14920526402552747920383678637

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