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Properties

Label 8-78e4-1.1-c3e4-0-1
Degree $8$
Conductor $37015056$
Sign $1$
Analytic cond. $448.582$
Root an. cond. $2.14526$
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  − 4·2-s − 6·3-s + 4·4-s + 26·5-s + 24·6-s − 9·7-s + 16·8-s + 9·9-s − 104·10-s − 38·11-s − 24·12-s + 12·13-s + 36·14-s − 156·15-s − 64·16-s − 99·17-s − 36·18-s + 16·19-s + 104·20-s + 54·21-s + 152·22-s + 14·23-s − 96·24-s + 259·25-s − 48·26-s + 54·27-s − 36·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 1/2·4-s + 2.32·5-s + 1.63·6-s − 0.485·7-s + 0.707·8-s + 1/3·9-s − 3.28·10-s − 1.04·11-s − 0.577·12-s + 0.256·13-s + 0.687·14-s − 2.68·15-s − 16-s − 1.41·17-s − 0.471·18-s + 0.193·19-s + 1.16·20-s + 0.561·21-s + 1.47·22-s + 0.126·23-s − 0.816·24-s + 2.07·25-s − 0.362·26-s + 0.384·27-s − 0.242·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(37015056\)    =    \(2^{4} \cdot 3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(448.582\)
Root analytic conductor: \(2.14526\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 37015056,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2513174528\)
\(L(\frac12)\) \(\approx\) \(0.2513174528\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 + p T + p^{2} T^{2} )^{2} \)
3$C_2$ \( ( 1 + p T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 12 T + 289 p T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \)
good5$D_{4}$ \( ( 1 - 13 T + 124 T^{2} - 13 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
7$D_4\times C_2$ \( 1 + 9 T - 457 T^{2} - 1332 T^{3} + 144012 T^{4} - 1332 p^{3} T^{5} - 457 p^{6} T^{6} + 9 p^{9} T^{7} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 + 38 T - 906 T^{2} - 11856 T^{3} + 1828975 T^{4} - 11856 p^{3} T^{5} - 906 p^{6} T^{6} + 38 p^{9} T^{7} + p^{12} T^{8} \)
17$D_4\times C_2$ \( 1 + 99 T - 961 T^{2} + 92664 T^{3} + 43530762 T^{4} + 92664 p^{3} T^{5} - 961 p^{6} T^{6} + 99 p^{9} T^{7} + p^{12} T^{8} \)
19$D_4\times C_2$ \( 1 - 16 T - 2758 T^{2} + 171264 T^{3} - 39717589 T^{4} + 171264 p^{3} T^{5} - 2758 p^{6} T^{6} - 16 p^{9} T^{7} + p^{12} T^{8} \)
23$D_4\times C_2$ \( 1 - 14 T - 23514 T^{2} + 8736 T^{3} + 411743479 T^{4} + 8736 p^{3} T^{5} - 23514 p^{6} T^{6} - 14 p^{9} T^{7} + p^{12} T^{8} \)
29$D_4\times C_2$ \( 1 + 121 T - 17439 T^{2} - 2020458 T^{3} + 77223730 T^{4} - 2020458 p^{3} T^{5} - 17439 p^{6} T^{6} + 121 p^{9} T^{7} + p^{12} T^{8} \)
31$D_{4}$ \( ( 1 + 117 T + 58798 T^{2} + 117 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + 389 T + 12353 T^{2} + 14650518 T^{3} + 9081516002 T^{4} + 14650518 p^{3} T^{5} + 12353 p^{6} T^{6} + 389 p^{9} T^{7} + p^{12} T^{8} \)
41$D_4\times C_2$ \( 1 - 333 T + 19523 T^{2} + 15476508 T^{3} - 4044939054 T^{4} + 15476508 p^{3} T^{5} + 19523 p^{6} T^{6} - 333 p^{9} T^{7} + p^{12} T^{8} \)
43$D_4\times C_2$ \( 1 + 15 p T + 173363 T^{2} + 1254720 p T^{3} + 19486583520 T^{4} + 1254720 p^{4} T^{5} + 173363 p^{6} T^{6} + 15 p^{10} T^{7} + p^{12} T^{8} \)
47$D_{4}$ \( ( 1 - 626 T + 288790 T^{2} - 626 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 + 729 T + 416986 T^{2} + 729 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 574 T + 79302 T^{2} - 92175216 T^{3} - 47260954769 T^{4} - 92175216 p^{3} T^{5} + 79302 p^{6} T^{6} + 574 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 - 846 T + 85517 T^{2} - 149096502 T^{3} + 183176471844 T^{4} - 149096502 p^{3} T^{5} + 85517 p^{6} T^{6} - 846 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 - 1059 T + 445691 T^{2} - 78645576 T^{3} + 28933751928 T^{4} - 78645576 p^{3} T^{5} + 445691 p^{6} T^{6} - 1059 p^{9} T^{7} + p^{12} T^{8} \)
71$D_4\times C_2$ \( 1 + 1346 T + 885918 T^{2} + 282627696 T^{3} + 80568567607 T^{4} + 282627696 p^{3} T^{5} + 885918 p^{6} T^{6} + 1346 p^{9} T^{7} + p^{12} T^{8} \)
73$D_{4}$ \( ( 1 - 1444 T + 1147893 T^{2} - 1444 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 599 T + 1037922 T^{2} + 599 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$D_{4}$ \( ( 1 - 312 T + 1070998 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 354 T - 1019158 T^{2} + 93974256 T^{3} + 724495550511 T^{4} + 93974256 p^{3} T^{5} - 1019158 p^{6} T^{6} - 354 p^{9} T^{7} + p^{12} T^{8} \)
97$D_4\times C_2$ \( 1 + 295 T - 1419371 T^{2} - 94090250 T^{3} + 1356831536062 T^{4} - 94090250 p^{3} T^{5} - 1419371 p^{6} T^{6} + 295 p^{9} T^{7} + p^{12} T^{8} \)
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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

−10.30462483959036235718567432462, −9.762077532707027562417171349903, −9.696925455186275040483270709984, −9.292971741545171573341203516837, −9.126770942146425641719166417730, −8.870264834808462857150988511084, −8.638304158275480261434151347365, −8.082116552404205257723602974051, −7.78895165339512901033941703499, −7.39825209719044816553916138842, −7.02578721452460982498784918947, −6.67902907871338489951617428268, −6.39371584272660332180667344622, −5.94580782260654655986270336596, −5.81842431568962360124467433954, −5.27546334445385344810752987956, −5.14894891055479695614511221409, −4.86340029711497740620006013701, −4.01307554649754611476719942155, −3.66260075091975604339148478908, −2.78222169229712279891917223412, −2.10594267477447696174069600541, −2.03325622244958561321049568458, −1.16780886813513214509614903904, −0.25194571050895366014290318185, 0.25194571050895366014290318185, 1.16780886813513214509614903904, 2.03325622244958561321049568458, 2.10594267477447696174069600541, 2.78222169229712279891917223412, 3.66260075091975604339148478908, 4.01307554649754611476719942155, 4.86340029711497740620006013701, 5.14894891055479695614511221409, 5.27546334445385344810752987956, 5.81842431568962360124467433954, 5.94580782260654655986270336596, 6.39371584272660332180667344622, 6.67902907871338489951617428268, 7.02578721452460982498784918947, 7.39825209719044816553916138842, 7.78895165339512901033941703499, 8.082116552404205257723602974051, 8.638304158275480261434151347365, 8.870264834808462857150988511084, 9.126770942146425641719166417730, 9.292971741545171573341203516837, 9.696925455186275040483270709984, 9.762077532707027562417171349903, 10.30462483959036235718567432462

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