Properties

Label 8-980e4-1.1-c3e4-0-0
Degree $8$
Conductor $922368160000$
Sign $1$
Analytic cond. $1.11781\times 10^{7}$
Root an. cond. $7.60406$
Motivic weight $3$
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  − 3-s + 10·5-s − 8·9-s + 47·11-s + 218·13-s − 10·15-s − 121·17-s + 156·19-s + 152·23-s + 25·25-s + 79·27-s − 30·29-s + 142·31-s − 47·33-s − 46·37-s − 218·39-s − 620·41-s − 1.89e3·43-s − 80·45-s − 131·47-s + 121·51-s − 344·53-s + 470·55-s − 156·57-s + 976·59-s − 898·61-s + 2.18e3·65-s + ⋯
L(s)  = 1  − 0.192·3-s + 0.894·5-s − 0.296·9-s + 1.28·11-s + 4.65·13-s − 0.172·15-s − 1.72·17-s + 1.88·19-s + 1.37·23-s + 1/5·25-s + 0.563·27-s − 0.192·29-s + 0.822·31-s − 0.247·33-s − 0.204·37-s − 0.895·39-s − 2.36·41-s − 6.70·43-s − 0.265·45-s − 0.406·47-s + 0.332·51-s − 0.891·53-s + 1.15·55-s − 0.362·57-s + 2.15·59-s − 1.88·61-s + 4.15·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.11781\times 10^{7}\)
Root analytic conductor: \(7.60406\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.7897736087\)
\(L(\frac12)\) \(\approx\) \(0.7897736087\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
7 \( 1 \)
good3$D_4\times C_2$ \( 1 + T + p^{2} T^{2} - 62 T^{3} - 692 T^{4} - 62 p^{3} T^{5} + p^{8} T^{6} + p^{9} T^{7} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 - 47 T + 551 T^{2} + 47188 T^{3} - 1962776 T^{4} + 47188 p^{3} T^{5} + 551 p^{6} T^{6} - 47 p^{9} T^{7} + p^{12} T^{8} \)
13$D_{4}$ \( ( 1 - 109 T + 6804 T^{2} - 109 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 121 T + 2711 T^{2} + 254584 T^{3} + 46256098 T^{4} + 254584 p^{3} T^{5} + 2711 p^{6} T^{6} + 121 p^{9} T^{7} + p^{12} T^{8} \)
19$D_4\times C_2$ \( 1 - 156 T + 8518 T^{2} - 327600 T^{3} + 36242619 T^{4} - 327600 p^{3} T^{5} + 8518 p^{6} T^{6} - 156 p^{9} T^{7} + p^{12} T^{8} \)
23$D_4\times C_2$ \( 1 - 152 T + 8930 T^{2} + 1544320 T^{3} - 228239981 T^{4} + 1544320 p^{3} T^{5} + 8930 p^{6} T^{6} - 152 p^{9} T^{7} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 + 15 T + 45784 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 142 T - 32258 T^{2} + 1016720 T^{3} + 1259856679 T^{4} + 1016720 p^{3} T^{5} - 32258 p^{6} T^{6} - 142 p^{9} T^{7} + p^{12} T^{8} \)
37$D_4\times C_2$ \( 1 + 46 T + 10090 T^{2} - 5026880 T^{3} - 2609323481 T^{4} - 5026880 p^{3} T^{5} + 10090 p^{6} T^{6} + 46 p^{9} T^{7} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 + 310 T + 155642 T^{2} + 310 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 22 p T + 382494 T^{2} + 22 p^{4} T^{3} + p^{6} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 131 T - 194713 T^{2} + 553868 T^{3} + 32329670044 T^{4} + 553868 p^{3} T^{5} - 194713 p^{6} T^{6} + 131 p^{9} T^{7} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 + 344 T + 78842 T^{2} - 88841440 T^{3} - 38222093765 T^{4} - 88841440 p^{3} T^{5} + 78842 p^{6} T^{6} + 344 p^{9} T^{7} + p^{12} T^{8} \)
59$D_4\times C_2$ \( 1 - 976 T + 312638 T^{2} - 223679680 T^{3} + 171701003899 T^{4} - 223679680 p^{3} T^{5} + 312638 p^{6} T^{6} - 976 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 + 898 T + 153082 T^{2} + 179025280 T^{3} + 192270874999 T^{4} + 179025280 p^{3} T^{5} + 153082 p^{6} T^{6} + 898 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 + 478 T - 358202 T^{2} - 7093520 T^{3} + 185022547135 T^{4} - 7093520 p^{3} T^{5} - 358202 p^{6} T^{6} + 478 p^{9} T^{7} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 - 1548 T + 1313902 T^{2} - 1548 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 530 T - 411734 T^{2} - 45262000 T^{3} + 219132756367 T^{4} - 45262000 p^{3} T^{5} - 411734 p^{6} T^{6} + 530 p^{9} T^{7} + p^{12} T^{8} \)
79$D_4\times C_2$ \( 1 + 623 T - 579881 T^{2} - 11256364 T^{3} + 502593170548 T^{4} - 11256364 p^{3} T^{5} - 579881 p^{6} T^{6} + 623 p^{9} T^{7} + p^{12} T^{8} \)
83$D_{4}$ \( ( 1 - 782 T + 487454 T^{2} - 782 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 484 T - 946402 T^{2} - 110971520 T^{3} + 731828803939 T^{4} - 110971520 p^{3} T^{5} - 946402 p^{6} T^{6} + 484 p^{9} T^{7} + p^{12} T^{8} \)
97$D_{4}$ \( ( 1 - 2253 T + 3091298 T^{2} - 2253 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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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

−6.60102993459839568023217998466, −6.51543336716288751012784473069, −6.37576627470528349305796832463, −6.30784472106603548417667138339, −5.98225119135760928015904377966, −5.50951404734997423175330551239, −5.49399836696333481282743469078, −5.09146399990351810416467739326, −4.97927748371890738241457731179, −4.83487605476339334537454647799, −4.45520535104960603157072202437, −4.17864652195124815167993863355, −3.56990511457249709157693774239, −3.46615567858516853497622442546, −3.45891064146480308559394586398, −3.34602383044603506391217240606, −3.20321957481027244723540622412, −2.47418968965267622388717551529, −1.94497341762125189881825869318, −1.91339751462981652662745939577, −1.60503727770440613656520035354, −1.13065143916138507968460452105, −1.05865943487470265054394809659, −0.985199418179161826135610278844, −0.080855269333075841304288471809, 0.080855269333075841304288471809, 0.985199418179161826135610278844, 1.05865943487470265054394809659, 1.13065143916138507968460452105, 1.60503727770440613656520035354, 1.91339751462981652662745939577, 1.94497341762125189881825869318, 2.47418968965267622388717551529, 3.20321957481027244723540622412, 3.34602383044603506391217240606, 3.45891064146480308559394586398, 3.46615567858516853497622442546, 3.56990511457249709157693774239, 4.17864652195124815167993863355, 4.45520535104960603157072202437, 4.83487605476339334537454647799, 4.97927748371890738241457731179, 5.09146399990351810416467739326, 5.49399836696333481282743469078, 5.50951404734997423175330551239, 5.98225119135760928015904377966, 6.30784472106603548417667138339, 6.37576627470528349305796832463, 6.51543336716288751012784473069, 6.60102993459839568023217998466

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