Properties

Degree 10
Conductor $ 1 $
Sign $1$
Motivic weight 59
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.49e8·2-s + 8.40e13·3-s − 4.70e17·4-s + 1.79e20·5-s − 3.77e22·6-s + 1.49e24·7-s + 2.13e26·8-s − 1.54e28·9-s − 8.09e28·10-s + 4.25e30·11-s − 3.95e31·12-s − 8.45e32·13-s − 6.73e32·14-s + 1.51e34·15-s + 4.33e34·16-s − 3.40e36·17-s + 6.95e36·18-s + 6.46e37·19-s − 8.46e37·20-s + 1.25e38·21-s − 1.91e39·22-s + 2.67e40·23-s + 1.79e40·24-s − 1.04e41·25-s + 3.80e41·26-s − 4.87e41·27-s − 7.05e41·28-s + ⋯
L(s)  = 1  − 0.592·2-s + 0.706·3-s − 0.816·4-s + 0.432·5-s − 0.418·6-s + 0.175·7-s + 0.487·8-s − 1.09·9-s − 0.255·10-s + 0.809·11-s − 0.577·12-s − 1.16·13-s − 0.104·14-s + 0.305·15-s + 0.130·16-s − 1.71·17-s + 0.648·18-s + 1.22·19-s − 0.352·20-s + 0.124·21-s − 0.479·22-s + 1.80·23-s + 0.344·24-s − 0.601·25-s + 0.689·26-s − 0.290·27-s − 0.143·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{5} \, L(s)\cr =\mathstrut & \,\Lambda(60-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+59/2)^{5} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(10\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(59\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(10,\ 1,\ (\ :59/2, 59/2, 59/2, 59/2, 59/2),\ 1)$
$L(30)$  $\approx$  $0.247569$
$L(\frac12)$  $\approx$  $0.247569$
$L(\frac{61}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, \(F_p\) is a polynomial of degree 10.
$p$$\Gal(F_p)$$F_p$
good2$C_2 \wr S_5$ \( 1 + 56211483 p^{3} T + 328541785870451 p^{11} T^{2} + 35871121944327534405 p^{23} T^{3} + \)\(22\!\cdots\!19\)\( p^{37} T^{4} + \)\(43\!\cdots\!37\)\( p^{56} T^{5} + \)\(22\!\cdots\!19\)\( p^{96} T^{6} + 35871121944327534405 p^{141} T^{7} + 328541785870451 p^{188} T^{8} + 56211483 p^{239} T^{9} + p^{295} T^{10} \)
3$C_2 \wr S_5$ \( 1 - 9335181305548 p^{2} T + \)\(10\!\cdots\!01\)\( p^{7} T^{2} - \)\(62\!\cdots\!40\)\( p^{16} T^{3} + \)\(67\!\cdots\!14\)\( p^{27} T^{4} - \)\(31\!\cdots\!96\)\( p^{40} T^{5} + \)\(67\!\cdots\!14\)\( p^{86} T^{6} - \)\(62\!\cdots\!40\)\( p^{134} T^{7} + \)\(10\!\cdots\!01\)\( p^{184} T^{8} - 9335181305548 p^{238} T^{9} + p^{295} T^{10} \)
5$C_2 \wr S_5$ \( 1 - 35992322430017211678 p T + \)\(43\!\cdots\!37\)\( p^{5} T^{2} + \)\(34\!\cdots\!52\)\( p^{10} T^{3} + \)\(32\!\cdots\!66\)\( p^{18} T^{4} + \)\(11\!\cdots\!84\)\( p^{27} T^{5} + \)\(32\!\cdots\!66\)\( p^{77} T^{6} + \)\(34\!\cdots\!52\)\( p^{128} T^{7} + \)\(43\!\cdots\!37\)\( p^{182} T^{8} - 35992322430017211678 p^{237} T^{9} + p^{295} T^{10} \)
7$C_2 \wr S_5$ \( 1 - \)\(21\!\cdots\!08\)\( p T + \)\(11\!\cdots\!43\)\( p^{4} T^{2} - \)\(10\!\cdots\!00\)\( p^{8} T^{3} + \)\(48\!\cdots\!02\)\( p^{14} T^{4} - \)\(12\!\cdots\!84\)\( p^{21} T^{5} + \)\(48\!\cdots\!02\)\( p^{73} T^{6} - \)\(10\!\cdots\!00\)\( p^{126} T^{7} + \)\(11\!\cdots\!43\)\( p^{181} T^{8} - \)\(21\!\cdots\!08\)\( p^{237} T^{9} + p^{295} T^{10} \)
11$C_2 \wr S_5$ \( 1 - \)\(38\!\cdots\!60\)\( p T + \)\(66\!\cdots\!45\)\( p^{3} T^{2} - \)\(20\!\cdots\!20\)\( p^{6} T^{3} + \)\(15\!\cdots\!10\)\( p^{10} T^{4} - \)\(35\!\cdots\!72\)\( p^{14} T^{5} + \)\(15\!\cdots\!10\)\( p^{69} T^{6} - \)\(20\!\cdots\!20\)\( p^{124} T^{7} + \)\(66\!\cdots\!45\)\( p^{180} T^{8} - \)\(38\!\cdots\!60\)\( p^{237} T^{9} + p^{295} T^{10} \)
13$C_2 \wr S_5$ \( 1 + \)\(84\!\cdots\!78\)\( T + \)\(13\!\cdots\!09\)\( p T^{2} + \)\(37\!\cdots\!60\)\( p^{3} T^{3} + \)\(25\!\cdots\!82\)\( p^{6} T^{4} + \)\(32\!\cdots\!76\)\( p^{10} T^{5} + \)\(25\!\cdots\!82\)\( p^{65} T^{6} + \)\(37\!\cdots\!60\)\( p^{121} T^{7} + \)\(13\!\cdots\!09\)\( p^{178} T^{8} + \)\(84\!\cdots\!78\)\( p^{236} T^{9} + p^{295} T^{10} \)
17$C_2 \wr S_5$ \( 1 + \)\(34\!\cdots\!54\)\( T + \)\(90\!\cdots\!49\)\( p T^{2} + \)\(66\!\cdots\!40\)\( p^{3} T^{3} + \)\(38\!\cdots\!82\)\( p^{6} T^{4} + \)\(12\!\cdots\!76\)\( p^{9} T^{5} + \)\(38\!\cdots\!82\)\( p^{65} T^{6} + \)\(66\!\cdots\!40\)\( p^{121} T^{7} + \)\(90\!\cdots\!49\)\( p^{178} T^{8} + \)\(34\!\cdots\!54\)\( p^{236} T^{9} + p^{295} T^{10} \)
19$C_2 \wr S_5$ \( 1 - \)\(34\!\cdots\!00\)\( p T + \)\(36\!\cdots\!95\)\( p^{2} T^{2} - \)\(47\!\cdots\!00\)\( p^{4} T^{3} + \)\(14\!\cdots\!10\)\( p^{6} T^{4} - \)\(14\!\cdots\!00\)\( p^{8} T^{5} + \)\(14\!\cdots\!10\)\( p^{65} T^{6} - \)\(47\!\cdots\!00\)\( p^{122} T^{7} + \)\(36\!\cdots\!95\)\( p^{179} T^{8} - \)\(34\!\cdots\!00\)\( p^{237} T^{9} + p^{295} T^{10} \)
23$C_2 \wr S_5$ \( 1 - \)\(26\!\cdots\!12\)\( T + \)\(29\!\cdots\!89\)\( p T^{2} - \)\(11\!\cdots\!60\)\( p^{3} T^{3} + \)\(40\!\cdots\!06\)\( p^{5} T^{4} - \)\(10\!\cdots\!48\)\( p^{7} T^{5} + \)\(40\!\cdots\!06\)\( p^{64} T^{6} - \)\(11\!\cdots\!60\)\( p^{121} T^{7} + \)\(29\!\cdots\!89\)\( p^{178} T^{8} - \)\(26\!\cdots\!12\)\( p^{236} T^{9} + p^{295} T^{10} \)
29$C_2 \wr S_5$ \( 1 + \)\(60\!\cdots\!50\)\( p T + \)\(95\!\cdots\!45\)\( p^{2} T^{2} + \)\(39\!\cdots\!00\)\( p^{3} T^{3} + \)\(37\!\cdots\!10\)\( p^{4} T^{4} + \)\(11\!\cdots\!00\)\( p^{5} T^{5} + \)\(37\!\cdots\!10\)\( p^{63} T^{6} + \)\(39\!\cdots\!00\)\( p^{121} T^{7} + \)\(95\!\cdots\!45\)\( p^{179} T^{8} + \)\(60\!\cdots\!50\)\( p^{237} T^{9} + p^{295} T^{10} \)
31$C_2 \wr S_5$ \( 1 + \)\(35\!\cdots\!40\)\( T + \)\(91\!\cdots\!95\)\( T^{2} + \)\(50\!\cdots\!80\)\( p T^{3} + \)\(23\!\cdots\!10\)\( p^{2} T^{4} + \)\(80\!\cdots\!28\)\( p^{3} T^{5} + \)\(23\!\cdots\!10\)\( p^{61} T^{6} + \)\(50\!\cdots\!80\)\( p^{119} T^{7} + \)\(91\!\cdots\!95\)\( p^{177} T^{8} + \)\(35\!\cdots\!40\)\( p^{236} T^{9} + p^{295} T^{10} \)
37$C_2 \wr S_5$ \( 1 - \)\(12\!\cdots\!26\)\( T + \)\(96\!\cdots\!13\)\( T^{2} - \)\(35\!\cdots\!20\)\( p T^{3} + \)\(36\!\cdots\!62\)\( p^{2} T^{4} - \)\(11\!\cdots\!36\)\( p^{3} T^{5} + \)\(36\!\cdots\!62\)\( p^{61} T^{6} - \)\(35\!\cdots\!20\)\( p^{119} T^{7} + \)\(96\!\cdots\!13\)\( p^{177} T^{8} - \)\(12\!\cdots\!26\)\( p^{236} T^{9} + p^{295} T^{10} \)
41$C_2 \wr S_5$ \( 1 + \)\(12\!\cdots\!90\)\( T + \)\(24\!\cdots\!45\)\( p T^{2} + \)\(37\!\cdots\!80\)\( p^{2} T^{3} + \)\(45\!\cdots\!10\)\( p^{3} T^{4} + \)\(44\!\cdots\!68\)\( p^{4} T^{5} + \)\(45\!\cdots\!10\)\( p^{62} T^{6} + \)\(37\!\cdots\!80\)\( p^{120} T^{7} + \)\(24\!\cdots\!45\)\( p^{178} T^{8} + \)\(12\!\cdots\!90\)\( p^{236} T^{9} + p^{295} T^{10} \)
43$C_2 \wr S_5$ \( 1 - \)\(13\!\cdots\!92\)\( T + \)\(19\!\cdots\!49\)\( p T^{2} - \)\(41\!\cdots\!00\)\( p^{2} T^{3} + \)\(41\!\cdots\!14\)\( p^{3} T^{4} - \)\(67\!\cdots\!16\)\( p^{4} T^{5} + \)\(41\!\cdots\!14\)\( p^{62} T^{6} - \)\(41\!\cdots\!00\)\( p^{120} T^{7} + \)\(19\!\cdots\!49\)\( p^{178} T^{8} - \)\(13\!\cdots\!92\)\( p^{236} T^{9} + p^{295} T^{10} \)
47$C_2 \wr S_5$ \( 1 - \)\(12\!\cdots\!28\)\( p T + \)\(13\!\cdots\!67\)\( p^{2} T^{2} - \)\(89\!\cdots\!40\)\( p^{3} T^{3} + \)\(55\!\cdots\!98\)\( p^{4} T^{4} - \)\(25\!\cdots\!64\)\( p^{5} T^{5} + \)\(55\!\cdots\!98\)\( p^{63} T^{6} - \)\(89\!\cdots\!40\)\( p^{121} T^{7} + \)\(13\!\cdots\!67\)\( p^{179} T^{8} - \)\(12\!\cdots\!28\)\( p^{237} T^{9} + p^{295} T^{10} \)
53$C_2 \wr S_5$ \( 1 - \)\(42\!\cdots\!94\)\( p T + \)\(14\!\cdots\!93\)\( p^{2} T^{2} - \)\(32\!\cdots\!20\)\( p^{3} T^{3} + \)\(61\!\cdots\!78\)\( p^{4} T^{4} - \)\(91\!\cdots\!72\)\( p^{5} T^{5} + \)\(61\!\cdots\!78\)\( p^{63} T^{6} - \)\(32\!\cdots\!20\)\( p^{121} T^{7} + \)\(14\!\cdots\!93\)\( p^{179} T^{8} - \)\(42\!\cdots\!94\)\( p^{237} T^{9} + p^{295} T^{10} \)
59$C_2 \wr S_5$ \( 1 + \)\(21\!\cdots\!00\)\( T + \)\(79\!\cdots\!95\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(32\!\cdots\!10\)\( T^{4} + \)\(40\!\cdots\!00\)\( T^{5} + \)\(32\!\cdots\!10\)\( p^{59} T^{6} + \)\(13\!\cdots\!00\)\( p^{118} T^{7} + \)\(79\!\cdots\!95\)\( p^{177} T^{8} + \)\(21\!\cdots\!00\)\( p^{236} T^{9} + p^{295} T^{10} \)
61$C_2 \wr S_5$ \( 1 + \)\(55\!\cdots\!90\)\( T + \)\(84\!\cdots\!45\)\( T^{2} + \)\(33\!\cdots\!80\)\( T^{3} + \)\(29\!\cdots\!10\)\( T^{4} + \)\(92\!\cdots\!48\)\( T^{5} + \)\(29\!\cdots\!10\)\( p^{59} T^{6} + \)\(33\!\cdots\!80\)\( p^{118} T^{7} + \)\(84\!\cdots\!45\)\( p^{177} T^{8} + \)\(55\!\cdots\!90\)\( p^{236} T^{9} + p^{295} T^{10} \)
67$C_2 \wr S_5$ \( 1 - \)\(25\!\cdots\!96\)\( T + \)\(20\!\cdots\!83\)\( T^{2} - \)\(32\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!58\)\( T^{4} - \)\(22\!\cdots\!28\)\( T^{5} + \)\(19\!\cdots\!58\)\( p^{59} T^{6} - \)\(32\!\cdots\!80\)\( p^{118} T^{7} + \)\(20\!\cdots\!83\)\( p^{177} T^{8} - \)\(25\!\cdots\!96\)\( p^{236} T^{9} + p^{295} T^{10} \)
71$C_2 \wr S_5$ \( 1 + \)\(63\!\cdots\!40\)\( T + \)\(82\!\cdots\!95\)\( T^{2} + \)\(37\!\cdots\!80\)\( T^{3} + \)\(26\!\cdots\!10\)\( T^{4} + \)\(89\!\cdots\!48\)\( T^{5} + \)\(26\!\cdots\!10\)\( p^{59} T^{6} + \)\(37\!\cdots\!80\)\( p^{118} T^{7} + \)\(82\!\cdots\!95\)\( p^{177} T^{8} + \)\(63\!\cdots\!40\)\( p^{236} T^{9} + p^{295} T^{10} \)
73$C_2 \wr S_5$ \( 1 - \)\(12\!\cdots\!62\)\( T + \)\(31\!\cdots\!97\)\( T^{2} - \)\(37\!\cdots\!20\)\( T^{3} + \)\(48\!\cdots\!58\)\( T^{4} - \)\(45\!\cdots\!56\)\( T^{5} + \)\(48\!\cdots\!58\)\( p^{59} T^{6} - \)\(37\!\cdots\!20\)\( p^{118} T^{7} + \)\(31\!\cdots\!97\)\( p^{177} T^{8} - \)\(12\!\cdots\!62\)\( p^{236} T^{9} + p^{295} T^{10} \)
79$C_2 \wr S_5$ \( 1 + \)\(19\!\cdots\!00\)\( T + \)\(33\!\cdots\!95\)\( T^{2} + \)\(34\!\cdots\!00\)\( T^{3} + \)\(36\!\cdots\!10\)\( T^{4} + \)\(30\!\cdots\!00\)\( T^{5} + \)\(36\!\cdots\!10\)\( p^{59} T^{6} + \)\(34\!\cdots\!00\)\( p^{118} T^{7} + \)\(33\!\cdots\!95\)\( p^{177} T^{8} + \)\(19\!\cdots\!00\)\( p^{236} T^{9} + p^{295} T^{10} \)
83$C_2 \wr S_5$ \( 1 - \)\(13\!\cdots\!52\)\( T + \)\(14\!\cdots\!27\)\( T^{2} - \)\(10\!\cdots\!60\)\( T^{3} + \)\(59\!\cdots\!78\)\( T^{4} - \)\(26\!\cdots\!36\)\( T^{5} + \)\(59\!\cdots\!78\)\( p^{59} T^{6} - \)\(10\!\cdots\!60\)\( p^{118} T^{7} + \)\(14\!\cdots\!27\)\( p^{177} T^{8} - \)\(13\!\cdots\!52\)\( p^{236} T^{9} + p^{295} T^{10} \)
89$C_2 \wr S_5$ \( 1 - \)\(41\!\cdots\!50\)\( T + \)\(38\!\cdots\!45\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(63\!\cdots\!10\)\( T^{4} - \)\(13\!\cdots\!00\)\( T^{5} + \)\(63\!\cdots\!10\)\( p^{59} T^{6} - \)\(10\!\cdots\!00\)\( p^{118} T^{7} + \)\(38\!\cdots\!45\)\( p^{177} T^{8} - \)\(41\!\cdots\!50\)\( p^{236} T^{9} + p^{295} T^{10} \)
97$C_2 \wr S_5$ \( 1 - \)\(11\!\cdots\!66\)\( T + \)\(92\!\cdots\!53\)\( T^{2} - \)\(60\!\cdots\!20\)\( T^{3} + \)\(32\!\cdots\!38\)\( T^{4} - \)\(13\!\cdots\!48\)\( T^{5} + \)\(32\!\cdots\!38\)\( p^{59} T^{6} - \)\(60\!\cdots\!20\)\( p^{118} T^{7} + \)\(92\!\cdots\!53\)\( p^{177} T^{8} - \)\(11\!\cdots\!66\)\( p^{236} T^{9} + p^{295} T^{10} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.423065804145231908188850616699, −9.245452262231743604694207948846, −9.060582200406625685943185371625, −8.802134135045037918398201960027, −8.698606407186667715640325704438, −7.80551940878668052863160991095, −7.53689042800403740235672257470, −7.14358773262112345305516870612, −6.93945325990773163531099102392, −6.19121117951191906941621640811, −5.85938177333788607586037303447, −5.32663091084914999311716507641, −5.03166302061618668622875081168, −4.82438506927423381864202585868, −4.26781622403418384419977402297, −3.55585079515958630395106621156, −3.42616382507351596843680429771, −3.25519075822932993127530371829, −2.43010475932392204575042668200, −2.11862684631809459824583932265, −2.02539191838660150640447250153, −1.39111825076257501596238050121, −1.03375234070354278860348613847, −0.38173724720063210159473700199, −0.11548805786913623004689854185, 0.11548805786913623004689854185, 0.38173724720063210159473700199, 1.03375234070354278860348613847, 1.39111825076257501596238050121, 2.02539191838660150640447250153, 2.11862684631809459824583932265, 2.43010475932392204575042668200, 3.25519075822932993127530371829, 3.42616382507351596843680429771, 3.55585079515958630395106621156, 4.26781622403418384419977402297, 4.82438506927423381864202585868, 5.03166302061618668622875081168, 5.32663091084914999311716507641, 5.85938177333788607586037303447, 6.19121117951191906941621640811, 6.93945325990773163531099102392, 7.14358773262112345305516870612, 7.53689042800403740235672257470, 7.80551940878668052863160991095, 8.698606407186667715640325704438, 8.802134135045037918398201960027, 9.060582200406625685943185371625, 9.245452262231743604694207948846, 9.423065804145231908188850616699

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