Properties

Label 10-1-1.1-c63e5-0-0
Degree $10$
Conductor $1$
Sign $1$
Analytic cond. $1.00343\times 10^{7}$
Root an. cond. $5.01359$
Motivic weight $63$
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  + 5.07e8·2-s + 9.53e14·3-s − 1.95e19·4-s − 5.01e20·5-s + 4.83e23·6-s + 3.76e26·7-s − 9.06e27·8-s − 2.08e30·9-s − 2.54e29·10-s − 5.40e32·11-s − 1.86e34·12-s + 1.08e35·13-s + 1.91e35·14-s − 4.77e35·15-s + 1.72e38·16-s + 2.33e38·17-s − 1.05e39·18-s − 7.86e39·19-s + 9.79e39·20-s + 3.59e41·21-s − 2.74e41·22-s + 1.57e43·23-s − 8.63e42·24-s − 1.24e44·25-s + 5.48e43·26-s − 6.44e44·27-s − 7.36e45·28-s + ⋯
L(s)  = 1  + 0.167·2-s + 0.891·3-s − 2.11·4-s − 0.0481·5-s + 0.148·6-s + 0.902·7-s − 0.323·8-s − 1.82·9-s − 0.00804·10-s − 0.848·11-s − 1.88·12-s + 0.880·13-s + 0.150·14-s − 0.0428·15-s + 2.02·16-s + 0.406·17-s − 0.304·18-s − 0.412·19-s + 0.101·20-s + 0.804·21-s − 0.141·22-s + 2.00·23-s − 0.288·24-s − 1.15·25-s + 0.147·26-s − 0.526·27-s − 1.91·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(64-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+63/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(1.00343\times 10^{7}\)
Root analytic conductor: \(5.01359\)
Motivic weight: \(63\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 1,\ (\ :63/2, 63/2, 63/2, 63/2, 63/2),\ 1)\)

Particular Values

\(L(32)\) \(\approx\) \(8.712291760\)
\(L(\frac12)\) \(\approx\) \(8.712291760\)
\(L(\frac{65}{2})\) 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)$
good2$C_2 \wr S_5$ \( 1 - 63414387 p^{3} T + 19336575150992617 p^{10} T^{2} - \)\(12\!\cdots\!25\)\( p^{23} T^{3} + \)\(78\!\cdots\!87\)\( p^{38} T^{4} - \)\(15\!\cdots\!91\)\( p^{55} T^{5} + \)\(78\!\cdots\!87\)\( p^{101} T^{6} - \)\(12\!\cdots\!25\)\( p^{149} T^{7} + 19336575150992617 p^{199} T^{8} - 63414387 p^{255} T^{9} + p^{315} T^{10} \)
3$C_2 \wr S_5$ \( 1 - 3922820374964 p^{5} T + \)\(45\!\cdots\!07\)\( p^{8} T^{2} - \)\(32\!\cdots\!00\)\( p^{17} T^{3} + \)\(24\!\cdots\!42\)\( p^{30} T^{4} - \)\(20\!\cdots\!08\)\( p^{43} T^{5} + \)\(24\!\cdots\!42\)\( p^{93} T^{6} - \)\(32\!\cdots\!00\)\( p^{143} T^{7} + \)\(45\!\cdots\!07\)\( p^{197} T^{8} - 3922820374964 p^{257} T^{9} + p^{315} T^{10} \)
5$C_2 \wr S_5$ \( 1 + \)\(10\!\cdots\!86\)\( p T + \)\(20\!\cdots\!49\)\( p^{4} T^{2} - \)\(14\!\cdots\!72\)\( p^{9} T^{3} + \)\(27\!\cdots\!94\)\( p^{17} T^{4} + \)\(75\!\cdots\!76\)\( p^{27} T^{5} + \)\(27\!\cdots\!94\)\( p^{80} T^{6} - \)\(14\!\cdots\!72\)\( p^{135} T^{7} + \)\(20\!\cdots\!49\)\( p^{193} T^{8} + \)\(10\!\cdots\!86\)\( p^{253} T^{9} + p^{315} T^{10} \)
7$C_2 \wr S_5$ \( 1 - \)\(53\!\cdots\!08\)\( p T + \)\(20\!\cdots\!49\)\( p^{5} T^{2} - \)\(11\!\cdots\!00\)\( p^{10} T^{3} + \)\(75\!\cdots\!98\)\( p^{16} T^{4} + \)\(30\!\cdots\!84\)\( p^{23} T^{5} + \)\(75\!\cdots\!98\)\( p^{79} T^{6} - \)\(11\!\cdots\!00\)\( p^{136} T^{7} + \)\(20\!\cdots\!49\)\( p^{194} T^{8} - \)\(53\!\cdots\!08\)\( p^{253} T^{9} + p^{315} T^{10} \)
11$C_2 \wr S_5$ \( 1 + \)\(49\!\cdots\!40\)\( p T + \)\(10\!\cdots\!95\)\( p^{2} T^{2} + \)\(26\!\cdots\!80\)\( p^{5} T^{3} + \)\(29\!\cdots\!10\)\( p^{9} T^{4} + \)\(47\!\cdots\!28\)\( p^{14} T^{5} + \)\(29\!\cdots\!10\)\( p^{72} T^{6} + \)\(26\!\cdots\!80\)\( p^{131} T^{7} + \)\(10\!\cdots\!95\)\( p^{191} T^{8} + \)\(49\!\cdots\!40\)\( p^{253} T^{9} + p^{315} T^{10} \)
13$C_2 \wr S_5$ \( 1 - \)\(83\!\cdots\!74\)\( p T + \)\(15\!\cdots\!13\)\( p^{2} T^{2} - \)\(99\!\cdots\!00\)\( p^{3} T^{3} + \)\(12\!\cdots\!02\)\( p^{6} T^{4} - \)\(45\!\cdots\!84\)\( p^{10} T^{5} + \)\(12\!\cdots\!02\)\( p^{69} T^{6} - \)\(99\!\cdots\!00\)\( p^{129} T^{7} + \)\(15\!\cdots\!13\)\( p^{191} T^{8} - \)\(83\!\cdots\!74\)\( p^{253} T^{9} + p^{315} T^{10} \)
17$C_2 \wr S_5$ \( 1 - \)\(23\!\cdots\!26\)\( T + \)\(48\!\cdots\!89\)\( p T^{2} - \)\(59\!\cdots\!00\)\( p^{3} T^{3} + \)\(32\!\cdots\!34\)\( p^{5} T^{4} - \)\(15\!\cdots\!68\)\( p^{8} T^{5} + \)\(32\!\cdots\!34\)\( p^{68} T^{6} - \)\(59\!\cdots\!00\)\( p^{129} T^{7} + \)\(48\!\cdots\!89\)\( p^{190} T^{8} - \)\(23\!\cdots\!26\)\( p^{252} T^{9} + p^{315} T^{10} \)
19$C_2 \wr S_5$ \( 1 + \)\(41\!\cdots\!00\)\( p T + \)\(40\!\cdots\!95\)\( p^{2} T^{2} + \)\(86\!\cdots\!00\)\( p^{4} T^{3} + \)\(10\!\cdots\!90\)\( p^{7} T^{4} + \)\(10\!\cdots\!00\)\( p^{10} T^{5} + \)\(10\!\cdots\!90\)\( p^{70} T^{6} + \)\(86\!\cdots\!00\)\( p^{130} T^{7} + \)\(40\!\cdots\!95\)\( p^{191} T^{8} + \)\(41\!\cdots\!00\)\( p^{253} T^{9} + p^{315} T^{10} \)
23$C_2 \wr S_5$ \( 1 - \)\(15\!\cdots\!72\)\( T + \)\(11\!\cdots\!29\)\( p T^{2} - \)\(36\!\cdots\!00\)\( p^{2} T^{3} + \)\(66\!\cdots\!58\)\( p^{4} T^{4} - \)\(66\!\cdots\!44\)\( p^{6} T^{5} + \)\(66\!\cdots\!58\)\( p^{67} T^{6} - \)\(36\!\cdots\!00\)\( p^{128} T^{7} + \)\(11\!\cdots\!29\)\( p^{190} T^{8} - \)\(15\!\cdots\!72\)\( p^{252} T^{9} + p^{315} T^{10} \)
29$C_2 \wr S_5$ \( 1 - \)\(17\!\cdots\!50\)\( p T + \)\(19\!\cdots\!45\)\( p^{2} T^{2} - \)\(14\!\cdots\!00\)\( p^{3} T^{3} + \)\(29\!\cdots\!90\)\( p^{5} T^{4} - \)\(45\!\cdots\!00\)\( p^{7} T^{5} + \)\(29\!\cdots\!90\)\( p^{68} T^{6} - \)\(14\!\cdots\!00\)\( p^{129} T^{7} + \)\(19\!\cdots\!45\)\( p^{191} T^{8} - \)\(17\!\cdots\!50\)\( p^{253} T^{9} + p^{315} T^{10} \)
31$C_2 \wr S_5$ \( 1 - \)\(16\!\cdots\!60\)\( p^{2} T + \)\(38\!\cdots\!95\)\( p^{2} T^{2} - \)\(13\!\cdots\!20\)\( p^{3} T^{3} + \)\(65\!\cdots\!10\)\( p^{4} T^{4} - \)\(17\!\cdots\!52\)\( p^{5} T^{5} + \)\(65\!\cdots\!10\)\( p^{67} T^{6} - \)\(13\!\cdots\!20\)\( p^{129} T^{7} + \)\(38\!\cdots\!95\)\( p^{191} T^{8} - \)\(16\!\cdots\!60\)\( p^{254} T^{9} + p^{315} T^{10} \)
37$C_2 \wr S_5$ \( 1 + \)\(17\!\cdots\!34\)\( T + \)\(27\!\cdots\!53\)\( T^{2} + \)\(10\!\cdots\!00\)\( p T^{3} + \)\(23\!\cdots\!22\)\( p^{2} T^{4} + \)\(67\!\cdots\!04\)\( p^{3} T^{5} + \)\(23\!\cdots\!22\)\( p^{65} T^{6} + \)\(10\!\cdots\!00\)\( p^{127} T^{7} + \)\(27\!\cdots\!53\)\( p^{189} T^{8} + \)\(17\!\cdots\!34\)\( p^{252} T^{9} + p^{315} T^{10} \)
41$C_2 \wr S_5$ \( 1 - \)\(15\!\cdots\!10\)\( T + \)\(26\!\cdots\!45\)\( T^{2} - \)\(61\!\cdots\!20\)\( p T^{3} + \)\(14\!\cdots\!10\)\( p^{2} T^{4} - \)\(21\!\cdots\!12\)\( p^{3} T^{5} + \)\(14\!\cdots\!10\)\( p^{65} T^{6} - \)\(61\!\cdots\!20\)\( p^{127} T^{7} + \)\(26\!\cdots\!45\)\( p^{189} T^{8} - \)\(15\!\cdots\!10\)\( p^{252} T^{9} + p^{315} T^{10} \)
43$C_2 \wr S_5$ \( 1 + \)\(29\!\cdots\!08\)\( T + \)\(83\!\cdots\!49\)\( p T^{2} + \)\(47\!\cdots\!00\)\( p^{2} T^{3} + \)\(70\!\cdots\!14\)\( p^{3} T^{4} + \)\(30\!\cdots\!84\)\( p^{4} T^{5} + \)\(70\!\cdots\!14\)\( p^{66} T^{6} + \)\(47\!\cdots\!00\)\( p^{128} T^{7} + \)\(83\!\cdots\!49\)\( p^{190} T^{8} + \)\(29\!\cdots\!08\)\( p^{252} T^{9} + p^{315} T^{10} \)
47$C_2 \wr S_5$ \( 1 + \)\(48\!\cdots\!64\)\( T + \)\(23\!\cdots\!09\)\( p T^{2} + \)\(18\!\cdots\!00\)\( p^{2} T^{3} + \)\(47\!\cdots\!46\)\( p^{3} T^{4} + \)\(27\!\cdots\!52\)\( p^{4} T^{5} + \)\(47\!\cdots\!46\)\( p^{66} T^{6} + \)\(18\!\cdots\!00\)\( p^{128} T^{7} + \)\(23\!\cdots\!09\)\( p^{190} T^{8} + \)\(48\!\cdots\!64\)\( p^{252} T^{9} + p^{315} T^{10} \)
53$C_2 \wr S_5$ \( 1 + \)\(69\!\cdots\!98\)\( T + \)\(24\!\cdots\!09\)\( p T^{2} + \)\(88\!\cdots\!00\)\( p^{2} T^{3} + \)\(48\!\cdots\!54\)\( p^{3} T^{4} - \)\(49\!\cdots\!36\)\( p^{4} T^{5} + \)\(48\!\cdots\!54\)\( p^{66} T^{6} + \)\(88\!\cdots\!00\)\( p^{128} T^{7} + \)\(24\!\cdots\!09\)\( p^{190} T^{8} + \)\(69\!\cdots\!98\)\( p^{252} T^{9} + p^{315} T^{10} \)
59$C_2 \wr S_5$ \( 1 - \)\(17\!\cdots\!00\)\( p T + \)\(51\!\cdots\!95\)\( p^{2} T^{2} - \)\(63\!\cdots\!00\)\( p^{3} T^{3} + \)\(10\!\cdots\!10\)\( p^{4} T^{4} - \)\(94\!\cdots\!00\)\( p^{5} T^{5} + \)\(10\!\cdots\!10\)\( p^{67} T^{6} - \)\(63\!\cdots\!00\)\( p^{129} T^{7} + \)\(51\!\cdots\!95\)\( p^{191} T^{8} - \)\(17\!\cdots\!00\)\( p^{253} T^{9} + p^{315} T^{10} \)
61$C_2 \wr S_5$ \( 1 - \)\(64\!\cdots\!10\)\( p T + \)\(26\!\cdots\!45\)\( p^{2} T^{2} - \)\(57\!\cdots\!20\)\( p^{3} T^{3} + \)\(12\!\cdots\!10\)\( p^{4} T^{4} - \)\(23\!\cdots\!52\)\( p^{5} T^{5} + \)\(12\!\cdots\!10\)\( p^{67} T^{6} - \)\(57\!\cdots\!20\)\( p^{129} T^{7} + \)\(26\!\cdots\!45\)\( p^{191} T^{8} - \)\(64\!\cdots\!10\)\( p^{253} T^{9} + p^{315} T^{10} \)
67$C_2 \wr S_5$ \( 1 + \)\(47\!\cdots\!24\)\( T + \)\(56\!\cdots\!63\)\( T^{2} + \)\(19\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!38\)\( T^{4} + \)\(31\!\cdots\!12\)\( T^{5} + \)\(12\!\cdots\!38\)\( p^{63} T^{6} + \)\(19\!\cdots\!00\)\( p^{126} T^{7} + \)\(56\!\cdots\!63\)\( p^{189} T^{8} + \)\(47\!\cdots\!24\)\( p^{252} T^{9} + p^{315} T^{10} \)
71$C_2 \wr S_5$ \( 1 - \)\(50\!\cdots\!60\)\( T + \)\(23\!\cdots\!95\)\( T^{2} - \)\(62\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!10\)\( T^{4} - \)\(32\!\cdots\!52\)\( T^{5} + \)\(16\!\cdots\!10\)\( p^{63} T^{6} - \)\(62\!\cdots\!20\)\( p^{126} T^{7} + \)\(23\!\cdots\!95\)\( p^{189} T^{8} - \)\(50\!\cdots\!60\)\( p^{252} T^{9} + p^{315} T^{10} \)
73$C_2 \wr S_5$ \( 1 + \)\(29\!\cdots\!78\)\( T + \)\(69\!\cdots\!17\)\( T^{2} + \)\(36\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!78\)\( T^{4} + \)\(13\!\cdots\!84\)\( T^{5} + \)\(23\!\cdots\!78\)\( p^{63} T^{6} + \)\(36\!\cdots\!00\)\( p^{126} T^{7} + \)\(69\!\cdots\!17\)\( p^{189} T^{8} + \)\(29\!\cdots\!78\)\( p^{252} T^{9} + p^{315} T^{10} \)
79$C_2 \wr S_5$ \( 1 - \)\(58\!\cdots\!00\)\( T + \)\(13\!\cdots\!95\)\( T^{2} - \)\(81\!\cdots\!00\)\( T^{3} + \)\(81\!\cdots\!10\)\( T^{4} - \)\(43\!\cdots\!00\)\( T^{5} + \)\(81\!\cdots\!10\)\( p^{63} T^{6} - \)\(81\!\cdots\!00\)\( p^{126} T^{7} + \)\(13\!\cdots\!95\)\( p^{189} T^{8} - \)\(58\!\cdots\!00\)\( p^{252} T^{9} + p^{315} T^{10} \)
83$C_2 \wr S_5$ \( 1 - \)\(27\!\cdots\!32\)\( T + \)\(25\!\cdots\!87\)\( T^{2} - \)\(58\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!38\)\( T^{4} - \)\(58\!\cdots\!16\)\( T^{5} + \)\(33\!\cdots\!38\)\( p^{63} T^{6} - \)\(58\!\cdots\!00\)\( p^{126} T^{7} + \)\(25\!\cdots\!87\)\( p^{189} T^{8} - \)\(27\!\cdots\!32\)\( p^{252} T^{9} + p^{315} T^{10} \)
89$C_2 \wr S_5$ \( 1 - \)\(32\!\cdots\!50\)\( T + \)\(78\!\cdots\!45\)\( T^{2} - \)\(14\!\cdots\!00\)\( T^{3} + \)\(71\!\cdots\!10\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{5} + \)\(71\!\cdots\!10\)\( p^{63} T^{6} - \)\(14\!\cdots\!00\)\( p^{126} T^{7} + \)\(78\!\cdots\!45\)\( p^{189} T^{8} - \)\(32\!\cdots\!50\)\( p^{252} T^{9} + p^{315} T^{10} \)
97$C_2 \wr S_5$ \( 1 + \)\(17\!\cdots\!14\)\( T + \)\(52\!\cdots\!73\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!58\)\( T^{4} + \)\(29\!\cdots\!12\)\( T^{5} + \)\(12\!\cdots\!58\)\( p^{63} T^{6} + \)\(12\!\cdots\!00\)\( p^{126} T^{7} + \)\(52\!\cdots\!73\)\( p^{189} T^{8} + \)\(17\!\cdots\!14\)\( p^{252} T^{9} + p^{315} T^{10} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.267756777022281022914557827508, −8.572137877718424829530249086383, −8.529931552949539470805252488505, −8.528091325463947058665729713447, −8.521125339432963339889268140313, −7.71513151385096268331106059192, −7.54164017761432726094566132719, −6.47207300135663117431492117326, −6.36932168417662856012186571069, −6.11308787721379987450502308199, −5.19326363783780132194148380768, −5.07839043795559135938970624058, −4.89862718473805193834153393419, −4.64415932938404842791853805292, −4.18209228791014334376063149687, −3.65374290931023038947315699063, −3.13322097782405085288527911067, −3.01726640481319240814008038374, −2.80685379906930605441069033119, −2.21468931928119782116171883813, −1.90718296516908783667460242487, −1.07806871839496369775651965234, −0.64534129392468597418821265851, −0.62694026687984783057215458063, −0.60824199369184346752501954201, 0.60824199369184346752501954201, 0.62694026687984783057215458063, 0.64534129392468597418821265851, 1.07806871839496369775651965234, 1.90718296516908783667460242487, 2.21468931928119782116171883813, 2.80685379906930605441069033119, 3.01726640481319240814008038374, 3.13322097782405085288527911067, 3.65374290931023038947315699063, 4.18209228791014334376063149687, 4.64415932938404842791853805292, 4.89862718473805193834153393419, 5.07839043795559135938970624058, 5.19326363783780132194148380768, 6.11308787721379987450502308199, 6.36932168417662856012186571069, 6.47207300135663117431492117326, 7.54164017761432726094566132719, 7.71513151385096268331106059192, 8.521125339432963339889268140313, 8.528091325463947058665729713447, 8.529931552949539470805252488505, 8.572137877718424829530249086383, 9.267756777022281022914557827508

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