Properties

Degree 10
Conductor $ 1 $
Sign $-1$
Motivic weight 73
Primitive no
Self-dual yes
Analytic rank 5

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.20e10·2-s − 1.29e17·3-s − 1.89e22·4-s + 2.30e25·5-s + 1.18e28·6-s − 4.35e30·7-s + 2.16e33·8-s − 1.44e35·9-s − 2.12e36·10-s + 5.00e37·11-s + 2.44e39·12-s + 4.75e39·13-s + 4.01e41·14-s − 2.98e42·15-s + 1.70e44·16-s + 6.63e44·17-s + 1.33e46·18-s + 3.13e46·19-s − 4.37e47·20-s + 5.62e47·21-s − 4.61e48·22-s − 4.11e49·23-s − 2.79e50·24-s − 2.21e51·25-s − 4.37e50·26-s + 1.81e52·27-s + 8.24e52·28-s + ⋯
L(s)  = 1  − 0.947·2-s − 0.496·3-s − 2.00·4-s + 0.709·5-s + 0.470·6-s − 0.620·7-s + 2.35·8-s − 2.13·9-s − 0.672·10-s + 0.488·11-s + 0.995·12-s + 0.104·13-s + 0.588·14-s − 0.352·15-s + 1.91·16-s + 0.813·17-s + 2.02·18-s + 0.664·19-s − 1.42·20-s + 0.308·21-s − 0.462·22-s − 0.815·23-s − 1.17·24-s − 2.09·25-s − 0.0987·26-s + 1.03·27-s + 1.24·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(74-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+73/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(10\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(73\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(5\)
Selberg data  =  \((10,\ 1,\ (\ :73/2, 73/2, 73/2, 73/2, 73/2),\ -1)\)
\(L(37)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{75}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where,\(F_p(T)\) is a polynomial of degree 10.
$p$$\Gal(F_p)$$F_p(T)$
good2$C_2 \wr S_5$ \( 1 + 5755583343 p^{4} T + 3345520000714170341 p^{13} T^{2} + \)\(62\!\cdots\!45\)\( p^{25} T^{3} + \)\(15\!\cdots\!69\)\( p^{41} T^{4} + \)\(10\!\cdots\!57\)\( p^{61} T^{5} + \)\(15\!\cdots\!69\)\( p^{114} T^{6} + \)\(62\!\cdots\!45\)\( p^{171} T^{7} + 3345520000714170341 p^{232} T^{8} + 5755583343 p^{296} T^{9} + p^{365} T^{10} \)
3$C_2 \wr S_5$ \( 1 + 4785029563937252 p^{3} T + \)\(90\!\cdots\!89\)\( p^{11} T^{2} + \)\(61\!\cdots\!40\)\( p^{20} T^{3} + \)\(50\!\cdots\!62\)\( p^{30} T^{4} + \)\(19\!\cdots\!08\)\( p^{46} T^{5} + \)\(50\!\cdots\!62\)\( p^{103} T^{6} + \)\(61\!\cdots\!40\)\( p^{166} T^{7} + \)\(90\!\cdots\!89\)\( p^{230} T^{8} + 4785029563937252 p^{295} T^{9} + p^{365} T^{10} \)
5$C_2 \wr S_5$ \( 1 - \)\(18\!\cdots\!14\)\( p^{3} T + \)\(70\!\cdots\!57\)\( p^{8} T^{2} - \)\(22\!\cdots\!16\)\( p^{14} T^{3} + \)\(61\!\cdots\!74\)\( p^{21} T^{4} + \)\(16\!\cdots\!96\)\( p^{30} T^{5} + \)\(61\!\cdots\!74\)\( p^{94} T^{6} - \)\(22\!\cdots\!16\)\( p^{160} T^{7} + \)\(70\!\cdots\!57\)\( p^{227} T^{8} - \)\(18\!\cdots\!14\)\( p^{295} T^{9} + p^{365} T^{10} \)
7$C_2 \wr S_5$ \( 1 + \)\(88\!\cdots\!92\)\( p^{2} T + \)\(93\!\cdots\!01\)\( p^{5} T^{2} + \)\(97\!\cdots\!00\)\( p^{10} T^{3} + \)\(45\!\cdots\!14\)\( p^{17} T^{4} + \)\(52\!\cdots\!12\)\( p^{25} T^{5} + \)\(45\!\cdots\!14\)\( p^{90} T^{6} + \)\(97\!\cdots\!00\)\( p^{156} T^{7} + \)\(93\!\cdots\!01\)\( p^{224} T^{8} + \)\(88\!\cdots\!92\)\( p^{294} T^{9} + p^{365} T^{10} \)
11$C_2 \wr S_5$ \( 1 - \)\(45\!\cdots\!60\)\( p T + \)\(25\!\cdots\!95\)\( p^{2} T^{2} - \)\(47\!\cdots\!20\)\( p^{5} T^{3} + \)\(18\!\cdots\!10\)\( p^{9} T^{4} - \)\(19\!\cdots\!72\)\( p^{14} T^{5} + \)\(18\!\cdots\!10\)\( p^{82} T^{6} - \)\(47\!\cdots\!20\)\( p^{151} T^{7} + \)\(25\!\cdots\!95\)\( p^{221} T^{8} - \)\(45\!\cdots\!60\)\( p^{293} T^{9} + p^{365} T^{10} \)
13$C_2 \wr S_5$ \( 1 - \)\(36\!\cdots\!22\)\( p T + \)\(15\!\cdots\!09\)\( p^{3} T^{2} - \)\(15\!\cdots\!80\)\( p^{6} T^{3} + \)\(76\!\cdots\!22\)\( p^{10} T^{4} - \)\(13\!\cdots\!64\)\( p^{15} T^{5} + \)\(76\!\cdots\!22\)\( p^{83} T^{6} - \)\(15\!\cdots\!80\)\( p^{152} T^{7} + \)\(15\!\cdots\!09\)\( p^{222} T^{8} - \)\(36\!\cdots\!22\)\( p^{293} T^{9} + p^{365} T^{10} \)
17$C_2 \wr S_5$ \( 1 - \)\(39\!\cdots\!06\)\( p T + \)\(24\!\cdots\!29\)\( p^{3} T^{2} - \)\(34\!\cdots\!40\)\( p^{5} T^{3} + \)\(10\!\cdots\!98\)\( p^{8} T^{4} - \)\(25\!\cdots\!16\)\( p^{12} T^{5} + \)\(10\!\cdots\!98\)\( p^{81} T^{6} - \)\(34\!\cdots\!40\)\( p^{151} T^{7} + \)\(24\!\cdots\!29\)\( p^{222} T^{8} - \)\(39\!\cdots\!06\)\( p^{293} T^{9} + p^{365} T^{10} \)
19$C_2 \wr S_5$ \( 1 - \)\(31\!\cdots\!00\)\( T + \)\(21\!\cdots\!05\)\( p T^{2} - \)\(61\!\cdots\!00\)\( p^{3} T^{3} + \)\(33\!\cdots\!90\)\( p^{5} T^{4} - \)\(86\!\cdots\!00\)\( p^{8} T^{5} + \)\(33\!\cdots\!90\)\( p^{78} T^{6} - \)\(61\!\cdots\!00\)\( p^{149} T^{7} + \)\(21\!\cdots\!05\)\( p^{220} T^{8} - \)\(31\!\cdots\!00\)\( p^{292} T^{9} + p^{365} T^{10} \)
23$C_2 \wr S_5$ \( 1 + \)\(17\!\cdots\!88\)\( p T + \)\(14\!\cdots\!47\)\( p^{2} T^{2} + \)\(81\!\cdots\!20\)\( p^{4} T^{3} + \)\(19\!\cdots\!62\)\( p^{6} T^{4} + \)\(36\!\cdots\!44\)\( p^{9} T^{5} + \)\(19\!\cdots\!62\)\( p^{79} T^{6} + \)\(81\!\cdots\!20\)\( p^{150} T^{7} + \)\(14\!\cdots\!47\)\( p^{221} T^{8} + \)\(17\!\cdots\!88\)\( p^{293} T^{9} + p^{365} T^{10} \)
29$C_2 \wr S_5$ \( 1 + \)\(21\!\cdots\!50\)\( T + \)\(71\!\cdots\!05\)\( p T^{2} + \)\(43\!\cdots\!00\)\( p^{2} T^{3} + \)\(27\!\cdots\!10\)\( p^{4} T^{4} + \)\(47\!\cdots\!00\)\( p^{6} T^{5} + \)\(27\!\cdots\!10\)\( p^{77} T^{6} + \)\(43\!\cdots\!00\)\( p^{148} T^{7} + \)\(71\!\cdots\!05\)\( p^{220} T^{8} + \)\(21\!\cdots\!50\)\( p^{292} T^{9} + p^{365} T^{10} \)
31$C_2 \wr S_5$ \( 1 + \)\(12\!\cdots\!40\)\( p T + \)\(32\!\cdots\!95\)\( p^{2} T^{2} + \)\(31\!\cdots\!80\)\( p^{3} T^{3} + \)\(15\!\cdots\!10\)\( p^{5} T^{4} + \)\(35\!\cdots\!68\)\( p^{7} T^{5} + \)\(15\!\cdots\!10\)\( p^{78} T^{6} + \)\(31\!\cdots\!80\)\( p^{149} T^{7} + \)\(32\!\cdots\!95\)\( p^{221} T^{8} + \)\(12\!\cdots\!40\)\( p^{293} T^{9} + p^{365} T^{10} \)
37$C_2 \wr S_5$ \( 1 + \)\(18\!\cdots\!94\)\( p T + \)\(24\!\cdots\!17\)\( T^{2} + \)\(17\!\cdots\!80\)\( p T^{3} + \)\(10\!\cdots\!82\)\( p^{2} T^{4} + \)\(52\!\cdots\!68\)\( p^{3} T^{5} + \)\(10\!\cdots\!82\)\( p^{75} T^{6} + \)\(17\!\cdots\!80\)\( p^{147} T^{7} + \)\(24\!\cdots\!17\)\( p^{219} T^{8} + \)\(18\!\cdots\!94\)\( p^{293} T^{9} + p^{365} T^{10} \)
41$C_2 \wr S_5$ \( 1 + \)\(89\!\cdots\!90\)\( T + \)\(21\!\cdots\!45\)\( T^{2} + \)\(38\!\cdots\!80\)\( p T^{3} + \)\(11\!\cdots\!10\)\( p^{2} T^{4} + \)\(17\!\cdots\!88\)\( p^{3} T^{5} + \)\(11\!\cdots\!10\)\( p^{75} T^{6} + \)\(38\!\cdots\!80\)\( p^{147} T^{7} + \)\(21\!\cdots\!45\)\( p^{219} T^{8} + \)\(89\!\cdots\!90\)\( p^{292} T^{9} + p^{365} T^{10} \)
43$C_2 \wr S_5$ \( 1 - \)\(11\!\cdots\!56\)\( T + \)\(12\!\cdots\!43\)\( T^{2} - \)\(19\!\cdots\!00\)\( p T^{3} + \)\(27\!\cdots\!02\)\( p^{2} T^{4} - \)\(28\!\cdots\!84\)\( p^{3} T^{5} + \)\(27\!\cdots\!02\)\( p^{75} T^{6} - \)\(19\!\cdots\!00\)\( p^{147} T^{7} + \)\(12\!\cdots\!43\)\( p^{219} T^{8} - \)\(11\!\cdots\!56\)\( p^{292} T^{9} + p^{365} T^{10} \)
47$C_2 \wr S_5$ \( 1 - \)\(26\!\cdots\!32\)\( T + \)\(64\!\cdots\!87\)\( T^{2} - \)\(21\!\cdots\!60\)\( p T^{3} + \)\(66\!\cdots\!42\)\( p^{2} T^{4} - \)\(16\!\cdots\!72\)\( p^{3} T^{5} + \)\(66\!\cdots\!42\)\( p^{75} T^{6} - \)\(21\!\cdots\!60\)\( p^{147} T^{7} + \)\(64\!\cdots\!87\)\( p^{219} T^{8} - \)\(26\!\cdots\!32\)\( p^{292} T^{9} + p^{365} T^{10} \)
53$C_2 \wr S_5$ \( 1 + \)\(22\!\cdots\!54\)\( T + \)\(10\!\cdots\!61\)\( p T^{2} + \)\(25\!\cdots\!60\)\( p^{2} T^{3} + \)\(61\!\cdots\!94\)\( p^{3} T^{4} + \)\(10\!\cdots\!72\)\( p^{4} T^{5} + \)\(61\!\cdots\!94\)\( p^{76} T^{6} + \)\(25\!\cdots\!60\)\( p^{148} T^{7} + \)\(10\!\cdots\!61\)\( p^{220} T^{8} + \)\(22\!\cdots\!54\)\( p^{292} T^{9} + p^{365} T^{10} \)
59$C_2 \wr S_5$ \( 1 - \)\(84\!\cdots\!00\)\( p T + \)\(13\!\cdots\!05\)\( p T^{2} - \)\(93\!\cdots\!00\)\( p^{2} T^{3} + \)\(13\!\cdots\!90\)\( p^{3} T^{4} - \)\(72\!\cdots\!00\)\( p^{4} T^{5} + \)\(13\!\cdots\!90\)\( p^{76} T^{6} - \)\(93\!\cdots\!00\)\( p^{148} T^{7} + \)\(13\!\cdots\!05\)\( p^{220} T^{8} - \)\(84\!\cdots\!00\)\( p^{293} T^{9} + p^{365} T^{10} \)
61$C_2 \wr S_5$ \( 1 + \)\(32\!\cdots\!90\)\( p T + \)\(29\!\cdots\!45\)\( p^{2} T^{2} + \)\(70\!\cdots\!80\)\( p^{3} T^{3} + \)\(33\!\cdots\!10\)\( p^{4} T^{4} + \)\(58\!\cdots\!48\)\( p^{5} T^{5} + \)\(33\!\cdots\!10\)\( p^{77} T^{6} + \)\(70\!\cdots\!80\)\( p^{149} T^{7} + \)\(29\!\cdots\!45\)\( p^{221} T^{8} + \)\(32\!\cdots\!90\)\( p^{293} T^{9} + p^{365} T^{10} \)
67$C_2 \wr S_5$ \( 1 - \)\(25\!\cdots\!56\)\( p T + \)\(10\!\cdots\!43\)\( p^{2} T^{2} - \)\(43\!\cdots\!60\)\( p^{3} T^{3} + \)\(71\!\cdots\!58\)\( p^{4} T^{4} - \)\(20\!\cdots\!68\)\( p^{5} T^{5} + \)\(71\!\cdots\!58\)\( p^{77} T^{6} - \)\(43\!\cdots\!60\)\( p^{149} T^{7} + \)\(10\!\cdots\!43\)\( p^{221} T^{8} - \)\(25\!\cdots\!56\)\( p^{293} T^{9} + p^{365} T^{10} \)
71$C_2 \wr S_5$ \( 1 - \)\(41\!\cdots\!60\)\( p T + \)\(87\!\cdots\!95\)\( p^{2} T^{2} - \)\(23\!\cdots\!20\)\( p^{3} T^{3} + \)\(36\!\cdots\!10\)\( p^{4} T^{4} - \)\(73\!\cdots\!52\)\( p^{5} T^{5} + \)\(36\!\cdots\!10\)\( p^{77} T^{6} - \)\(23\!\cdots\!20\)\( p^{149} T^{7} + \)\(87\!\cdots\!95\)\( p^{221} T^{8} - \)\(41\!\cdots\!60\)\( p^{293} T^{9} + p^{365} T^{10} \)
73$C_2 \wr S_5$ \( 1 + \)\(23\!\cdots\!74\)\( T + \)\(55\!\cdots\!13\)\( T^{2} + \)\(77\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!18\)\( T^{4} + \)\(11\!\cdots\!72\)\( T^{5} + \)\(10\!\cdots\!18\)\( p^{73} T^{6} + \)\(77\!\cdots\!20\)\( p^{146} T^{7} + \)\(55\!\cdots\!13\)\( p^{219} T^{8} + \)\(23\!\cdots\!74\)\( p^{292} T^{9} + p^{365} T^{10} \)
79$C_2 \wr S_5$ \( 1 - \)\(12\!\cdots\!00\)\( T + \)\(10\!\cdots\!95\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!10\)\( T^{4} - \)\(42\!\cdots\!00\)\( T^{5} + \)\(55\!\cdots\!10\)\( p^{73} T^{6} - \)\(10\!\cdots\!00\)\( p^{146} T^{7} + \)\(10\!\cdots\!95\)\( p^{219} T^{8} - \)\(12\!\cdots\!00\)\( p^{292} T^{9} + p^{365} T^{10} \)
83$C_2 \wr S_5$ \( 1 - \)\(10\!\cdots\!16\)\( T + \)\(23\!\cdots\!03\)\( T^{2} - \)\(14\!\cdots\!40\)\( T^{3} + \)\(46\!\cdots\!58\)\( T^{4} - \)\(38\!\cdots\!08\)\( T^{5} + \)\(46\!\cdots\!58\)\( p^{73} T^{6} - \)\(14\!\cdots\!40\)\( p^{146} T^{7} + \)\(23\!\cdots\!03\)\( p^{219} T^{8} - \)\(10\!\cdots\!16\)\( p^{292} T^{9} + p^{365} T^{10} \)
89$C_2 \wr S_5$ \( 1 + \)\(44\!\cdots\!50\)\( T + \)\(62\!\cdots\!45\)\( T^{2} + \)\(39\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!10\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{5} + \)\(17\!\cdots\!10\)\( p^{73} T^{6} + \)\(39\!\cdots\!00\)\( p^{146} T^{7} + \)\(62\!\cdots\!45\)\( p^{219} T^{8} + \)\(44\!\cdots\!50\)\( p^{292} T^{9} + p^{365} T^{10} \)
97$C_2 \wr S_5$ \( 1 + \)\(47\!\cdots\!18\)\( T + \)\(59\!\cdots\!37\)\( T^{2} + \)\(20\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!78\)\( T^{4} + \)\(32\!\cdots\!44\)\( T^{5} + \)\(13\!\cdots\!78\)\( p^{73} T^{6} + \)\(20\!\cdots\!80\)\( p^{146} T^{7} + \)\(59\!\cdots\!37\)\( p^{219} T^{8} + \)\(47\!\cdots\!18\)\( p^{292} T^{9} + p^{365} T^{10} \)
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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

−9.336242290136873608398234029104, −9.268826832423104002057256829430, −8.962355286761724857259983449662, −8.763827905758262063495101613042, −8.391836326963413296838111280111, −7.85227310763809429256627856750, −7.60589516847797714562159159172, −7.47979629862534823040583339353, −6.58119676321529783002704159117, −6.24241685386205450248069218885, −5.92356223185394185889005904993, −5.73303297228531362946256654245, −5.16327254886260277742658361087, −5.12999744947285585775759446480, −5.06991403691409709398322010035, −4.01676214092327570276345851221, −3.79574254300370297129996634068, −3.78210053216481678964923402046, −3.36258067312358559770568212185, −2.90788868250565303219938558231, −2.37016929090892543548118267949, −2.11654228559532010207876022589, −1.41489081260714454570483608629, −1.28192350893123960744080206343, −1.16525561455138451385061388305, 0, 0, 0, 0, 0, 1.16525561455138451385061388305, 1.28192350893123960744080206343, 1.41489081260714454570483608629, 2.11654228559532010207876022589, 2.37016929090892543548118267949, 2.90788868250565303219938558231, 3.36258067312358559770568212185, 3.78210053216481678964923402046, 3.79574254300370297129996634068, 4.01676214092327570276345851221, 5.06991403691409709398322010035, 5.12999744947285585775759446480, 5.16327254886260277742658361087, 5.73303297228531362946256654245, 5.92356223185394185889005904993, 6.24241685386205450248069218885, 6.58119676321529783002704159117, 7.47979629862534823040583339353, 7.60589516847797714562159159172, 7.85227310763809429256627856750, 8.391836326963413296838111280111, 8.763827905758262063495101613042, 8.962355286761724857259983449662, 9.268826832423104002057256829430, 9.336242290136873608398234029104

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