Properties

Degree 10
Conductor $ 1 $
Sign $1$
Motivic weight 67
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.55e9·2-s + 3.44e15·3-s − 1.78e20·4-s + 3.30e23·5-s + 1.91e25·6-s + 3.36e28·7-s − 2.33e29·8-s − 2.11e32·9-s + 1.83e33·10-s + 2.03e35·11-s − 6.13e35·12-s + 1.77e35·13-s + 1.86e38·14-s + 1.13e39·15-s + 1.19e40·16-s + 7.52e40·17-s − 1.17e42·18-s + 3.99e42·19-s − 5.89e43·20-s + 1.15e44·21-s + 1.12e45·22-s − 4.19e44·23-s − 8.03e44·24-s − 2.70e46·25-s + 9.87e44·26-s − 2.20e47·27-s − 5.99e48·28-s + ⋯
L(s)  = 1  + 0.457·2-s + 0.357·3-s − 1.20·4-s + 1.27·5-s + 0.163·6-s + 1.64·7-s − 0.130·8-s − 2.28·9-s + 0.580·10-s + 2.63·11-s − 0.432·12-s + 0.00856·13-s + 0.751·14-s + 0.454·15-s + 0.549·16-s + 0.453·17-s − 1.04·18-s + 0.580·19-s − 1.53·20-s + 0.588·21-s + 1.20·22-s − 0.101·23-s − 0.0465·24-s − 0.399·25-s + 0.00391·26-s − 0.247·27-s − 1.98·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(68-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+67/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(10\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(67\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((10,\ 1,\ (\ :67/2, 67/2, 67/2, 67/2, 67/2),\ 1)\)
\(L(34)\)  \(\approx\)  \(16.87265373\)
\(L(\frac12)\)  \(\approx\)  \(16.87265373\)
\(L(\frac{69}{2})\)   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$C_2 \wr S_5$ \( 1 - 694362657 p^{3} T + 51056480289306133 p^{12} T^{2} - \)\(91\!\cdots\!45\)\( p^{21} T^{3} + \)\(12\!\cdots\!27\)\( p^{38} T^{4} - \)\(15\!\cdots\!81\)\( p^{55} T^{5} + \)\(12\!\cdots\!27\)\( p^{105} T^{6} - \)\(91\!\cdots\!45\)\( p^{155} T^{7} + 51056480289306133 p^{213} T^{8} - 694362657 p^{271} T^{9} + p^{335} T^{10} \)
3$C_2 \wr S_5$ \( 1 - 14170206868804 p^{5} T + \)\(12\!\cdots\!61\)\( p^{11} T^{2} - \)\(33\!\cdots\!40\)\( p^{18} T^{3} + \)\(49\!\cdots\!34\)\( p^{31} T^{4} - \)\(18\!\cdots\!84\)\( p^{46} T^{5} + \)\(49\!\cdots\!34\)\( p^{98} T^{6} - \)\(33\!\cdots\!40\)\( p^{152} T^{7} + \)\(12\!\cdots\!61\)\( p^{212} T^{8} - 14170206868804 p^{273} T^{9} + p^{335} T^{10} \)
5$C_2 \wr S_5$ \( 1 - \)\(13\!\cdots\!58\)\( p^{2} T + \)\(43\!\cdots\!17\)\( p^{5} T^{2} - \)\(30\!\cdots\!28\)\( p^{9} T^{3} + \)\(11\!\cdots\!94\)\( p^{16} T^{4} - \)\(27\!\cdots\!68\)\( p^{26} T^{5} + \)\(11\!\cdots\!94\)\( p^{83} T^{6} - \)\(30\!\cdots\!28\)\( p^{143} T^{7} + \)\(43\!\cdots\!17\)\( p^{206} T^{8} - \)\(13\!\cdots\!58\)\( p^{270} T^{9} + p^{335} T^{10} \)
7$C_2 \wr S_5$ \( 1 - \)\(68\!\cdots\!44\)\( p^{2} T + \)\(16\!\cdots\!07\)\( p^{6} T^{2} - \)\(25\!\cdots\!00\)\( p^{11} T^{3} + \)\(10\!\cdots\!02\)\( p^{18} T^{4} - \)\(32\!\cdots\!12\)\( p^{26} T^{5} + \)\(10\!\cdots\!02\)\( p^{85} T^{6} - \)\(25\!\cdots\!00\)\( p^{145} T^{7} + \)\(16\!\cdots\!07\)\( p^{207} T^{8} - \)\(68\!\cdots\!44\)\( p^{270} T^{9} + p^{335} T^{10} \)
11$C_2 \wr S_5$ \( 1 - \)\(18\!\cdots\!60\)\( p T + \)\(20\!\cdots\!45\)\( p^{3} T^{2} - \)\(14\!\cdots\!20\)\( p^{6} T^{3} + \)\(83\!\cdots\!10\)\( p^{9} T^{4} - \)\(41\!\cdots\!92\)\( p^{13} T^{5} + \)\(83\!\cdots\!10\)\( p^{76} T^{6} - \)\(14\!\cdots\!20\)\( p^{140} T^{7} + \)\(20\!\cdots\!45\)\( p^{204} T^{8} - \)\(18\!\cdots\!60\)\( p^{269} T^{9} + p^{335} T^{10} \)
13$C_2 \wr S_5$ \( 1 - \)\(13\!\cdots\!54\)\( p T + \)\(57\!\cdots\!41\)\( p^{3} T^{2} + \)\(92\!\cdots\!20\)\( p^{6} T^{3} + \)\(74\!\cdots\!26\)\( p^{9} T^{4} + \)\(17\!\cdots\!24\)\( p^{12} T^{5} + \)\(74\!\cdots\!26\)\( p^{76} T^{6} + \)\(92\!\cdots\!20\)\( p^{140} T^{7} + \)\(57\!\cdots\!41\)\( p^{204} T^{8} - \)\(13\!\cdots\!54\)\( p^{269} T^{9} + p^{335} T^{10} \)
17$C_2 \wr S_5$ \( 1 - \)\(44\!\cdots\!18\)\( p T + \)\(30\!\cdots\!37\)\( p^{2} T^{2} - \)\(17\!\cdots\!40\)\( p^{3} T^{3} + \)\(46\!\cdots\!58\)\( p^{4} T^{4} - \)\(84\!\cdots\!76\)\( p^{7} T^{5} + \)\(46\!\cdots\!58\)\( p^{71} T^{6} - \)\(17\!\cdots\!40\)\( p^{137} T^{7} + \)\(30\!\cdots\!37\)\( p^{203} T^{8} - \)\(44\!\cdots\!18\)\( p^{269} T^{9} + p^{335} T^{10} \)
19$C_2 \wr S_5$ \( 1 - \)\(21\!\cdots\!00\)\( p T + \)\(40\!\cdots\!95\)\( p^{2} T^{2} - \)\(52\!\cdots\!00\)\( p^{4} T^{3} + \)\(12\!\cdots\!90\)\( p^{7} T^{4} - \)\(74\!\cdots\!00\)\( p^{10} T^{5} + \)\(12\!\cdots\!90\)\( p^{74} T^{6} - \)\(52\!\cdots\!00\)\( p^{138} T^{7} + \)\(40\!\cdots\!95\)\( p^{203} T^{8} - \)\(21\!\cdots\!00\)\( p^{269} T^{9} + p^{335} T^{10} \)
23$C_2 \wr S_5$ \( 1 + \)\(41\!\cdots\!68\)\( T + \)\(73\!\cdots\!87\)\( T^{2} + \)\(13\!\cdots\!40\)\( p T^{3} + \)\(19\!\cdots\!94\)\( p^{3} T^{4} + \)\(12\!\cdots\!68\)\( p^{5} T^{5} + \)\(19\!\cdots\!94\)\( p^{70} T^{6} + \)\(13\!\cdots\!40\)\( p^{135} T^{7} + \)\(73\!\cdots\!87\)\( p^{201} T^{8} + \)\(41\!\cdots\!68\)\( p^{268} T^{9} + p^{335} T^{10} \)
29$C_2 \wr S_5$ \( 1 - \)\(18\!\cdots\!50\)\( T + \)\(10\!\cdots\!05\)\( p T^{2} - \)\(39\!\cdots\!00\)\( p^{2} T^{3} + \)\(68\!\cdots\!10\)\( p^{4} T^{4} - \)\(81\!\cdots\!00\)\( p^{6} T^{5} + \)\(68\!\cdots\!10\)\( p^{71} T^{6} - \)\(39\!\cdots\!00\)\( p^{136} T^{7} + \)\(10\!\cdots\!05\)\( p^{202} T^{8} - \)\(18\!\cdots\!50\)\( p^{268} T^{9} + p^{335} T^{10} \)
31$C_2 \wr S_5$ \( 1 - \)\(11\!\cdots\!60\)\( p T + \)\(84\!\cdots\!95\)\( p^{2} T^{2} - \)\(13\!\cdots\!20\)\( p^{4} T^{3} + \)\(16\!\cdots\!10\)\( p^{6} T^{4} - \)\(17\!\cdots\!72\)\( p^{8} T^{5} + \)\(16\!\cdots\!10\)\( p^{73} T^{6} - \)\(13\!\cdots\!20\)\( p^{138} T^{7} + \)\(84\!\cdots\!95\)\( p^{203} T^{8} - \)\(11\!\cdots\!60\)\( p^{269} T^{9} + p^{335} T^{10} \)
37$C_2 \wr S_5$ \( 1 + \)\(56\!\cdots\!94\)\( T + \)\(47\!\cdots\!93\)\( T^{2} + \)\(57\!\cdots\!20\)\( p T^{3} + \)\(78\!\cdots\!82\)\( p^{2} T^{4} + \)\(67\!\cdots\!44\)\( p^{3} T^{5} + \)\(78\!\cdots\!82\)\( p^{69} T^{6} + \)\(57\!\cdots\!20\)\( p^{135} T^{7} + \)\(47\!\cdots\!93\)\( p^{201} T^{8} + \)\(56\!\cdots\!94\)\( p^{268} T^{9} + p^{335} T^{10} \)
41$C_2 \wr S_5$ \( 1 - \)\(11\!\cdots\!10\)\( T + \)\(30\!\cdots\!45\)\( T^{2} + \)\(23\!\cdots\!80\)\( p T^{3} + \)\(25\!\cdots\!10\)\( p^{2} T^{4} + \)\(70\!\cdots\!88\)\( p^{3} T^{5} + \)\(25\!\cdots\!10\)\( p^{69} T^{6} + \)\(23\!\cdots\!80\)\( p^{135} T^{7} + \)\(30\!\cdots\!45\)\( p^{201} T^{8} - \)\(11\!\cdots\!10\)\( p^{268} T^{9} + p^{335} T^{10} \)
43$C_2 \wr S_5$ \( 1 - \)\(65\!\cdots\!92\)\( T + \)\(25\!\cdots\!49\)\( p T^{2} - \)\(30\!\cdots\!00\)\( p^{2} T^{3} + \)\(70\!\cdots\!14\)\( p^{3} T^{4} - \)\(63\!\cdots\!16\)\( p^{4} T^{5} + \)\(70\!\cdots\!14\)\( p^{70} T^{6} - \)\(30\!\cdots\!00\)\( p^{136} T^{7} + \)\(25\!\cdots\!49\)\( p^{202} T^{8} - \)\(65\!\cdots\!92\)\( p^{268} T^{9} + p^{335} T^{10} \)
47$C_2 \wr S_5$ \( 1 + \)\(12\!\cdots\!44\)\( T + \)\(90\!\cdots\!69\)\( p T^{2} + \)\(19\!\cdots\!80\)\( p^{2} T^{3} + \)\(79\!\cdots\!86\)\( p^{3} T^{4} + \)\(12\!\cdots\!12\)\( p^{4} T^{5} + \)\(79\!\cdots\!86\)\( p^{70} T^{6} + \)\(19\!\cdots\!80\)\( p^{136} T^{7} + \)\(90\!\cdots\!69\)\( p^{202} T^{8} + \)\(12\!\cdots\!44\)\( p^{268} T^{9} + p^{335} T^{10} \)
53$C_2 \wr S_5$ \( 1 - \)\(10\!\cdots\!22\)\( T + \)\(14\!\cdots\!89\)\( p T^{2} + \)\(39\!\cdots\!60\)\( p^{2} T^{3} + \)\(20\!\cdots\!74\)\( p^{3} T^{4} + \)\(80\!\cdots\!44\)\( p^{4} T^{5} + \)\(20\!\cdots\!74\)\( p^{70} T^{6} + \)\(39\!\cdots\!60\)\( p^{136} T^{7} + \)\(14\!\cdots\!89\)\( p^{202} T^{8} - \)\(10\!\cdots\!22\)\( p^{268} T^{9} + p^{335} T^{10} \)
59$C_2 \wr S_5$ \( 1 + \)\(51\!\cdots\!00\)\( p T + \)\(39\!\cdots\!95\)\( p^{2} T^{2} + \)\(14\!\cdots\!00\)\( p^{3} T^{3} + \)\(72\!\cdots\!10\)\( p^{4} T^{4} + \)\(20\!\cdots\!00\)\( p^{5} T^{5} + \)\(72\!\cdots\!10\)\( p^{71} T^{6} + \)\(14\!\cdots\!00\)\( p^{137} T^{7} + \)\(39\!\cdots\!95\)\( p^{203} T^{8} + \)\(51\!\cdots\!00\)\( p^{269} T^{9} + p^{335} T^{10} \)
61$C_2 \wr S_5$ \( 1 + \)\(18\!\cdots\!90\)\( p T + \)\(57\!\cdots\!45\)\( p^{2} T^{2} + \)\(70\!\cdots\!80\)\( p^{3} T^{3} + \)\(12\!\cdots\!10\)\( p^{4} T^{4} + \)\(11\!\cdots\!48\)\( p^{5} T^{5} + \)\(12\!\cdots\!10\)\( p^{71} T^{6} + \)\(70\!\cdots\!80\)\( p^{137} T^{7} + \)\(57\!\cdots\!45\)\( p^{203} T^{8} + \)\(18\!\cdots\!90\)\( p^{269} T^{9} + p^{335} T^{10} \)
67$C_2 \wr S_5$ \( 1 + \)\(11\!\cdots\!44\)\( T + \)\(30\!\cdots\!43\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(58\!\cdots\!18\)\( T^{4} - \)\(37\!\cdots\!48\)\( T^{5} + \)\(58\!\cdots\!18\)\( p^{67} T^{6} + \)\(16\!\cdots\!80\)\( p^{134} T^{7} + \)\(30\!\cdots\!43\)\( p^{201} T^{8} + \)\(11\!\cdots\!44\)\( p^{268} T^{9} + p^{335} T^{10} \)
71$C_2 \wr S_5$ \( 1 + \)\(11\!\cdots\!40\)\( T + \)\(43\!\cdots\!95\)\( T^{2} + \)\(37\!\cdots\!80\)\( T^{3} + \)\(80\!\cdots\!10\)\( T^{4} + \)\(54\!\cdots\!48\)\( T^{5} + \)\(80\!\cdots\!10\)\( p^{67} T^{6} + \)\(37\!\cdots\!80\)\( p^{134} T^{7} + \)\(43\!\cdots\!95\)\( p^{201} T^{8} + \)\(11\!\cdots\!40\)\( p^{268} T^{9} + p^{335} T^{10} \)
73$C_2 \wr S_5$ \( 1 + \)\(30\!\cdots\!18\)\( T + \)\(36\!\cdots\!37\)\( T^{2} + \)\(83\!\cdots\!20\)\( T^{3} + \)\(52\!\cdots\!98\)\( T^{4} + \)\(86\!\cdots\!24\)\( T^{5} + \)\(52\!\cdots\!98\)\( p^{67} T^{6} + \)\(83\!\cdots\!20\)\( p^{134} T^{7} + \)\(36\!\cdots\!37\)\( p^{201} T^{8} + \)\(30\!\cdots\!18\)\( p^{268} T^{9} + p^{335} T^{10} \)
79$C_2 \wr S_5$ \( 1 - \)\(29\!\cdots\!00\)\( T + \)\(51\!\cdots\!95\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!10\)\( T^{4} - \)\(26\!\cdots\!00\)\( T^{5} + \)\(12\!\cdots\!10\)\( p^{67} T^{6} - \)\(13\!\cdots\!00\)\( p^{134} T^{7} + \)\(51\!\cdots\!95\)\( p^{201} T^{8} - \)\(29\!\cdots\!00\)\( p^{268} T^{9} + p^{335} T^{10} \)
83$C_2 \wr S_5$ \( 1 + \)\(47\!\cdots\!88\)\( T + \)\(21\!\cdots\!47\)\( T^{2} + \)\(64\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!06\)\( p T^{4} + \)\(35\!\cdots\!04\)\( T^{5} + \)\(20\!\cdots\!06\)\( p^{68} T^{6} + \)\(64\!\cdots\!60\)\( p^{134} T^{7} + \)\(21\!\cdots\!47\)\( p^{201} T^{8} + \)\(47\!\cdots\!88\)\( p^{268} T^{9} + p^{335} T^{10} \)
89$C_2 \wr S_5$ \( 1 + \)\(11\!\cdots\!50\)\( T + \)\(98\!\cdots\!45\)\( T^{2} + \)\(38\!\cdots\!00\)\( T^{3} + \)\(39\!\cdots\!10\)\( T^{4} + \)\(27\!\cdots\!00\)\( T^{5} + \)\(39\!\cdots\!10\)\( p^{67} T^{6} + \)\(38\!\cdots\!00\)\( p^{134} T^{7} + \)\(98\!\cdots\!45\)\( p^{201} T^{8} + \)\(11\!\cdots\!50\)\( p^{268} T^{9} + p^{335} T^{10} \)
97$C_2 \wr S_5$ \( 1 - \)\(28\!\cdots\!06\)\( T + \)\(36\!\cdots\!93\)\( T^{2} - \)\(13\!\cdots\!80\)\( T^{3} + \)\(66\!\cdots\!78\)\( T^{4} - \)\(24\!\cdots\!28\)\( T^{5} + \)\(66\!\cdots\!78\)\( p^{67} T^{6} - \)\(13\!\cdots\!80\)\( p^{134} T^{7} + \)\(36\!\cdots\!93\)\( p^{201} T^{8} - \)\(28\!\cdots\!06\)\( p^{268} T^{9} + p^{335} T^{10} \)
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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

−8.858675758657522720164921571811, −8.778273531595656465865546320133, −8.398297559593765716015857212247, −8.391833007584314591458868850339, −7.79914147839544650450516891186, −7.43183866614746778765322778546, −6.54311877586695736437341601593, −6.37818207318158337389829469495, −6.09078674297176093913354331349, −5.94469613750623119616267567635, −5.29599700427440966724136933913, −4.85449009180606295178890287890, −4.63299702481245344620824972548, −4.38977130502919495322892275986, −4.36670364333491102955122907493, −3.37379077599699753982702523477, −3.15911341946688949251242986902, −2.94488524404003807235366340415, −2.60914844321186924080308460842, −1.78866306257034809070773597280, −1.59737227155102043920021945378, −1.57226775393031713484103606981, −0.971299662601215759494365511330, −0.60391611597935734051522552277, −0.46173712903443472817543896296, 0.46173712903443472817543896296, 0.60391611597935734051522552277, 0.971299662601215759494365511330, 1.57226775393031713484103606981, 1.59737227155102043920021945378, 1.78866306257034809070773597280, 2.60914844321186924080308460842, 2.94488524404003807235366340415, 3.15911341946688949251242986902, 3.37379077599699753982702523477, 4.36670364333491102955122907493, 4.38977130502919495322892275986, 4.63299702481245344620824972548, 4.85449009180606295178890287890, 5.29599700427440966724136933913, 5.94469613750623119616267567635, 6.09078674297176093913354331349, 6.37818207318158337389829469495, 6.54311877586695736437341601593, 7.43183866614746778765322778546, 7.79914147839544650450516891186, 8.391833007584314591458868850339, 8.398297559593765716015857212247, 8.778273531595656465865546320133, 8.858675758657522720164921571811

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