Properties

Degree 10
Conductor $ 1 $
Sign $-1$
Motivic weight 69
Primitive no
Self-dual yes
Analytic rank 5

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.80e10·2-s − 4.85e15·3-s − 6.84e20·4-s − 1.86e24·5-s + 8.74e25·6-s + 7.67e28·7-s + 2.51e30·8-s − 2.23e33·9-s + 3.35e34·10-s − 6.06e34·11-s + 3.32e36·12-s + 2.41e38·13-s − 1.38e39·14-s + 9.05e39·15-s + 4.91e41·16-s − 3.40e42·17-s + 4.02e43·18-s + 5.04e43·19-s + 1.27e45·20-s − 3.73e44·21-s + 1.09e45·22-s + 4.95e46·23-s − 1.22e46·24-s − 2.26e48·25-s − 4.35e48·26-s + 1.06e49·27-s − 5.25e49·28-s + ⋯
L(s)  = 1  − 0.741·2-s − 0.168·3-s − 1.15·4-s − 1.43·5-s + 0.124·6-s + 0.536·7-s + 0.175·8-s − 2.67·9-s + 1.06·10-s − 0.0715·11-s + 0.195·12-s + 0.896·13-s − 0.397·14-s + 0.240·15-s + 1.41·16-s − 1.20·17-s + 1.98·18-s + 0.385·19-s + 1.66·20-s − 0.0902·21-s + 0.0530·22-s + 0.519·23-s − 0.0294·24-s − 1.33·25-s − 0.664·26-s + 0.443·27-s − 0.622·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{5} \, L(s)\cr =\mathstrut & -\,\Lambda(70-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+69/2)^{5} \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(10\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(69\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  5
Selberg data  =  $(10,\ 1,\ (\ :69/2, 69/2, 69/2, 69/2, 69/2),\ -1)$
$L(35)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{71}{2})$   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$C_2 \wr S_5$ \( 1 + 562679199 p^{5} T + 123136181005504961 p^{13} T^{2} + \)\(41\!\cdots\!75\)\( p^{26} T^{3} + \)\(74\!\cdots\!61\)\( p^{43} T^{4} + \)\(53\!\cdots\!71\)\( p^{62} T^{5} + \)\(74\!\cdots\!61\)\( p^{112} T^{6} + \)\(41\!\cdots\!75\)\( p^{164} T^{7} + 123136181005504961 p^{220} T^{8} + 562679199 p^{281} T^{9} + p^{345} T^{10} \)
3$C_2 \wr S_5$ \( 1 + 179928975067252 p^{3} T + \)\(38\!\cdots\!67\)\( p^{10} T^{2} + \)\(86\!\cdots\!00\)\( p^{17} T^{3} + \)\(40\!\cdots\!66\)\( p^{29} T^{4} + \)\(16\!\cdots\!52\)\( p^{44} T^{5} + \)\(40\!\cdots\!66\)\( p^{98} T^{6} + \)\(86\!\cdots\!00\)\( p^{155} T^{7} + \)\(38\!\cdots\!67\)\( p^{217} T^{8} + 179928975067252 p^{279} T^{9} + p^{345} T^{10} \)
5$C_2 \wr S_5$ \( 1 + \)\(14\!\cdots\!94\)\( p^{3} T + \)\(73\!\cdots\!61\)\( p^{7} T^{2} + \)\(33\!\cdots\!68\)\( p^{12} T^{3} + \)\(35\!\cdots\!42\)\( p^{21} T^{4} + \)\(41\!\cdots\!28\)\( p^{31} T^{5} + \)\(35\!\cdots\!42\)\( p^{90} T^{6} + \)\(33\!\cdots\!68\)\( p^{150} T^{7} + \)\(73\!\cdots\!61\)\( p^{214} T^{8} + \)\(14\!\cdots\!94\)\( p^{279} T^{9} + p^{345} T^{10} \)
7$C_2 \wr S_5$ \( 1 - \)\(10\!\cdots\!56\)\( p T + \)\(26\!\cdots\!07\)\( p^{4} T^{2} - \)\(12\!\cdots\!00\)\( p^{9} T^{3} + \)\(44\!\cdots\!86\)\( p^{15} T^{4} - \)\(37\!\cdots\!84\)\( p^{22} T^{5} + \)\(44\!\cdots\!86\)\( p^{84} T^{6} - \)\(12\!\cdots\!00\)\( p^{147} T^{7} + \)\(26\!\cdots\!07\)\( p^{211} T^{8} - \)\(10\!\cdots\!56\)\( p^{277} T^{9} + p^{345} T^{10} \)
11$C_2 \wr S_5$ \( 1 + \)\(55\!\cdots\!40\)\( p T + \)\(13\!\cdots\!45\)\( p^{3} T^{2} - \)\(11\!\cdots\!20\)\( p^{6} T^{3} + \)\(66\!\cdots\!10\)\( p^{10} T^{4} - \)\(97\!\cdots\!72\)\( p^{14} T^{5} + \)\(66\!\cdots\!10\)\( p^{79} T^{6} - \)\(11\!\cdots\!20\)\( p^{144} T^{7} + \)\(13\!\cdots\!45\)\( p^{210} T^{8} + \)\(55\!\cdots\!40\)\( p^{277} T^{9} + p^{345} T^{10} \)
13$C_2 \wr S_5$ \( 1 - \)\(18\!\cdots\!22\)\( p T + \)\(92\!\cdots\!09\)\( p^{3} T^{2} - \)\(59\!\cdots\!00\)\( p^{6} T^{3} + \)\(11\!\cdots\!42\)\( p^{10} T^{4} - \)\(43\!\cdots\!92\)\( p^{14} T^{5} + \)\(11\!\cdots\!42\)\( p^{79} T^{6} - \)\(59\!\cdots\!00\)\( p^{144} T^{7} + \)\(92\!\cdots\!09\)\( p^{210} T^{8} - \)\(18\!\cdots\!22\)\( p^{277} T^{9} + p^{345} T^{10} \)
17$C_2 \wr S_5$ \( 1 + \)\(20\!\cdots\!14\)\( p T + \)\(37\!\cdots\!69\)\( p^{3} T^{2} + \)\(10\!\cdots\!00\)\( p^{5} T^{3} + \)\(67\!\cdots\!66\)\( p^{7} T^{4} - \)\(19\!\cdots\!28\)\( p^{9} T^{5} + \)\(67\!\cdots\!66\)\( p^{76} T^{6} + \)\(10\!\cdots\!00\)\( p^{143} T^{7} + \)\(37\!\cdots\!69\)\( p^{210} T^{8} + \)\(20\!\cdots\!14\)\( p^{277} T^{9} + p^{345} T^{10} \)
19$C_2 \wr S_5$ \( 1 - \)\(50\!\cdots\!00\)\( T + \)\(34\!\cdots\!05\)\( p T^{2} - \)\(38\!\cdots\!00\)\( p^{3} T^{3} + \)\(81\!\cdots\!90\)\( p^{5} T^{4} - \)\(38\!\cdots\!00\)\( p^{8} T^{5} + \)\(81\!\cdots\!90\)\( p^{74} T^{6} - \)\(38\!\cdots\!00\)\( p^{141} T^{7} + \)\(34\!\cdots\!05\)\( p^{208} T^{8} - \)\(50\!\cdots\!00\)\( p^{276} T^{9} + p^{345} T^{10} \)
23$C_2 \wr S_5$ \( 1 - \)\(21\!\cdots\!12\)\( p T + \)\(27\!\cdots\!47\)\( p^{2} T^{2} - \)\(50\!\cdots\!00\)\( p^{3} T^{3} + \)\(30\!\cdots\!18\)\( p^{4} T^{4} - \)\(36\!\cdots\!16\)\( p^{5} T^{5} + \)\(30\!\cdots\!18\)\( p^{73} T^{6} - \)\(50\!\cdots\!00\)\( p^{141} T^{7} + \)\(27\!\cdots\!47\)\( p^{209} T^{8} - \)\(21\!\cdots\!12\)\( p^{277} T^{9} + p^{345} T^{10} \)
29$C_2 \wr S_5$ \( 1 + \)\(21\!\cdots\!50\)\( p T + \)\(52\!\cdots\!45\)\( p^{2} T^{2} + \)\(79\!\cdots\!00\)\( p^{3} T^{3} + \)\(35\!\cdots\!90\)\( p^{5} T^{4} + \)\(13\!\cdots\!00\)\( p^{7} T^{5} + \)\(35\!\cdots\!90\)\( p^{74} T^{6} + \)\(79\!\cdots\!00\)\( p^{141} T^{7} + \)\(52\!\cdots\!45\)\( p^{209} T^{8} + \)\(21\!\cdots\!50\)\( p^{277} T^{9} + p^{345} T^{10} \)
31$C_2 \wr S_5$ \( 1 + \)\(25\!\cdots\!40\)\( p T + \)\(50\!\cdots\!95\)\( p^{2} T^{2} + \)\(73\!\cdots\!80\)\( p^{3} T^{3} + \)\(28\!\cdots\!10\)\( p^{5} T^{4} + \)\(93\!\cdots\!68\)\( p^{7} T^{5} + \)\(28\!\cdots\!10\)\( p^{74} T^{6} + \)\(73\!\cdots\!80\)\( p^{141} T^{7} + \)\(50\!\cdots\!95\)\( p^{209} T^{8} + \)\(25\!\cdots\!40\)\( p^{277} T^{9} + p^{345} T^{10} \)
37$C_2 \wr S_5$ \( 1 - \)\(11\!\cdots\!02\)\( T + \)\(41\!\cdots\!77\)\( T^{2} - \)\(11\!\cdots\!00\)\( p T^{3} + \)\(71\!\cdots\!82\)\( p^{2} T^{4} - \)\(16\!\cdots\!72\)\( p^{3} T^{5} + \)\(71\!\cdots\!82\)\( p^{71} T^{6} - \)\(11\!\cdots\!00\)\( p^{139} T^{7} + \)\(41\!\cdots\!77\)\( p^{207} T^{8} - \)\(11\!\cdots\!02\)\( p^{276} T^{9} + p^{345} T^{10} \)
41$C_2 \wr S_5$ \( 1 - \)\(12\!\cdots\!10\)\( T + \)\(13\!\cdots\!45\)\( T^{2} - \)\(23\!\cdots\!20\)\( p T^{3} + \)\(87\!\cdots\!10\)\( p^{3} T^{4} - \)\(40\!\cdots\!12\)\( p^{3} T^{5} + \)\(87\!\cdots\!10\)\( p^{72} T^{6} - \)\(23\!\cdots\!20\)\( p^{139} T^{7} + \)\(13\!\cdots\!45\)\( p^{207} T^{8} - \)\(12\!\cdots\!10\)\( p^{276} T^{9} + p^{345} T^{10} \)
43$C_2 \wr S_5$ \( 1 - \)\(18\!\cdots\!56\)\( T + \)\(11\!\cdots\!43\)\( T^{2} - \)\(69\!\cdots\!00\)\( p T^{3} + \)\(49\!\cdots\!02\)\( p^{2} T^{4} - \)\(25\!\cdots\!84\)\( p^{3} T^{5} + \)\(49\!\cdots\!02\)\( p^{71} T^{6} - \)\(69\!\cdots\!00\)\( p^{139} T^{7} + \)\(11\!\cdots\!43\)\( p^{207} T^{8} - \)\(18\!\cdots\!56\)\( p^{276} T^{9} + p^{345} T^{10} \)
47$C_2 \wr S_5$ \( 1 + \)\(10\!\cdots\!28\)\( T + \)\(95\!\cdots\!67\)\( T^{2} + \)\(13\!\cdots\!00\)\( p T^{3} + \)\(18\!\cdots\!42\)\( p^{2} T^{4} + \)\(18\!\cdots\!08\)\( p^{3} T^{5} + \)\(18\!\cdots\!42\)\( p^{71} T^{6} + \)\(13\!\cdots\!00\)\( p^{139} T^{7} + \)\(95\!\cdots\!67\)\( p^{207} T^{8} + \)\(10\!\cdots\!28\)\( p^{276} T^{9} + p^{345} T^{10} \)
53$C_2 \wr S_5$ \( 1 - \)\(63\!\cdots\!46\)\( T + \)\(87\!\cdots\!61\)\( p T^{2} - \)\(68\!\cdots\!00\)\( p^{2} T^{3} + \)\(56\!\cdots\!14\)\( p^{3} T^{4} - \)\(32\!\cdots\!48\)\( p^{4} T^{5} + \)\(56\!\cdots\!14\)\( p^{72} T^{6} - \)\(68\!\cdots\!00\)\( p^{140} T^{7} + \)\(87\!\cdots\!61\)\( p^{208} T^{8} - \)\(63\!\cdots\!46\)\( p^{276} T^{9} + p^{345} T^{10} \)
59$C_2 \wr S_5$ \( 1 + \)\(33\!\cdots\!00\)\( T + \)\(17\!\cdots\!05\)\( p T^{2} + \)\(56\!\cdots\!00\)\( p^{2} T^{3} + \)\(16\!\cdots\!90\)\( p^{3} T^{4} + \)\(37\!\cdots\!00\)\( p^{4} T^{5} + \)\(16\!\cdots\!90\)\( p^{72} T^{6} + \)\(56\!\cdots\!00\)\( p^{140} T^{7} + \)\(17\!\cdots\!05\)\( p^{208} T^{8} + \)\(33\!\cdots\!00\)\( p^{276} T^{9} + p^{345} T^{10} \)
61$C_2 \wr S_5$ \( 1 - \)\(43\!\cdots\!10\)\( p T + \)\(78\!\cdots\!45\)\( p^{2} T^{2} - \)\(13\!\cdots\!20\)\( p^{3} T^{3} + \)\(85\!\cdots\!10\)\( p^{5} T^{4} - \)\(15\!\cdots\!52\)\( p^{5} T^{5} + \)\(85\!\cdots\!10\)\( p^{74} T^{6} - \)\(13\!\cdots\!20\)\( p^{141} T^{7} + \)\(78\!\cdots\!45\)\( p^{209} T^{8} - \)\(43\!\cdots\!10\)\( p^{277} T^{9} + p^{345} T^{10} \)
67$C_2 \wr S_5$ \( 1 + \)\(18\!\cdots\!64\)\( p T + \)\(75\!\cdots\!23\)\( p^{2} T^{2} + \)\(10\!\cdots\!00\)\( p^{3} T^{3} + \)\(27\!\cdots\!58\)\( p^{4} T^{4} + \)\(31\!\cdots\!12\)\( p^{5} T^{5} + \)\(27\!\cdots\!58\)\( p^{73} T^{6} + \)\(10\!\cdots\!00\)\( p^{141} T^{7} + \)\(75\!\cdots\!23\)\( p^{209} T^{8} + \)\(18\!\cdots\!64\)\( p^{277} T^{9} + p^{345} T^{10} \)
71$C_2 \wr S_5$ \( 1 + \)\(11\!\cdots\!40\)\( T + \)\(17\!\cdots\!95\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!10\)\( T^{4} + \)\(11\!\cdots\!48\)\( T^{5} + \)\(17\!\cdots\!10\)\( p^{69} T^{6} + \)\(15\!\cdots\!80\)\( p^{138} T^{7} + \)\(17\!\cdots\!95\)\( p^{207} T^{8} + \)\(11\!\cdots\!40\)\( p^{276} T^{9} + p^{345} T^{10} \)
73$C_2 \wr S_5$ \( 1 - \)\(33\!\cdots\!26\)\( T + \)\(16\!\cdots\!13\)\( T^{2} - \)\(45\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!38\)\( T^{4} - \)\(24\!\cdots\!88\)\( T^{5} + \)\(12\!\cdots\!38\)\( p^{69} T^{6} - \)\(45\!\cdots\!00\)\( p^{138} T^{7} + \)\(16\!\cdots\!13\)\( p^{207} T^{8} - \)\(33\!\cdots\!26\)\( p^{276} T^{9} + p^{345} T^{10} \)
79$C_2 \wr S_5$ \( 1 + \)\(36\!\cdots\!00\)\( T + \)\(35\!\cdots\!95\)\( T^{2} + \)\(80\!\cdots\!00\)\( T^{3} + \)\(47\!\cdots\!10\)\( T^{4} + \)\(80\!\cdots\!00\)\( T^{5} + \)\(47\!\cdots\!10\)\( p^{69} T^{6} + \)\(80\!\cdots\!00\)\( p^{138} T^{7} + \)\(35\!\cdots\!95\)\( p^{207} T^{8} + \)\(36\!\cdots\!00\)\( p^{276} T^{9} + p^{345} T^{10} \)
83$C_2 \wr S_5$ \( 1 - \)\(11\!\cdots\!16\)\( T + \)\(11\!\cdots\!03\)\( T^{2} - \)\(95\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!18\)\( T^{4} - \)\(34\!\cdots\!88\)\( T^{5} + \)\(55\!\cdots\!18\)\( p^{69} T^{6} - \)\(95\!\cdots\!00\)\( p^{138} T^{7} + \)\(11\!\cdots\!03\)\( p^{207} T^{8} - \)\(11\!\cdots\!16\)\( p^{276} T^{9} + p^{345} T^{10} \)
89$C_2 \wr S_5$ \( 1 + \)\(18\!\cdots\!50\)\( T + \)\(14\!\cdots\!45\)\( T^{2} + \)\(18\!\cdots\!00\)\( T^{3} + \)\(81\!\cdots\!10\)\( T^{4} + \)\(76\!\cdots\!00\)\( T^{5} + \)\(81\!\cdots\!10\)\( p^{69} T^{6} + \)\(18\!\cdots\!00\)\( p^{138} T^{7} + \)\(14\!\cdots\!45\)\( p^{207} T^{8} + \)\(18\!\cdots\!50\)\( p^{276} T^{9} + p^{345} T^{10} \)
97$C_2 \wr S_5$ \( 1 - \)\(36\!\cdots\!22\)\( T + \)\(28\!\cdots\!17\)\( T^{2} - \)\(62\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!78\)\( T^{4} - \)\(13\!\cdots\!16\)\( T^{5} + \)\(55\!\cdots\!78\)\( p^{69} T^{6} - \)\(62\!\cdots\!00\)\( p^{138} T^{7} + \)\(28\!\cdots\!17\)\( p^{207} T^{8} - \)\(36\!\cdots\!22\)\( p^{276} T^{9} + p^{345} T^{10} \)
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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

−9.367970717525828935737881679877, −9.281628834778413159500848067949, −9.167861774318045912540525546355, −8.919332482578869035286588611111, −8.570633730192148129111991215485, −7.87951929313009344124672408927, −7.85327507052930294636656172219, −7.81097344317321505890215721680, −7.26931435417719647869548790433, −6.36950965213024748672164192648, −6.27825358075793592534840856417, −5.64015087618336625205524727929, −5.62343814090929368146225190322, −5.31032359578194051714993233073, −4.84086002681136528295761321043, −4.24124945859646473349903099146, −3.86294287331884541733476343675, −3.84643910427900934906866316490, −3.38110934497104234346407917904, −3.10273195639955045316355560619, −2.44941283318093265885292563093, −2.37531976789441885212991925895, −1.74239420542143451773655268590, −1.16933579666952814635707694032, −1.12778507481101958553075888661, 0, 0, 0, 0, 0, 1.12778507481101958553075888661, 1.16933579666952814635707694032, 1.74239420542143451773655268590, 2.37531976789441885212991925895, 2.44941283318093265885292563093, 3.10273195639955045316355560619, 3.38110934497104234346407917904, 3.84643910427900934906866316490, 3.86294287331884541733476343675, 4.24124945859646473349903099146, 4.84086002681136528295761321043, 5.31032359578194051714993233073, 5.62343814090929368146225190322, 5.64015087618336625205524727929, 6.27825358075793592534840856417, 6.36950965213024748672164192648, 7.26931435417719647869548790433, 7.81097344317321505890215721680, 7.85327507052930294636656172219, 7.87951929313009344124672408927, 8.570633730192148129111991215485, 8.919332482578869035286588611111, 9.167861774318045912540525546355, 9.281628834778413159500848067949, 9.367970717525828935737881679877

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