Properties

Degree 6
Conductor $ 1 $
Sign $-1$
Motivic weight 41
Primitive no
Self-dual yes
Analytic rank 3

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.44e5·2-s − 1.08e10·3-s − 1.03e11·4-s − 2.12e14·5-s + 3.72e15·6-s + 5.78e16·7-s − 8.83e17·8-s + 1.04e19·9-s + 7.31e19·10-s − 3.06e21·11-s + 1.11e21·12-s − 9.85e22·13-s − 1.99e22·14-s + 2.29e24·15-s + 1.82e23·16-s + 3.55e25·17-s − 3.61e24·18-s − 2.33e26·19-s + 2.19e25·20-s − 6.26e26·21-s + 1.05e27·22-s + 2.81e27·23-s + 9.55e27·24-s − 5.19e28·25-s + 3.39e28·26-s + 4.54e29·27-s − 5.97e27·28-s + ⋯
L(s)  = 1  − 0.232·2-s − 1.79·3-s − 0.0469·4-s − 0.995·5-s + 0.416·6-s + 0.274·7-s − 0.270·8-s + 0.287·9-s + 0.231·10-s − 1.37·11-s + 0.0841·12-s − 1.43·13-s − 0.0637·14-s + 1.78·15-s + 0.0376·16-s + 2.12·17-s − 0.0667·18-s − 1.42·19-s + 0.0467·20-s − 0.491·21-s + 0.319·22-s + 0.341·23-s + 0.485·24-s − 1.14·25-s + 0.334·26-s + 2.06·27-s − 0.0128·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\,\Lambda(42-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+41/2)^{3} \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(41\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(6,\ 1,\ (\ :41/2, 41/2, 41/2),\ -1)$
$L(21)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{43}{2})$   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 6.
$p$$\Gal(F_p)$$F_p$
good2$S_4\times C_2$ \( 1 + 21543 p^{4} T + 27110271 p^{13} T^{2} + 29668758195 p^{25} T^{3} + 27110271 p^{54} T^{4} + 21543 p^{86} T^{5} + p^{123} T^{6} \)
3$S_4\times C_2$ \( 1 + 1202328116 p^{2} T + 48750573799961899 p^{7} T^{2} + \)\(13\!\cdots\!40\)\( p^{16} T^{3} + 48750573799961899 p^{48} T^{4} + 1202328116 p^{84} T^{5} + p^{123} T^{6} \)
5$S_4\times C_2$ \( 1 + 8492094011262 p^{2} T + \)\(12\!\cdots\!71\)\( p^{7} T^{2} + \)\(16\!\cdots\!68\)\( p^{13} T^{3} + \)\(12\!\cdots\!71\)\( p^{48} T^{4} + 8492094011262 p^{84} T^{5} + p^{123} T^{6} \)
7$S_4\times C_2$ \( 1 - 8268345179748456 p T + \)\(34\!\cdots\!99\)\( p^{3} T^{2} - \)\(46\!\cdots\!00\)\( p^{6} T^{3} + \)\(34\!\cdots\!99\)\( p^{44} T^{4} - 8268345179748456 p^{83} T^{5} + p^{123} T^{6} \)
11$S_4\times C_2$ \( 1 + \)\(30\!\cdots\!64\)\( T + \)\(80\!\cdots\!65\)\( p^{2} T^{2} + \)\(10\!\cdots\!80\)\( p^{5} T^{3} + \)\(80\!\cdots\!65\)\( p^{43} T^{4} + \)\(30\!\cdots\!64\)\( p^{82} T^{5} + p^{123} T^{6} \)
13$S_4\times C_2$ \( 1 + \)\(75\!\cdots\!38\)\( p T + \)\(25\!\cdots\!39\)\( p^{3} T^{2} + \)\(26\!\cdots\!60\)\( p^{5} T^{3} + \)\(25\!\cdots\!39\)\( p^{44} T^{4} + \)\(75\!\cdots\!38\)\( p^{83} T^{5} + p^{123} T^{6} \)
17$S_4\times C_2$ \( 1 - \)\(20\!\cdots\!06\)\( p T + \)\(24\!\cdots\!39\)\( p^{3} T^{2} - \)\(14\!\cdots\!40\)\( p^{5} T^{3} + \)\(24\!\cdots\!39\)\( p^{44} T^{4} - \)\(20\!\cdots\!06\)\( p^{83} T^{5} + p^{123} T^{6} \)
19$S_4\times C_2$ \( 1 + \)\(12\!\cdots\!20\)\( p T + \)\(24\!\cdots\!37\)\( p^{2} T^{2} + \)\(90\!\cdots\!40\)\( p^{4} T^{3} + \)\(24\!\cdots\!37\)\( p^{43} T^{4} + \)\(12\!\cdots\!20\)\( p^{83} T^{5} + p^{123} T^{6} \)
23$S_4\times C_2$ \( 1 - \)\(28\!\cdots\!56\)\( T + \)\(37\!\cdots\!11\)\( p T^{2} - \)\(13\!\cdots\!20\)\( p^{2} T^{3} + \)\(37\!\cdots\!11\)\( p^{42} T^{4} - \)\(28\!\cdots\!56\)\( p^{82} T^{5} + p^{123} T^{6} \)
29$S_4\times C_2$ \( 1 + \)\(43\!\cdots\!30\)\( p T + \)\(18\!\cdots\!07\)\( p^{2} T^{2} - \)\(16\!\cdots\!60\)\( p^{3} T^{3} + \)\(18\!\cdots\!07\)\( p^{43} T^{4} + \)\(43\!\cdots\!30\)\( p^{83} T^{5} + p^{123} T^{6} \)
31$S_4\times C_2$ \( 1 + \)\(55\!\cdots\!04\)\( T + \)\(11\!\cdots\!15\)\( p T^{2} + \)\(48\!\cdots\!80\)\( p^{2} T^{3} + \)\(11\!\cdots\!15\)\( p^{42} T^{4} + \)\(55\!\cdots\!04\)\( p^{82} T^{5} + p^{123} T^{6} \)
37$S_4\times C_2$ \( 1 - \)\(13\!\cdots\!06\)\( p T + \)\(34\!\cdots\!03\)\( p^{2} T^{2} - \)\(46\!\cdots\!80\)\( p^{3} T^{3} + \)\(34\!\cdots\!03\)\( p^{43} T^{4} - \)\(13\!\cdots\!06\)\( p^{83} T^{5} + p^{123} T^{6} \)
41$S_4\times C_2$ \( 1 + \)\(31\!\cdots\!74\)\( T + \)\(70\!\cdots\!15\)\( T^{2} + \)\(92\!\cdots\!80\)\( T^{3} + \)\(70\!\cdots\!15\)\( p^{41} T^{4} + \)\(31\!\cdots\!74\)\( p^{82} T^{5} + p^{123} T^{6} \)
43$S_4\times C_2$ \( 1 - \)\(14\!\cdots\!56\)\( T + \)\(27\!\cdots\!93\)\( T^{2} - \)\(28\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!93\)\( p^{41} T^{4} - \)\(14\!\cdots\!56\)\( p^{82} T^{5} + p^{123} T^{6} \)
47$S_4\times C_2$ \( 1 + \)\(63\!\cdots\!68\)\( T + \)\(22\!\cdots\!57\)\( T^{2} + \)\(51\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!57\)\( p^{41} T^{4} + \)\(63\!\cdots\!68\)\( p^{82} T^{5} + p^{123} T^{6} \)
53$S_4\times C_2$ \( 1 - \)\(79\!\cdots\!06\)\( T + \)\(59\!\cdots\!63\)\( T^{2} - \)\(16\!\cdots\!60\)\( T^{3} + \)\(59\!\cdots\!63\)\( p^{41} T^{4} - \)\(79\!\cdots\!06\)\( p^{82} T^{5} + p^{123} T^{6} \)
59$S_4\times C_2$ \( 1 - \)\(19\!\cdots\!60\)\( T + \)\(77\!\cdots\!77\)\( T^{2} - \)\(18\!\cdots\!80\)\( T^{3} + \)\(77\!\cdots\!77\)\( p^{41} T^{4} - \)\(19\!\cdots\!60\)\( p^{82} T^{5} + p^{123} T^{6} \)
61$S_4\times C_2$ \( 1 - \)\(87\!\cdots\!86\)\( T + \)\(62\!\cdots\!15\)\( T^{2} - \)\(27\!\cdots\!20\)\( T^{3} + \)\(62\!\cdots\!15\)\( p^{41} T^{4} - \)\(87\!\cdots\!86\)\( p^{82} T^{5} + p^{123} T^{6} \)
67$S_4\times C_2$ \( 1 - \)\(11\!\cdots\!52\)\( T + \)\(78\!\cdots\!57\)\( T^{2} + \)\(10\!\cdots\!20\)\( T^{3} + \)\(78\!\cdots\!57\)\( p^{41} T^{4} - \)\(11\!\cdots\!52\)\( p^{82} T^{5} + p^{123} T^{6} \)
71$S_4\times C_2$ \( 1 + \)\(14\!\cdots\!84\)\( T + \)\(22\!\cdots\!65\)\( T^{2} + \)\(18\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!65\)\( p^{41} T^{4} + \)\(14\!\cdots\!84\)\( p^{82} T^{5} + p^{123} T^{6} \)
73$S_4\times C_2$ \( 1 - \)\(45\!\cdots\!06\)\( T + \)\(18\!\cdots\!03\)\( T^{2} - \)\(58\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!03\)\( p^{41} T^{4} - \)\(45\!\cdots\!06\)\( p^{82} T^{5} + p^{123} T^{6} \)
79$S_4\times C_2$ \( 1 + \)\(52\!\cdots\!20\)\( T + \)\(83\!\cdots\!37\)\( T^{2} + \)\(82\!\cdots\!60\)\( T^{3} + \)\(83\!\cdots\!37\)\( p^{41} T^{4} + \)\(52\!\cdots\!20\)\( p^{82} T^{5} + p^{123} T^{6} \)
83$S_4\times C_2$ \( 1 + \)\(61\!\cdots\!44\)\( T + \)\(61\!\cdots\!73\)\( T^{2} + \)\(12\!\cdots\!60\)\( T^{3} + \)\(61\!\cdots\!73\)\( p^{41} T^{4} + \)\(61\!\cdots\!44\)\( p^{82} T^{5} + p^{123} T^{6} \)
89$S_4\times C_2$ \( 1 + \)\(14\!\cdots\!10\)\( T + \)\(20\!\cdots\!67\)\( T^{2} + \)\(53\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!67\)\( p^{41} T^{4} + \)\(14\!\cdots\!10\)\( p^{82} T^{5} + p^{123} T^{6} \)
97$S_4\times C_2$ \( 1 - \)\(11\!\cdots\!82\)\( T + \)\(11\!\cdots\!07\)\( T^{2} - \)\(67\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!07\)\( p^{41} T^{4} - \)\(11\!\cdots\!82\)\( p^{82} T^{5} + p^{123} T^{6} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−20.92918365859342746781592699153, −19.75391418162021441356110899221, −19.21081244188731159242930442973, −18.50764456138899202751180438173, −17.48033887266875676383394906611, −17.38772926926616988119887193712, −16.49379218817783972206351253264, −16.09743473091875402325370095204, −14.86679917546178749798549819934, −14.61898485256595024349163130251, −13.05589820423207455440841210847, −12.31317004652992076923206127811, −11.55581316214425608002854778495, −11.51375731605963753238051158912, −10.34711477872763083081943080374, −9.886708198321619448957698846734, −8.152569295302234085979200853491, −8.050931476300633943553916436279, −6.86604869845762650166994838645, −5.94895438381446622073970476791, −5.09857215754134194960555388559, −5.00816212639601221312502909711, −3.53110805246023592341654273008, −2.71348491106079875612499419113, −1.49703521748310075275001255323, 0, 0, 0, 1.49703521748310075275001255323, 2.71348491106079875612499419113, 3.53110805246023592341654273008, 5.00816212639601221312502909711, 5.09857215754134194960555388559, 5.94895438381446622073970476791, 6.86604869845762650166994838645, 8.050931476300633943553916436279, 8.152569295302234085979200853491, 9.886708198321619448957698846734, 10.34711477872763083081943080374, 11.51375731605963753238051158912, 11.55581316214425608002854778495, 12.31317004652992076923206127811, 13.05589820423207455440841210847, 14.61898485256595024349163130251, 14.86679917546178749798549819934, 16.09743473091875402325370095204, 16.49379218817783972206351253264, 17.38772926926616988119887193712, 17.48033887266875676383394906611, 18.50764456138899202751180438173, 19.21081244188731159242930442973, 19.75391418162021441356110899221, 20.92918365859342746781592699153

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