Properties

Degree 6
Conductor $ 2^{3} $
Sign $-1$
Motivic weight 81
Primitive no
Self-dual yes
Analytic rank 3

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.29e12·2-s + 1.26e19·3-s + 7.25e24·4-s − 2.09e28·5-s − 4.15e31·6-s − 5.52e32·7-s − 1.32e37·8-s − 6.44e38·9-s + 6.92e40·10-s + 1.28e42·11-s + 9.14e43·12-s − 2.03e44·13-s + 1.82e45·14-s − 2.64e47·15-s + 2.19e49·16-s + 1.37e49·17-s + 2.12e51·18-s + 4.79e51·19-s − 1.52e53·20-s − 6.96e51·21-s − 4.22e54·22-s − 3.01e55·23-s − 1.67e56·24-s − 2.59e56·25-s + 6.69e56·26-s − 1.08e58·27-s − 4.00e57·28-s + ⋯
L(s)  = 1  − 2.12·2-s + 0.598·3-s + 3·4-s − 1.03·5-s − 1.26·6-s − 0.0328·7-s − 3.53·8-s − 1.45·9-s + 2.18·10-s + 0.853·11-s + 1.79·12-s − 0.155·13-s + 0.0696·14-s − 0.617·15-s + 15/4·16-s + 0.201·17-s + 3.08·18-s + 0.778·19-s − 3.09·20-s − 0.0196·21-s − 1.80·22-s − 2.13·23-s − 2.11·24-s − 0.628·25-s + 0.330·26-s − 1.16·27-s − 0.0984·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(82-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+81/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(81\)
character  :  induced by $\chi_{2} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(6,\ 8,\ (\ :81/2, 81/2, 81/2),\ -1)$
$L(41)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{83}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) is a polynomial of degree 6. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + p^{40} T )^{3} \)
good3$S_4\times C_2$ \( 1 - 155609391017163148 p^{4} T + \)\(55\!\cdots\!51\)\( p^{15} T^{2} - \)\(35\!\cdots\!36\)\( p^{30} T^{3} + \)\(55\!\cdots\!51\)\( p^{96} T^{4} - 155609391017163148 p^{166} T^{5} + p^{243} T^{6} \)
5$S_4\times C_2$ \( 1 + \)\(83\!\cdots\!06\)\( p^{2} T + \)\(35\!\cdots\!71\)\( p^{9} T^{2} + \)\(83\!\cdots\!68\)\( p^{19} T^{3} + \)\(35\!\cdots\!71\)\( p^{90} T^{4} + \)\(83\!\cdots\!06\)\( p^{164} T^{5} + p^{243} T^{6} \)
7$S_4\times C_2$ \( 1 + \)\(78\!\cdots\!32\)\( p T + \)\(22\!\cdots\!59\)\( p^{5} T^{2} + \)\(94\!\cdots\!48\)\( p^{12} T^{3} + \)\(22\!\cdots\!59\)\( p^{86} T^{4} + \)\(78\!\cdots\!32\)\( p^{163} T^{5} + p^{243} T^{6} \)
11$S_4\times C_2$ \( 1 - \)\(11\!\cdots\!76\)\( p T + \)\(41\!\cdots\!15\)\( p^{5} T^{2} - \)\(22\!\cdots\!20\)\( p^{10} T^{3} + \)\(41\!\cdots\!15\)\( p^{86} T^{4} - \)\(11\!\cdots\!76\)\( p^{163} T^{5} + p^{243} T^{6} \)
13$S_4\times C_2$ \( 1 + \)\(15\!\cdots\!34\)\( p T + \)\(11\!\cdots\!07\)\( p^{4} T^{2} + \)\(65\!\cdots\!16\)\( p^{8} T^{3} + \)\(11\!\cdots\!07\)\( p^{85} T^{4} + \)\(15\!\cdots\!34\)\( p^{163} T^{5} + p^{243} T^{6} \)
17$S_4\times C_2$ \( 1 - \)\(80\!\cdots\!38\)\( p T + \)\(83\!\cdots\!71\)\( p^{3} T^{2} - \)\(84\!\cdots\!84\)\( p^{7} T^{3} + \)\(83\!\cdots\!71\)\( p^{84} T^{4} - \)\(80\!\cdots\!38\)\( p^{163} T^{5} + p^{243} T^{6} \)
19$S_4\times C_2$ \( 1 - \)\(25\!\cdots\!40\)\( p T + \)\(79\!\cdots\!23\)\( p^{3} T^{2} - \)\(15\!\cdots\!20\)\( p^{7} T^{3} + \)\(79\!\cdots\!23\)\( p^{84} T^{4} - \)\(25\!\cdots\!40\)\( p^{163} T^{5} + p^{243} T^{6} \)
23$S_4\times C_2$ \( 1 + \)\(30\!\cdots\!72\)\( T + \)\(35\!\cdots\!39\)\( p T^{2} + \)\(43\!\cdots\!96\)\( p^{4} T^{3} + \)\(35\!\cdots\!39\)\( p^{82} T^{4} + \)\(30\!\cdots\!72\)\( p^{162} T^{5} + p^{243} T^{6} \)
29$S_4\times C_2$ \( 1 - \)\(21\!\cdots\!90\)\( T + \)\(20\!\cdots\!03\)\( p T^{2} + \)\(19\!\cdots\!20\)\( p^{3} T^{3} + \)\(20\!\cdots\!03\)\( p^{82} T^{4} - \)\(21\!\cdots\!90\)\( p^{162} T^{5} + p^{243} T^{6} \)
31$S_4\times C_2$ \( 1 + \)\(38\!\cdots\!04\)\( T + \)\(23\!\cdots\!15\)\( p T^{2} - \)\(53\!\cdots\!20\)\( p^{3} T^{3} + \)\(23\!\cdots\!15\)\( p^{82} T^{4} + \)\(38\!\cdots\!04\)\( p^{162} T^{5} + p^{243} T^{6} \)
37$S_4\times C_2$ \( 1 + \)\(26\!\cdots\!22\)\( p T + \)\(14\!\cdots\!47\)\( p^{2} T^{2} + \)\(15\!\cdots\!28\)\( p^{4} T^{3} + \)\(14\!\cdots\!47\)\( p^{83} T^{4} + \)\(26\!\cdots\!22\)\( p^{163} T^{5} + p^{243} T^{6} \)
41$S_4\times C_2$ \( 1 - \)\(31\!\cdots\!26\)\( T + \)\(28\!\cdots\!15\)\( p T^{2} - \)\(12\!\cdots\!20\)\( p^{2} T^{3} + \)\(28\!\cdots\!15\)\( p^{82} T^{4} - \)\(31\!\cdots\!26\)\( p^{162} T^{5} + p^{243} T^{6} \)
43$S_4\times C_2$ \( 1 - \)\(90\!\cdots\!68\)\( T + \)\(67\!\cdots\!59\)\( p T^{2} - \)\(12\!\cdots\!36\)\( p^{2} T^{3} + \)\(67\!\cdots\!59\)\( p^{82} T^{4} - \)\(90\!\cdots\!68\)\( p^{162} T^{5} + p^{243} T^{6} \)
47$S_4\times C_2$ \( 1 - \)\(16\!\cdots\!56\)\( T + \)\(31\!\cdots\!99\)\( p T^{2} - \)\(41\!\cdots\!08\)\( p^{2} T^{3} + \)\(31\!\cdots\!99\)\( p^{82} T^{4} - \)\(16\!\cdots\!56\)\( p^{162} T^{5} + p^{243} T^{6} \)
53$S_4\times C_2$ \( 1 + \)\(90\!\cdots\!62\)\( T + \)\(25\!\cdots\!19\)\( p T^{2} + \)\(26\!\cdots\!04\)\( p^{2} T^{3} + \)\(25\!\cdots\!19\)\( p^{82} T^{4} + \)\(90\!\cdots\!62\)\( p^{162} T^{5} + p^{243} T^{6} \)
59$S_4\times C_2$ \( 1 - \)\(29\!\cdots\!20\)\( p T + \)\(16\!\cdots\!63\)\( p^{3} T^{2} - \)\(63\!\cdots\!60\)\( p^{3} T^{3} + \)\(16\!\cdots\!63\)\( p^{84} T^{4} - \)\(29\!\cdots\!20\)\( p^{163} T^{5} + p^{243} T^{6} \)
61$S_4\times C_2$ \( 1 - \)\(36\!\cdots\!86\)\( T + \)\(25\!\cdots\!15\)\( p T^{2} - \)\(83\!\cdots\!20\)\( p^{2} T^{3} + \)\(25\!\cdots\!15\)\( p^{82} T^{4} - \)\(36\!\cdots\!86\)\( p^{162} T^{5} + p^{243} T^{6} \)
67$S_4\times C_2$ \( 1 + \)\(67\!\cdots\!12\)\( p T + \)\(52\!\cdots\!57\)\( p^{2} T^{2} + \)\(24\!\cdots\!36\)\( p^{3} T^{3} + \)\(52\!\cdots\!57\)\( p^{83} T^{4} + \)\(67\!\cdots\!12\)\( p^{163} T^{5} + p^{243} T^{6} \)
71$S_4\times C_2$ \( 1 + \)\(93\!\cdots\!04\)\( p T + \)\(39\!\cdots\!65\)\( p^{2} T^{2} + \)\(35\!\cdots\!80\)\( p^{3} T^{3} + \)\(39\!\cdots\!65\)\( p^{83} T^{4} + \)\(93\!\cdots\!04\)\( p^{163} T^{5} + p^{243} T^{6} \)
73$S_4\times C_2$ \( 1 - \)\(39\!\cdots\!86\)\( p T + \)\(27\!\cdots\!43\)\( p^{2} T^{2} - \)\(10\!\cdots\!92\)\( p^{3} T^{3} + \)\(27\!\cdots\!43\)\( p^{83} T^{4} - \)\(39\!\cdots\!86\)\( p^{163} T^{5} + p^{243} T^{6} \)
79$S_4\times C_2$ \( 1 + \)\(20\!\cdots\!40\)\( p T + \)\(38\!\cdots\!57\)\( p^{2} T^{2} + \)\(36\!\cdots\!20\)\( p^{3} T^{3} + \)\(38\!\cdots\!57\)\( p^{83} T^{4} + \)\(20\!\cdots\!40\)\( p^{163} T^{5} + p^{243} T^{6} \)
83$S_4\times C_2$ \( 1 + \)\(15\!\cdots\!52\)\( T + \)\(15\!\cdots\!17\)\( T^{2} + \)\(98\!\cdots\!36\)\( T^{3} + \)\(15\!\cdots\!17\)\( p^{81} T^{4} + \)\(15\!\cdots\!52\)\( p^{162} T^{5} + p^{243} T^{6} \)
89$S_4\times C_2$ \( 1 + \)\(86\!\cdots\!30\)\( T + \)\(15\!\cdots\!67\)\( T^{2} + \)\(13\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!67\)\( p^{81} T^{4} + \)\(86\!\cdots\!30\)\( p^{162} T^{5} + p^{243} T^{6} \)
97$S_4\times C_2$ \( 1 + \)\(53\!\cdots\!94\)\( T + \)\(31\!\cdots\!03\)\( T^{2} + \)\(91\!\cdots\!28\)\( T^{3} + \)\(31\!\cdots\!03\)\( p^{81} T^{4} + \)\(53\!\cdots\!94\)\( p^{162} T^{5} + p^{243} T^{6} \)
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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

−12.06536689271228537383935783759, −11.39808348000426711159930058706, −11.33276885695560761944184492947, −10.91540996963695945676992130567, −10.08397471696831798285837330471, −9.679392999521842746831868360653, −9.494518375178867106013539250488, −8.723154818811695291405901533350, −8.435405160896127290094455419338, −8.421978727899651524210412131409, −7.47186979821420211745410067144, −7.43518670554110222039130674780, −7.20307415043015495547717835981, −6.01276076126792328979619132152, −5.94217770458994949981878820727, −5.70631700038199326996320614006, −4.58984046791382573935465836002, −3.93042105477405562264675147715, −3.60622958348746428171113004427, −3.24056467226841452092943992394, −2.47516467190448767621510586119, −2.36424398822134285468909051906, −1.85240261465840691084710946396, −1.11177672784169054483642633318, −1.00739705264753796023107134327, 0, 0, 0, 1.00739705264753796023107134327, 1.11177672784169054483642633318, 1.85240261465840691084710946396, 2.36424398822134285468909051906, 2.47516467190448767621510586119, 3.24056467226841452092943992394, 3.60622958348746428171113004427, 3.93042105477405562264675147715, 4.58984046791382573935465836002, 5.70631700038199326996320614006, 5.94217770458994949981878820727, 6.01276076126792328979619132152, 7.20307415043015495547717835981, 7.43518670554110222039130674780, 7.47186979821420211745410067144, 8.421978727899651524210412131409, 8.435405160896127290094455419338, 8.723154818811695291405901533350, 9.494518375178867106013539250488, 9.679392999521842746831868360653, 10.08397471696831798285837330471, 10.91540996963695945676992130567, 11.33276885695560761944184492947, 11.39808348000426711159930058706, 12.06536689271228537383935783759

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