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Properties

Label 6-3e3-1.1-c33e3-0-0
Degree $6$
Conductor $27$
Sign $1$
Analytic cond. $8863.12$
Root an. cond. $4.54915$
Motivic weight $33$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.12e4·2-s + 1.29e8·3-s − 1.25e10·4-s + 5.12e10·5-s + 5.32e12·6-s + 7.61e13·7-s − 2.75e14·8-s + 1.11e16·9-s + 2.11e15·10-s − 1.03e17·11-s − 1.62e18·12-s + 1.91e18·13-s + 3.13e18·14-s + 6.61e18·15-s + 8.00e19·16-s − 1.69e19·17-s + 4.58e20·18-s + 2.63e21·19-s − 6.45e20·20-s + 9.82e21·21-s − 4.26e21·22-s + 7.95e22·23-s − 3.55e22·24-s + 3.82e22·25-s + 7.89e22·26-s + 7.97e23·27-s − 9.58e23·28-s + ⋯
L(s)  = 1  + 0.444·2-s + 1.73·3-s − 1.46·4-s + 0.150·5-s + 0.769·6-s + 0.865·7-s − 0.345·8-s + 2·9-s + 0.0667·10-s − 0.679·11-s − 2.53·12-s + 0.798·13-s + 0.384·14-s + 0.260·15-s + 1.08·16-s − 0.0844·17-s + 0.889·18-s + 2.09·19-s − 0.220·20-s + 1.49·21-s − 0.301·22-s + 2.70·23-s − 0.598·24-s + 0.328·25-s + 0.354·26-s + 1.92·27-s − 1.26·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+33/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $1$
Analytic conductor: \(8863.12\)
Root analytic conductor: \(4.54915\)
Motivic weight: \(33\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 27,\ (\ :33/2, 33/2, 33/2),\ 1)\)

Particular Values

\(L(17)\) \(\approx\) \(10.75814646\)
\(L(\frac12)\) \(\approx\) \(10.75814646\)
\(L(\frac{35}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 - p^{16} T )^{3} \)
good2$S_4\times C_2$ \( 1 - 20601 p T + 446482443 p^{5} T^{2} - 101574513081 p^{13} T^{3} + 446482443 p^{38} T^{4} - 20601 p^{67} T^{5} + p^{99} T^{6} \)
5$S_4\times C_2$ \( 1 - 10252364778 p T - 56999390506149574101 p^{4} T^{2} + \)\(15\!\cdots\!64\)\( p^{8} T^{3} - 56999390506149574101 p^{37} T^{4} - 10252364778 p^{67} T^{5} + p^{99} T^{6} \)
7$S_4\times C_2$ \( 1 - 10873390573608 p T + \)\(45\!\cdots\!85\)\( p^{2} T^{2} - \)\(91\!\cdots\!36\)\( p^{6} T^{3} + \)\(45\!\cdots\!85\)\( p^{35} T^{4} - 10873390573608 p^{67} T^{5} + p^{99} T^{6} \)
11$S_4\times C_2$ \( 1 + 9411249898677780 p T + \)\(43\!\cdots\!41\)\( p^{2} T^{2} + \)\(24\!\cdots\!16\)\( p^{3} T^{3} + \)\(43\!\cdots\!41\)\( p^{35} T^{4} + 9411249898677780 p^{67} T^{5} + p^{99} T^{6} \)
13$S_4\times C_2$ \( 1 - 1915412601357848490 T + \)\(50\!\cdots\!87\)\( p T^{2} - \)\(11\!\cdots\!16\)\( p^{3} T^{3} + \)\(50\!\cdots\!87\)\( p^{34} T^{4} - 1915412601357848490 p^{66} T^{5} + p^{99} T^{6} \)
17$S_4\times C_2$ \( 1 + 16951254031996646346 T + \)\(64\!\cdots\!99\)\( p T^{2} + \)\(39\!\cdots\!48\)\( p^{2} T^{3} + \)\(64\!\cdots\!99\)\( p^{34} T^{4} + 16951254031996646346 p^{66} T^{5} + p^{99} T^{6} \)
19$S_4\times C_2$ \( 1 - \)\(26\!\cdots\!64\)\( T + \)\(23\!\cdots\!23\)\( p T^{2} - \)\(14\!\cdots\!32\)\( p^{2} T^{3} + \)\(23\!\cdots\!23\)\( p^{34} T^{4} - \)\(26\!\cdots\!64\)\( p^{66} T^{5} + p^{99} T^{6} \)
23$S_4\times C_2$ \( 1 - \)\(15\!\cdots\!92\)\( p^{2} T + \)\(85\!\cdots\!65\)\( p^{2} T^{2} - \)\(12\!\cdots\!24\)\( p^{3} T^{3} + \)\(85\!\cdots\!65\)\( p^{35} T^{4} - \)\(15\!\cdots\!92\)\( p^{68} T^{5} + p^{99} T^{6} \)
29$S_4\times C_2$ \( 1 - \)\(40\!\cdots\!02\)\( p T + \)\(46\!\cdots\!07\)\( p^{2} T^{2} - \)\(10\!\cdots\!16\)\( p^{3} T^{3} + \)\(46\!\cdots\!07\)\( p^{35} T^{4} - \)\(40\!\cdots\!02\)\( p^{67} T^{5} + p^{99} T^{6} \)
31$S_4\times C_2$ \( 1 + \)\(55\!\cdots\!96\)\( p T + \)\(29\!\cdots\!97\)\( p^{2} T^{2} + \)\(16\!\cdots\!52\)\( p^{3} T^{3} + \)\(29\!\cdots\!97\)\( p^{35} T^{4} + \)\(55\!\cdots\!96\)\( p^{67} T^{5} + p^{99} T^{6} \)
37$S_4\times C_2$ \( 1 - \)\(23\!\cdots\!86\)\( T + \)\(11\!\cdots\!95\)\( T^{2} - \)\(65\!\cdots\!24\)\( T^{3} + \)\(11\!\cdots\!95\)\( p^{33} T^{4} - \)\(23\!\cdots\!86\)\( p^{66} T^{5} + p^{99} T^{6} \)
41$S_4\times C_2$ \( 1 - \)\(42\!\cdots\!14\)\( T + \)\(39\!\cdots\!27\)\( T^{2} - \)\(10\!\cdots\!68\)\( T^{3} + \)\(39\!\cdots\!27\)\( p^{33} T^{4} - \)\(42\!\cdots\!14\)\( p^{66} T^{5} + p^{99} T^{6} \)
43$S_4\times C_2$ \( 1 - \)\(93\!\cdots\!44\)\( T + \)\(46\!\cdots\!71\)\( p T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(46\!\cdots\!71\)\( p^{34} T^{4} - \)\(93\!\cdots\!44\)\( p^{66} T^{5} + p^{99} T^{6} \)
47$S_4\times C_2$ \( 1 + \)\(87\!\cdots\!52\)\( T + \)\(64\!\cdots\!57\)\( T^{2} + \)\(26\!\cdots\!60\)\( T^{3} + \)\(64\!\cdots\!57\)\( p^{33} T^{4} + \)\(87\!\cdots\!52\)\( p^{66} T^{5} + p^{99} T^{6} \)
53$S_4\times C_2$ \( 1 + \)\(11\!\cdots\!02\)\( T + \)\(18\!\cdots\!15\)\( T^{2} + \)\(16\!\cdots\!12\)\( T^{3} + \)\(18\!\cdots\!15\)\( p^{33} T^{4} + \)\(11\!\cdots\!02\)\( p^{66} T^{5} + p^{99} T^{6} \)
59$S_4\times C_2$ \( 1 - \)\(12\!\cdots\!56\)\( T + \)\(82\!\cdots\!37\)\( T^{2} - \)\(68\!\cdots\!48\)\( T^{3} + \)\(82\!\cdots\!37\)\( p^{33} T^{4} - \)\(12\!\cdots\!56\)\( p^{66} T^{5} + p^{99} T^{6} \)
61$S_4\times C_2$ \( 1 - \)\(36\!\cdots\!62\)\( T + \)\(12\!\cdots\!59\)\( T^{2} - \)\(22\!\cdots\!36\)\( T^{3} + \)\(12\!\cdots\!59\)\( p^{33} T^{4} - \)\(36\!\cdots\!62\)\( p^{66} T^{5} + p^{99} T^{6} \)
67$S_4\times C_2$ \( 1 + \)\(15\!\cdots\!96\)\( T + \)\(11\!\cdots\!33\)\( T^{2} + \)\(29\!\cdots\!72\)\( T^{3} + \)\(11\!\cdots\!33\)\( p^{33} T^{4} + \)\(15\!\cdots\!96\)\( p^{66} T^{5} + p^{99} T^{6} \)
71$S_4\times C_2$ \( 1 + \)\(66\!\cdots\!84\)\( T + \)\(51\!\cdots\!85\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} + \)\(51\!\cdots\!85\)\( p^{33} T^{4} + \)\(66\!\cdots\!84\)\( p^{66} T^{5} + p^{99} T^{6} \)
73$S_4\times C_2$ \( 1 - \)\(17\!\cdots\!18\)\( T + \)\(16\!\cdots\!55\)\( T^{2} - \)\(10\!\cdots\!88\)\( T^{3} + \)\(16\!\cdots\!55\)\( p^{33} T^{4} - \)\(17\!\cdots\!18\)\( p^{66} T^{5} + p^{99} T^{6} \)
79$S_4\times C_2$ \( 1 - \)\(47\!\cdots\!40\)\( T + \)\(18\!\cdots\!17\)\( T^{2} - \)\(41\!\cdots\!20\)\( T^{3} + \)\(18\!\cdots\!17\)\( p^{33} T^{4} - \)\(47\!\cdots\!40\)\( p^{66} T^{5} + p^{99} T^{6} \)
83$S_4\times C_2$ \( 1 + \)\(10\!\cdots\!64\)\( T + \)\(92\!\cdots\!69\)\( T^{2} + \)\(46\!\cdots\!56\)\( T^{3} + \)\(92\!\cdots\!69\)\( p^{33} T^{4} + \)\(10\!\cdots\!64\)\( p^{66} T^{5} + p^{99} T^{6} \)
89$S_4\times C_2$ \( 1 - \)\(28\!\cdots\!94\)\( T + \)\(81\!\cdots\!67\)\( T^{2} - \)\(12\!\cdots\!72\)\( T^{3} + \)\(81\!\cdots\!67\)\( p^{33} T^{4} - \)\(28\!\cdots\!94\)\( p^{66} T^{5} + p^{99} T^{6} \)
97$S_4\times C_2$ \( 1 - \)\(81\!\cdots\!54\)\( T + \)\(65\!\cdots\!03\)\( T^{2} - \)\(25\!\cdots\!48\)\( T^{3} + \)\(65\!\cdots\!03\)\( p^{33} T^{4} - \)\(81\!\cdots\!54\)\( p^{66} T^{5} + p^{99} T^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.55348101968547170877074281033, −14.63536795233339899390254528590, −14.31315554302090612309140912103, −14.15235717831377510403502321747, −13.28737009726142012692179128624, −13.07749128373282373052612590530, −12.80876022028587968584738325113, −11.43137430135910842444497293478, −10.95430614240344230614971354364, −10.00653831383892929429930472047, −9.421622027924170441613547079917, −9.004134883315293782655764648184, −8.534621937754753143385854627653, −7.82439988114468064056682900960, −7.52954939004550486519450736355, −6.58434272921466909657985326117, −5.25159371430610392723995867494, −4.90374921472849735004398444109, −4.62157657850927502396275335657, −3.47687482460424071039093416890, −3.40763278698834802794877756713, −2.64063185623969695195543043138, −1.75188171337547453550145898254, −1.05114857384417310432380055168, −0.73785809413213367116209908545, 0.73785809413213367116209908545, 1.05114857384417310432380055168, 1.75188171337547453550145898254, 2.64063185623969695195543043138, 3.40763278698834802794877756713, 3.47687482460424071039093416890, 4.62157657850927502396275335657, 4.90374921472849735004398444109, 5.25159371430610392723995867494, 6.58434272921466909657985326117, 7.52954939004550486519450736355, 7.82439988114468064056682900960, 8.534621937754753143385854627653, 9.004134883315293782655764648184, 9.421622027924170441613547079917, 10.00653831383892929429930472047, 10.95430614240344230614971354364, 11.43137430135910842444497293478, 12.80876022028587968584738325113, 13.07749128373282373052612590530, 13.28737009726142012692179128624, 14.15235717831377510403502321747, 14.31315554302090612309140912103, 14.63536795233339899390254528590, 15.55348101968547170877074281033

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