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Properties

Label 6-3e3-1.1-c33e3-0-1
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 $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.36e5·2-s − 1.29e8·3-s + 3.74e9·4-s − 2.60e11·5-s − 1.76e13·6-s + 1.07e13·7-s − 1.27e14·8-s + 1.11e16·9-s − 3.55e16·10-s − 3.45e17·11-s − 4.83e17·12-s + 7.30e17·13-s + 1.47e18·14-s + 3.36e19·15-s − 1.02e19·16-s − 2.41e20·17-s + 1.51e21·18-s − 3.02e20·19-s − 9.74e20·20-s − 1.38e21·21-s − 4.71e22·22-s − 6.21e22·23-s + 1.65e22·24-s − 1.97e23·25-s + 9.97e22·26-s − 7.97e23·27-s + 4.02e22·28-s + ⋯
L(s)  = 1  + 1.47·2-s − 1.73·3-s + 0.435·4-s − 0.763·5-s − 2.55·6-s + 0.122·7-s − 0.160·8-s + 2·9-s − 1.12·10-s − 2.26·11-s − 0.754·12-s + 0.304·13-s + 0.180·14-s + 1.32·15-s − 0.139·16-s − 1.20·17-s + 2.94·18-s − 0.240·19-s − 0.332·20-s − 0.211·21-s − 3.34·22-s − 2.11·23-s + 0.278·24-s − 1.69·25-s + 0.448·26-s − 1.92·27-s + 0.0533·28-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: \(3\)
Selberg data: \((6,\ 27,\ (\ :33/2, 33/2, 33/2),\ -1)\)

Particular Values

\(L(17)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{35}{2})\) 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$S_4\times C_2$ \( 1 - 34155 p^{2} T + 14572887 p^{10} T^{2} - 5339268063 p^{18} T^{3} + 14572887 p^{43} T^{4} - 34155 p^{68} T^{5} + p^{99} T^{6} \)
5$S_4\times C_2$ \( 1 + 260488036134 T + \)\(10\!\cdots\!67\)\( p^{2} T^{2} + \)\(73\!\cdots\!48\)\( p^{7} T^{3} + \)\(10\!\cdots\!67\)\( p^{35} T^{4} + 260488036134 p^{66} T^{5} + p^{99} T^{6} \)
7$S_4\times C_2$ \( 1 - 1537242698976 p T + \)\(52\!\cdots\!89\)\( p^{2} T^{2} - \)\(50\!\cdots\!52\)\( p^{6} T^{3} + \)\(52\!\cdots\!89\)\( p^{35} T^{4} - 1537242698976 p^{67} T^{5} + p^{99} T^{6} \)
11$S_4\times C_2$ \( 1 + 2855226191742036 p^{2} T + \)\(53\!\cdots\!93\)\( p^{2} T^{2} + \)\(68\!\cdots\!80\)\( p^{3} T^{3} + \)\(53\!\cdots\!93\)\( p^{35} T^{4} + 2855226191742036 p^{68} T^{5} + p^{99} T^{6} \)
13$S_4\times C_2$ \( 1 - 730047702981793722 T + \)\(45\!\cdots\!83\)\( p T^{2} + \)\(25\!\cdots\!32\)\( p^{3} T^{3} + \)\(45\!\cdots\!83\)\( p^{34} T^{4} - 730047702981793722 p^{66} T^{5} + p^{99} T^{6} \)
17$S_4\times C_2$ \( 1 + \)\(24\!\cdots\!34\)\( T + \)\(13\!\cdots\!63\)\( T^{2} + \)\(11\!\cdots\!04\)\( p T^{3} + \)\(13\!\cdots\!63\)\( p^{33} T^{4} + \)\(24\!\cdots\!34\)\( p^{66} T^{5} + p^{99} T^{6} \)
19$S_4\times C_2$ \( 1 + \)\(30\!\cdots\!32\)\( T + \)\(24\!\cdots\!27\)\( p T^{2} + \)\(26\!\cdots\!96\)\( p^{2} T^{3} + \)\(24\!\cdots\!27\)\( p^{34} T^{4} + \)\(30\!\cdots\!32\)\( p^{66} T^{5} + p^{99} T^{6} \)
23$S_4\times C_2$ \( 1 + \)\(62\!\cdots\!24\)\( T + \)\(14\!\cdots\!03\)\( p T^{2} + \)\(19\!\cdots\!96\)\( p^{2} T^{3} + \)\(14\!\cdots\!03\)\( p^{34} T^{4} + \)\(62\!\cdots\!24\)\( p^{66} T^{5} + p^{99} T^{6} \)
29$S_4\times C_2$ \( 1 + \)\(12\!\cdots\!94\)\( T + \)\(16\!\cdots\!19\)\( p T^{2} + \)\(55\!\cdots\!52\)\( p^{2} T^{3} + \)\(16\!\cdots\!19\)\( p^{34} T^{4} + \)\(12\!\cdots\!94\)\( p^{66} T^{5} + p^{99} T^{6} \)
31$S_4\times C_2$ \( 1 - \)\(35\!\cdots\!92\)\( p T + \)\(90\!\cdots\!93\)\( p^{2} T^{2} - \)\(13\!\cdots\!04\)\( p^{3} T^{3} + \)\(90\!\cdots\!93\)\( p^{35} T^{4} - \)\(35\!\cdots\!92\)\( p^{67} T^{5} + p^{99} T^{6} \)
37$S_4\times C_2$ \( 1 + \)\(47\!\cdots\!18\)\( T + \)\(52\!\cdots\!51\)\( T^{2} - \)\(22\!\cdots\!08\)\( T^{3} + \)\(52\!\cdots\!51\)\( p^{33} T^{4} + \)\(47\!\cdots\!18\)\( p^{66} T^{5} + p^{99} T^{6} \)
41$S_4\times C_2$ \( 1 + \)\(51\!\cdots\!18\)\( T + \)\(18\!\cdots\!83\)\( T^{2} + \)\(26\!\cdots\!56\)\( T^{3} + \)\(18\!\cdots\!83\)\( p^{33} T^{4} + \)\(51\!\cdots\!18\)\( p^{66} T^{5} + p^{99} T^{6} \)
43$S_4\times C_2$ \( 1 + \)\(24\!\cdots\!76\)\( T + \)\(13\!\cdots\!33\)\( T^{2} + \)\(42\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!33\)\( p^{33} T^{4} + \)\(24\!\cdots\!76\)\( p^{66} T^{5} + p^{99} T^{6} \)
47$S_4\times C_2$ \( 1 - \)\(19\!\cdots\!28\)\( T + \)\(37\!\cdots\!17\)\( T^{2} - \)\(43\!\cdots\!20\)\( T^{3} + \)\(37\!\cdots\!17\)\( p^{33} T^{4} - \)\(19\!\cdots\!28\)\( p^{66} T^{5} + p^{99} T^{6} \)
53$S_4\times C_2$ \( 1 - \)\(80\!\cdots\!66\)\( T + \)\(17\!\cdots\!19\)\( T^{2} - \)\(16\!\cdots\!36\)\( T^{3} + \)\(17\!\cdots\!19\)\( p^{33} T^{4} - \)\(80\!\cdots\!66\)\( p^{66} T^{5} + p^{99} T^{6} \)
59$S_4\times C_2$ \( 1 + \)\(24\!\cdots\!68\)\( T + \)\(92\!\cdots\!73\)\( T^{2} + \)\(13\!\cdots\!64\)\( T^{3} + \)\(92\!\cdots\!73\)\( p^{33} T^{4} + \)\(24\!\cdots\!68\)\( p^{66} T^{5} + p^{99} T^{6} \)
61$S_4\times C_2$ \( 1 - \)\(37\!\cdots\!18\)\( T + \)\(22\!\cdots\!99\)\( T^{2} - \)\(75\!\cdots\!24\)\( T^{3} + \)\(22\!\cdots\!99\)\( p^{33} T^{4} - \)\(37\!\cdots\!18\)\( p^{66} T^{5} + p^{99} T^{6} \)
67$S_4\times C_2$ \( 1 - \)\(10\!\cdots\!96\)\( T + \)\(24\!\cdots\!33\)\( T^{2} - \)\(23\!\cdots\!72\)\( T^{3} + \)\(24\!\cdots\!33\)\( p^{33} T^{4} - \)\(10\!\cdots\!96\)\( p^{66} T^{5} + p^{99} T^{6} \)
71$S_4\times C_2$ \( 1 + \)\(13\!\cdots\!84\)\( T + \)\(94\!\cdots\!85\)\( T^{2} + \)\(40\!\cdots\!00\)\( T^{3} + \)\(94\!\cdots\!85\)\( p^{33} T^{4} + \)\(13\!\cdots\!84\)\( p^{66} T^{5} + p^{99} T^{6} \)
73$S_4\times C_2$ \( 1 + \)\(21\!\cdots\!94\)\( T + \)\(76\!\cdots\!79\)\( T^{2} + \)\(14\!\cdots\!04\)\( T^{3} + \)\(76\!\cdots\!79\)\( p^{33} T^{4} + \)\(21\!\cdots\!94\)\( p^{66} T^{5} + p^{99} T^{6} \)
79$S_4\times C_2$ \( 1 - \)\(22\!\cdots\!00\)\( T + \)\(70\!\cdots\!17\)\( T^{2} - \)\(23\!\cdots\!00\)\( T^{3} + \)\(70\!\cdots\!17\)\( p^{33} T^{4} - \)\(22\!\cdots\!00\)\( p^{66} T^{5} + p^{99} T^{6} \)
83$S_4\times C_2$ \( 1 + \)\(36\!\cdots\!60\)\( T + \)\(97\!\cdots\!57\)\( T^{2} - \)\(13\!\cdots\!24\)\( T^{3} + \)\(97\!\cdots\!57\)\( p^{33} T^{4} + \)\(36\!\cdots\!60\)\( p^{66} T^{5} + p^{99} T^{6} \)
89$S_4\times C_2$ \( 1 + \)\(30\!\cdots\!22\)\( T + \)\(93\!\cdots\!23\)\( T^{2} + \)\(13\!\cdots\!56\)\( T^{3} + \)\(93\!\cdots\!23\)\( p^{33} T^{4} + \)\(30\!\cdots\!22\)\( p^{66} T^{5} + p^{99} T^{6} \)
97$S_4\times C_2$ \( 1 + \)\(87\!\cdots\!94\)\( T + \)\(67\!\cdots\!43\)\( T^{2} + \)\(41\!\cdots\!68\)\( T^{3} + \)\(67\!\cdots\!43\)\( p^{33} T^{4} + \)\(87\!\cdots\!94\)\( p^{66} T^{5} + p^{99} T^{6} \)
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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

−16.01879125677040186893938816810, −15.94348163227884232240622695556, −15.57047626467868140839571500443, −15.00328542007060913660329522520, −13.66674508064452252117834308574, −13.53385783499617849588113184065, −13.28777455876486443575413651794, −12.28796642080302454944614244056, −12.06394838807788824159713110598, −11.44877366421240981001886355255, −10.86896361566120947149658507482, −10.21922508713911823311475238587, −9.873132899281669554709942652041, −8.337221878198501816246778948455, −7.988815170797356720565846935624, −7.23896615403389723999887620709, −6.39807142629599475039708588895, −5.76705856078090832702916308966, −5.43460478094096448163200830958, −4.72396043192429427116059972957, −4.18594927259881472595598639142, −4.15232597037379101069777629725, −2.96137119819843436375014686914, −2.14786426208390986970260053324, −1.41779393831383572931708372171, 0, 0, 0, 1.41779393831383572931708372171, 2.14786426208390986970260053324, 2.96137119819843436375014686914, 4.15232597037379101069777629725, 4.18594927259881472595598639142, 4.72396043192429427116059972957, 5.43460478094096448163200830958, 5.76705856078090832702916308966, 6.39807142629599475039708588895, 7.23896615403389723999887620709, 7.988815170797356720565846935624, 8.337221878198501816246778948455, 9.873132899281669554709942652041, 10.21922508713911823311475238587, 10.86896361566120947149658507482, 11.44877366421240981001886355255, 12.06394838807788824159713110598, 12.28796642080302454944614244056, 13.28777455876486443575413651794, 13.53385783499617849588113184065, 13.66674508064452252117834308574, 15.00328542007060913660329522520, 15.57047626467868140839571500443, 15.94348163227884232240622695556, 16.01879125677040186893938816810

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