Properties

Degree $6$
Conductor $27$
Sign $-1$
Motivic weight $43$
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.85e6·2-s + 3.13e10·3-s − 2.28e12·4-s − 5.07e14·5-s + 1.52e17·6-s − 1.63e18·7-s − 5.98e19·8-s + 6.56e20·9-s − 2.46e21·10-s − 2.77e22·11-s − 7.18e22·12-s − 9.94e23·13-s − 7.93e24·14-s − 1.59e25·15-s − 8.49e25·16-s − 1.60e26·17-s + 3.18e27·18-s − 3.22e27·19-s + 1.16e27·20-s − 5.12e28·21-s − 1.34e29·22-s + 1.14e29·23-s − 1.87e30·24-s − 6.63e29·25-s − 4.83e30·26-s + 1.14e31·27-s + 3.73e30·28-s + ⋯
L(s)  = 1  + 1.63·2-s + 1.73·3-s − 0.260·4-s − 0.476·5-s + 2.83·6-s − 1.10·7-s − 2.29·8-s + 2·9-s − 0.779·10-s − 1.13·11-s − 0.450·12-s − 1.11·13-s − 1.80·14-s − 0.824·15-s − 1.09·16-s − 0.563·17-s + 3.27·18-s − 1.03·19-s + 0.123·20-s − 1.91·21-s − 1.85·22-s + 0.606·23-s − 3.97·24-s − 0.583·25-s − 1.82·26-s + 1.92·27-s + 0.287·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+43/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $-1$
Motivic weight: \(43\)
Character: induced by $\chi_{3} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 27,\ (\ :43/2, 43/2, 43/2),\ -1)\)

Particular Values

\(L(22)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{45}{2})\) 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$C_1$ \( ( 1 - p^{21} T )^{3} \)
good2$S_4\times C_2$ \( 1 - 75891 p^{6} T + 25273446891 p^{10} T^{2} - 18357626592849 p^{22} T^{3} + 25273446891 p^{53} T^{4} - 75891 p^{92} T^{5} + p^{129} T^{6} \)
5$S_4\times C_2$ \( 1 + 101550664221054 p T + \)\(14\!\cdots\!51\)\( p^{4} T^{2} - \)\(76\!\cdots\!76\)\( p^{10} T^{3} + \)\(14\!\cdots\!51\)\( p^{47} T^{4} + 101550664221054 p^{87} T^{5} + p^{129} T^{6} \)
7$S_4\times C_2$ \( 1 + 233309900529584184 p T + \)\(16\!\cdots\!85\)\( p^{4} T^{2} + \)\(49\!\cdots\!36\)\( p^{7} T^{3} + \)\(16\!\cdots\!85\)\( p^{47} T^{4} + 233309900529584184 p^{87} T^{5} + p^{129} T^{6} \)
11$S_4\times C_2$ \( 1 + \)\(25\!\cdots\!20\)\( p T + \)\(10\!\cdots\!41\)\( p^{2} T^{2} + \)\(25\!\cdots\!36\)\( p^{3} T^{3} + \)\(10\!\cdots\!41\)\( p^{45} T^{4} + \)\(25\!\cdots\!20\)\( p^{87} T^{5} + p^{129} T^{6} \)
13$S_4\times C_2$ \( 1 + \)\(76\!\cdots\!50\)\( p T + \)\(80\!\cdots\!47\)\( p^{3} T^{2} + \)\(24\!\cdots\!84\)\( p^{6} T^{3} + \)\(80\!\cdots\!47\)\( p^{46} T^{4} + \)\(76\!\cdots\!50\)\( p^{87} T^{5} + p^{129} T^{6} \)
17$S_4\times C_2$ \( 1 + \)\(94\!\cdots\!46\)\( p T + \)\(42\!\cdots\!23\)\( p^{2} T^{2} + \)\(55\!\cdots\!96\)\( p^{4} T^{3} + \)\(42\!\cdots\!23\)\( p^{45} T^{4} + \)\(94\!\cdots\!46\)\( p^{87} T^{5} + p^{129} T^{6} \)
19$S_4\times C_2$ \( 1 + \)\(16\!\cdots\!56\)\( p T + \)\(67\!\cdots\!17\)\( p^{2} T^{2} + \)\(39\!\cdots\!12\)\( p^{4} T^{3} + \)\(67\!\cdots\!17\)\( p^{45} T^{4} + \)\(16\!\cdots\!56\)\( p^{87} T^{5} + p^{129} T^{6} \)
23$S_4\times C_2$ \( 1 - \)\(11\!\cdots\!04\)\( T + \)\(79\!\cdots\!65\)\( T^{2} - \)\(37\!\cdots\!72\)\( p T^{3} + \)\(79\!\cdots\!65\)\( p^{43} T^{4} - \)\(11\!\cdots\!04\)\( p^{86} T^{5} + p^{129} T^{6} \)
29$S_4\times C_2$ \( 1 + \)\(28\!\cdots\!58\)\( T + \)\(20\!\cdots\!63\)\( p T^{2} - \)\(75\!\cdots\!36\)\( p^{2} T^{3} + \)\(20\!\cdots\!63\)\( p^{44} T^{4} + \)\(28\!\cdots\!58\)\( p^{86} T^{5} + p^{129} T^{6} \)
31$S_4\times C_2$ \( 1 + \)\(56\!\cdots\!36\)\( T + \)\(88\!\cdots\!87\)\( p T^{2} + \)\(33\!\cdots\!32\)\( p^{2} T^{3} + \)\(88\!\cdots\!87\)\( p^{44} T^{4} + \)\(56\!\cdots\!36\)\( p^{86} T^{5} + p^{129} T^{6} \)
37$S_4\times C_2$ \( 1 + \)\(21\!\cdots\!94\)\( p T + \)\(12\!\cdots\!75\)\( p^{3} T^{2} + \)\(80\!\cdots\!16\)\( p^{3} T^{3} + \)\(12\!\cdots\!75\)\( p^{46} T^{4} + \)\(21\!\cdots\!94\)\( p^{87} T^{5} + p^{129} T^{6} \)
41$S_4\times C_2$ \( 1 + \)\(36\!\cdots\!26\)\( p T + \)\(84\!\cdots\!47\)\( p^{2} T^{2} + \)\(11\!\cdots\!72\)\( p^{3} T^{3} + \)\(84\!\cdots\!47\)\( p^{45} T^{4} + \)\(36\!\cdots\!26\)\( p^{87} T^{5} + p^{129} T^{6} \)
43$S_4\times C_2$ \( 1 + \)\(50\!\cdots\!88\)\( T + \)\(13\!\cdots\!17\)\( T^{2} + \)\(21\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!17\)\( p^{43} T^{4} + \)\(50\!\cdots\!88\)\( p^{86} T^{5} + p^{129} T^{6} \)
47$S_4\times C_2$ \( 1 - \)\(51\!\cdots\!36\)\( T + \)\(63\!\cdots\!93\)\( T^{2} - \)\(14\!\cdots\!20\)\( T^{3} + \)\(63\!\cdots\!93\)\( p^{43} T^{4} - \)\(51\!\cdots\!36\)\( p^{86} T^{5} + p^{129} T^{6} \)
53$S_4\times C_2$ \( 1 - \)\(31\!\cdots\!34\)\( T + \)\(11\!\cdots\!15\)\( T^{2} - \)\(18\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!15\)\( p^{43} T^{4} - \)\(31\!\cdots\!34\)\( p^{86} T^{5} + p^{129} T^{6} \)
59$S_4\times C_2$ \( 1 + \)\(17\!\cdots\!76\)\( T + \)\(51\!\cdots\!17\)\( T^{2} + \)\(51\!\cdots\!08\)\( T^{3} + \)\(51\!\cdots\!17\)\( p^{43} T^{4} + \)\(17\!\cdots\!76\)\( p^{86} T^{5} + p^{129} T^{6} \)
61$S_4\times C_2$ \( 1 + \)\(12\!\cdots\!18\)\( T + \)\(74\!\cdots\!59\)\( T^{2} + \)\(22\!\cdots\!04\)\( T^{3} + \)\(74\!\cdots\!59\)\( p^{43} T^{4} + \)\(12\!\cdots\!18\)\( p^{86} T^{5} + p^{129} T^{6} \)
67$S_4\times C_2$ \( 1 - \)\(12\!\cdots\!48\)\( T + \)\(60\!\cdots\!57\)\( T^{2} - \)\(84\!\cdots\!44\)\( T^{3} + \)\(60\!\cdots\!57\)\( p^{43} T^{4} - \)\(12\!\cdots\!48\)\( p^{86} T^{5} + p^{129} T^{6} \)
71$S_4\times C_2$ \( 1 - \)\(22\!\cdots\!16\)\( T + \)\(28\!\cdots\!85\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(28\!\cdots\!85\)\( p^{43} T^{4} - \)\(22\!\cdots\!16\)\( p^{86} T^{5} + p^{129} T^{6} \)
73$S_4\times C_2$ \( 1 - \)\(31\!\cdots\!54\)\( T + \)\(67\!\cdots\!75\)\( T^{2} - \)\(89\!\cdots\!76\)\( T^{3} + \)\(67\!\cdots\!75\)\( p^{43} T^{4} - \)\(31\!\cdots\!54\)\( p^{86} T^{5} + p^{129} T^{6} \)
79$S_4\times C_2$ \( 1 + \)\(43\!\cdots\!80\)\( T + \)\(13\!\cdots\!23\)\( p T^{2} + \)\(34\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!23\)\( p^{44} T^{4} + \)\(43\!\cdots\!80\)\( p^{86} T^{5} + p^{129} T^{6} \)
83$S_4\times C_2$ \( 1 - \)\(89\!\cdots\!68\)\( T + \)\(32\!\cdots\!21\)\( T^{2} - \)\(24\!\cdots\!88\)\( T^{3} + \)\(32\!\cdots\!21\)\( p^{43} T^{4} - \)\(89\!\cdots\!68\)\( p^{86} T^{5} + p^{129} T^{6} \)
89$S_4\times C_2$ \( 1 - \)\(65\!\cdots\!06\)\( T + \)\(14\!\cdots\!07\)\( T^{2} + \)\(34\!\cdots\!72\)\( T^{3} + \)\(14\!\cdots\!07\)\( p^{43} T^{4} - \)\(65\!\cdots\!06\)\( p^{86} T^{5} + p^{129} T^{6} \)
97$S_4\times C_2$ \( 1 - \)\(61\!\cdots\!78\)\( T + \)\(84\!\cdots\!47\)\( T^{2} - \)\(32\!\cdots\!64\)\( T^{3} + \)\(84\!\cdots\!47\)\( p^{43} T^{4} - \)\(61\!\cdots\!78\)\( p^{86} T^{5} + p^{129} T^{6} \)
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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

−14.76106598841272696882443536687, −13.86114570105810332994260477440, −13.53244704435461337073574164602, −13.47258992607920183743076766243, −12.77781380583756211841277665729, −12.57588880279262339539561938014, −12.00930564359947662396997850230, −10.89221785549526099792673120290, −10.11157787148284646165772175715, −9.702647959702834978861160925650, −9.296499235348970975407832377435, −8.478431318381445473061522867460, −8.358323044125470451079358831124, −7.65736410488286676200001533813, −6.70818696674847612958340526260, −6.65277210365325693052747921469, −5.25975086481495194044518627305, −4.93026348265395101803088471747, −4.85715559143026608315069480264, −3.66289620441689658929329871020, −3.62575963961869245448965658226, −3.46990581254968739945337984940, −2.63625675068958808595944468931, −2.12413853973690171786007794616, −1.56479521037483426254455630649, 0, 0, 0, 1.56479521037483426254455630649, 2.12413853973690171786007794616, 2.63625675068958808595944468931, 3.46990581254968739945337984940, 3.62575963961869245448965658226, 3.66289620441689658929329871020, 4.85715559143026608315069480264, 4.93026348265395101803088471747, 5.25975086481495194044518627305, 6.65277210365325693052747921469, 6.70818696674847612958340526260, 7.65736410488286676200001533813, 8.358323044125470451079358831124, 8.478431318381445473061522867460, 9.296499235348970975407832377435, 9.702647959702834978861160925650, 10.11157787148284646165772175715, 10.89221785549526099792673120290, 12.00930564359947662396997850230, 12.57588880279262339539561938014, 12.77781380583756211841277665729, 13.47258992607920183743076766243, 13.53244704435461337073574164602, 13.86114570105810332994260477440, 14.76106598841272696882443536687

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