Properties

Degree $6$
Conductor $27$
Sign $1$
Motivic weight $39$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.33e5·2-s − 3.48e9·3-s − 2.03e11·4-s − 5.33e13·5-s − 1.86e15·6-s − 1.57e15·7-s − 1.07e17·8-s + 8.10e18·9-s − 2.84e19·10-s − 5.31e20·11-s + 7.09e20·12-s + 5.47e21·13-s − 8.41e20·14-s + 1.86e23·15-s − 8.72e22·16-s + 7.23e23·17-s + 4.32e24·18-s − 1.06e25·19-s + 1.08e25·20-s + 5.49e24·21-s − 2.83e26·22-s + 4.16e26·23-s + 3.76e26·24-s − 1.90e27·25-s + 2.91e27·26-s − 1.57e28·27-s + 3.21e26·28-s + ⋯
L(s)  = 1  + 0.719·2-s − 1.73·3-s − 0.370·4-s − 1.25·5-s − 1.24·6-s − 0.0523·7-s − 0.264·8-s + 2·9-s − 0.900·10-s − 2.61·11-s + 0.641·12-s + 1.03·13-s − 0.0376·14-s + 2.16·15-s − 0.288·16-s + 0.734·17-s + 1.43·18-s − 1.23·19-s + 0.463·20-s + 0.0905·21-s − 1.88·22-s + 1.16·23-s + 0.458·24-s − 1.04·25-s + 0.747·26-s − 1.92·27-s + 0.0193·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+39/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $1$
Motivic weight: \(39\)
Character: induced by $\chi_{3} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 27,\ (\ :39/2, 39/2, 39/2),\ 1)\)

Particular Values

\(L(20)\) \(\approx\) \(0.02280952003\)
\(L(\frac12)\) \(\approx\) \(0.02280952003\)
\(L(\frac{41}{2})\) 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$C_1$ \( ( 1 + p^{19} T )^{3} \)
good2$S_4\times C_2$ \( 1 - 266787 p T + 3814554693 p^{7} T^{2} - 3985936932681 p^{16} T^{3} + 3814554693 p^{46} T^{4} - 266787 p^{79} T^{5} + p^{117} T^{6} \)
5$S_4\times C_2$ \( 1 + 10676394788886 p T + \)\(15\!\cdots\!23\)\( p^{5} T^{2} + \)\(38\!\cdots\!44\)\( p^{11} T^{3} + \)\(15\!\cdots\!23\)\( p^{44} T^{4} + 10676394788886 p^{79} T^{5} + p^{117} T^{6} \)
7$S_4\times C_2$ \( 1 + 225330382317504 p T + \)\(18\!\cdots\!65\)\( p^{2} T^{2} + \)\(10\!\cdots\!52\)\( p^{6} T^{3} + \)\(18\!\cdots\!65\)\( p^{41} T^{4} + 225330382317504 p^{79} T^{5} + p^{117} T^{6} \)
11$S_4\times C_2$ \( 1 + \)\(53\!\cdots\!40\)\( T + \)\(19\!\cdots\!11\)\( p T^{2} + \)\(33\!\cdots\!76\)\( p^{4} T^{3} + \)\(19\!\cdots\!11\)\( p^{40} T^{4} + \)\(53\!\cdots\!40\)\( p^{78} T^{5} + p^{117} T^{6} \)
13$S_4\times C_2$ \( 1 - \)\(42\!\cdots\!30\)\( p T + \)\(45\!\cdots\!31\)\( p^{2} T^{2} - \)\(13\!\cdots\!92\)\( p^{3} T^{3} + \)\(45\!\cdots\!31\)\( p^{41} T^{4} - \)\(42\!\cdots\!30\)\( p^{79} T^{5} + p^{117} T^{6} \)
17$S_4\times C_2$ \( 1 - \)\(72\!\cdots\!58\)\( T + \)\(17\!\cdots\!23\)\( p^{2} T^{2} - \)\(75\!\cdots\!24\)\( p^{4} T^{3} + \)\(17\!\cdots\!23\)\( p^{41} T^{4} - \)\(72\!\cdots\!58\)\( p^{78} T^{5} + p^{117} T^{6} \)
19$S_4\times C_2$ \( 1 + \)\(56\!\cdots\!96\)\( p T + \)\(40\!\cdots\!77\)\( p^{2} T^{2} + \)\(19\!\cdots\!88\)\( p^{3} T^{3} + \)\(40\!\cdots\!77\)\( p^{41} T^{4} + \)\(56\!\cdots\!96\)\( p^{79} T^{5} + p^{117} T^{6} \)
23$S_4\times C_2$ \( 1 - \)\(41\!\cdots\!64\)\( T + \)\(25\!\cdots\!55\)\( p T^{2} + \)\(64\!\cdots\!76\)\( p^{2} T^{3} + \)\(25\!\cdots\!55\)\( p^{40} T^{4} - \)\(41\!\cdots\!64\)\( p^{78} T^{5} + p^{117} T^{6} \)
29$S_4\times C_2$ \( 1 - \)\(25\!\cdots\!78\)\( p T + \)\(58\!\cdots\!07\)\( p^{2} T^{2} - \)\(69\!\cdots\!04\)\( p^{3} T^{3} + \)\(58\!\cdots\!07\)\( p^{41} T^{4} - \)\(25\!\cdots\!78\)\( p^{79} T^{5} + p^{117} T^{6} \)
31$S_4\times C_2$ \( 1 + \)\(38\!\cdots\!56\)\( T + \)\(65\!\cdots\!87\)\( p T^{2} + \)\(79\!\cdots\!32\)\( p^{2} T^{3} + \)\(65\!\cdots\!87\)\( p^{40} T^{4} + \)\(38\!\cdots\!56\)\( p^{78} T^{5} + p^{117} T^{6} \)
37$S_4\times C_2$ \( 1 - \)\(78\!\cdots\!06\)\( p T + \)\(17\!\cdots\!15\)\( p^{2} T^{2} - \)\(59\!\cdots\!44\)\( p^{3} T^{3} + \)\(17\!\cdots\!15\)\( p^{41} T^{4} - \)\(78\!\cdots\!06\)\( p^{79} T^{5} + p^{117} T^{6} \)
41$S_4\times C_2$ \( 1 + \)\(13\!\cdots\!26\)\( T + \)\(11\!\cdots\!27\)\( T^{2} + \)\(14\!\cdots\!72\)\( T^{3} + \)\(11\!\cdots\!27\)\( p^{39} T^{4} + \)\(13\!\cdots\!26\)\( p^{78} T^{5} + p^{117} T^{6} \)
43$S_4\times C_2$ \( 1 + \)\(24\!\cdots\!28\)\( T + \)\(33\!\cdots\!77\)\( T^{2} + \)\(28\!\cdots\!60\)\( T^{3} + \)\(33\!\cdots\!77\)\( p^{39} T^{4} + \)\(24\!\cdots\!28\)\( p^{78} T^{5} + p^{117} T^{6} \)
47$S_4\times C_2$ \( 1 + \)\(10\!\cdots\!04\)\( T + \)\(86\!\cdots\!53\)\( T^{2} + \)\(39\!\cdots\!80\)\( T^{3} + \)\(86\!\cdots\!53\)\( p^{39} T^{4} + \)\(10\!\cdots\!04\)\( p^{78} T^{5} + p^{117} T^{6} \)
53$S_4\times C_2$ \( 1 - \)\(62\!\cdots\!34\)\( T + \)\(47\!\cdots\!55\)\( T^{2} - \)\(22\!\cdots\!36\)\( T^{3} + \)\(47\!\cdots\!55\)\( p^{39} T^{4} - \)\(62\!\cdots\!34\)\( p^{78} T^{5} + p^{117} T^{6} \)
59$S_4\times C_2$ \( 1 + \)\(75\!\cdots\!56\)\( T + \)\(29\!\cdots\!37\)\( T^{2} + \)\(16\!\cdots\!68\)\( T^{3} + \)\(29\!\cdots\!37\)\( p^{39} T^{4} + \)\(75\!\cdots\!56\)\( p^{78} T^{5} + p^{117} T^{6} \)
61$S_4\times C_2$ \( 1 + \)\(71\!\cdots\!98\)\( T + \)\(73\!\cdots\!99\)\( T^{2} + \)\(33\!\cdots\!84\)\( T^{3} + \)\(73\!\cdots\!99\)\( p^{39} T^{4} + \)\(71\!\cdots\!98\)\( p^{78} T^{5} + p^{117} T^{6} \)
67$S_4\times C_2$ \( 1 + \)\(51\!\cdots\!72\)\( T + \)\(32\!\cdots\!37\)\( T^{2} + \)\(82\!\cdots\!56\)\( T^{3} + \)\(32\!\cdots\!37\)\( p^{39} T^{4} + \)\(51\!\cdots\!72\)\( p^{78} T^{5} + p^{117} T^{6} \)
71$S_4\times C_2$ \( 1 - \)\(84\!\cdots\!76\)\( T + \)\(40\!\cdots\!85\)\( T^{2} - \)\(20\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!85\)\( p^{39} T^{4} - \)\(84\!\cdots\!76\)\( p^{78} T^{5} + p^{117} T^{6} \)
73$S_4\times C_2$ \( 1 - \)\(63\!\cdots\!14\)\( T + \)\(59\!\cdots\!15\)\( T^{2} - \)\(10\!\cdots\!96\)\( T^{3} + \)\(59\!\cdots\!15\)\( p^{39} T^{4} - \)\(63\!\cdots\!14\)\( p^{78} T^{5} + p^{117} T^{6} \)
79$S_4\times C_2$ \( 1 + \)\(16\!\cdots\!00\)\( T + \)\(16\!\cdots\!57\)\( T^{2} + \)\(94\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!57\)\( p^{39} T^{4} + \)\(16\!\cdots\!00\)\( p^{78} T^{5} + p^{117} T^{6} \)
83$S_4\times C_2$ \( 1 - \)\(59\!\cdots\!48\)\( T + \)\(32\!\cdots\!21\)\( T^{2} - \)\(91\!\cdots\!88\)\( T^{3} + \)\(32\!\cdots\!21\)\( p^{39} T^{4} - \)\(59\!\cdots\!48\)\( p^{78} T^{5} + p^{117} T^{6} \)
89$S_4\times C_2$ \( 1 - \)\(18\!\cdots\!86\)\( T + \)\(36\!\cdots\!47\)\( T^{2} - \)\(36\!\cdots\!48\)\( T^{3} + \)\(36\!\cdots\!47\)\( p^{39} T^{4} - \)\(18\!\cdots\!86\)\( p^{78} T^{5} + p^{117} T^{6} \)
97$S_4\times C_2$ \( 1 + \)\(99\!\cdots\!42\)\( T + \)\(88\!\cdots\!87\)\( T^{2} + \)\(45\!\cdots\!16\)\( T^{3} + \)\(88\!\cdots\!87\)\( p^{39} T^{4} + \)\(99\!\cdots\!42\)\( p^{78} T^{5} + p^{117} T^{6} \)
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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

−14.81522468805728535591554272751, −13.45959797543461921094243692178, −13.27865707835231704204784554121, −13.10662426979785944794974144451, −12.08736538840444288721523391180, −11.96525737344024458868288953903, −11.15666885629649966618385242900, −10.88450905266892113272519849528, −10.11272813802674521204998487875, −9.980572378766690460606941198006, −8.451738440920814554881711828451, −8.202488053188601532799446059622, −7.67905762148281441319403498354, −6.84169261746180358147463070705, −6.29219727325417041678329990124, −5.74503436072033728824460289475, −4.87852124076909544237890976039, −4.86601351642958178363589531726, −4.46148901211808124548251909046, −3.35303034384620127315404256382, −3.29631035790004653621081044578, −2.15129261232880695177346048504, −1.37779641069758828311165351227, −0.59979934643266272629309286477, −0.05062235783673003003673606429, 0.05062235783673003003673606429, 0.59979934643266272629309286477, 1.37779641069758828311165351227, 2.15129261232880695177346048504, 3.29631035790004653621081044578, 3.35303034384620127315404256382, 4.46148901211808124548251909046, 4.86601351642958178363589531726, 4.87852124076909544237890976039, 5.74503436072033728824460289475, 6.29219727325417041678329990124, 6.84169261746180358147463070705, 7.67905762148281441319403498354, 8.202488053188601532799446059622, 8.451738440920814554881711828451, 9.980572378766690460606941198006, 10.11272813802674521204998487875, 10.88450905266892113272519849528, 11.15666885629649966618385242900, 11.96525737344024458868288953903, 12.08736538840444288721523391180, 13.10662426979785944794974144451, 13.27865707835231704204784554121, 13.45959797543461921094243692178, 14.81522468805728535591554272751

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