Properties

Label 12-3e6-1.1-c69e6-0-0
Degree $12$
Conductor $729$
Sign $1$
Analytic cond. $5.47747\times 10^{11}$
Root an. cond. $9.51075$
Motivic weight $69$
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  + 1.97e10·2-s + 1.00e17·3-s − 4.69e20·4-s + 6.28e23·5-s + 1.97e27·6-s + 4.79e28·7-s − 7.51e30·8-s + 5.84e33·9-s + 1.23e34·10-s + 1.39e36·11-s − 4.69e37·12-s + 1.33e38·13-s + 9.43e38·14-s + 6.28e40·15-s − 7.08e40·16-s − 2.74e41·17-s + 1.15e44·18-s + 1.38e44·19-s − 2.94e44·20-s + 4.79e45·21-s + 2.73e46·22-s + 2.18e47·23-s − 7.51e47·24-s − 3.30e48·25-s + 2.63e48·26-s + 2.59e50·27-s − 2.24e49·28-s + ⋯
L(s)  = 1  + 0.810·2-s + 3.46·3-s − 0.794·4-s + 0.482·5-s + 2.80·6-s + 0.334·7-s − 0.523·8-s + 7·9-s + 0.391·10-s + 1.64·11-s − 2.75·12-s + 0.494·13-s + 0.271·14-s + 1.67·15-s − 0.203·16-s − 0.0971·17-s + 5.67·18-s + 1.05·19-s − 0.383·20-s + 1.15·21-s + 1.33·22-s + 2.28·23-s − 1.81·24-s − 1.95·25-s + 0.401·26-s + 10.7·27-s − 0.265·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(70-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+69/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $1$
Analytic conductor: \(5.47747\times 10^{11}\)
Root analytic conductor: \(9.51075\)
Motivic weight: \(69\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 729,\ (\ :[69/2]^{6}),\ 1)\)

Particular Values

\(L(35)\) \(\approx\) \(238.9140121\)
\(L(\frac12)\) \(\approx\) \(238.9140121\)
\(L(\frac{71}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - p^{34} T )^{6} \)
good2 \( 1 - 9850481469 p T + 26785721671933348257 p^{5} T^{2} - \)\(56\!\cdots\!13\)\( p^{15} T^{3} + \)\(20\!\cdots\!81\)\( p^{25} T^{4} - \)\(14\!\cdots\!47\)\( p^{43} T^{5} + \)\(22\!\cdots\!17\)\( p^{61} T^{6} - \)\(14\!\cdots\!47\)\( p^{112} T^{7} + \)\(20\!\cdots\!81\)\( p^{163} T^{8} - \)\(56\!\cdots\!13\)\( p^{222} T^{9} + 26785721671933348257 p^{281} T^{10} - 9850481469 p^{346} T^{11} + p^{414} T^{12} \)
5 \( 1 - \)\(62\!\cdots\!36\)\( T + \)\(29\!\cdots\!22\)\( p^{3} T^{2} - \)\(12\!\cdots\!52\)\( p^{7} T^{3} + \)\(38\!\cdots\!47\)\( p^{12} T^{4} - \)\(74\!\cdots\!52\)\( p^{21} T^{5} + \)\(90\!\cdots\!76\)\( p^{32} T^{6} - \)\(74\!\cdots\!52\)\( p^{90} T^{7} + \)\(38\!\cdots\!47\)\( p^{150} T^{8} - \)\(12\!\cdots\!52\)\( p^{214} T^{9} + \)\(29\!\cdots\!22\)\( p^{279} T^{10} - \)\(62\!\cdots\!36\)\( p^{345} T^{11} + p^{414} T^{12} \)
7 \( 1 - \)\(68\!\cdots\!48\)\( p T + \)\(12\!\cdots\!46\)\( p^{2} T^{2} - \)\(41\!\cdots\!56\)\( p^{7} T^{3} + \)\(16\!\cdots\!91\)\( p^{12} T^{4} - \)\(10\!\cdots\!36\)\( p^{19} T^{5} + \)\(84\!\cdots\!76\)\( p^{27} T^{6} - \)\(10\!\cdots\!36\)\( p^{88} T^{7} + \)\(16\!\cdots\!91\)\( p^{150} T^{8} - \)\(41\!\cdots\!56\)\( p^{214} T^{9} + \)\(12\!\cdots\!46\)\( p^{278} T^{10} - \)\(68\!\cdots\!48\)\( p^{346} T^{11} + p^{414} T^{12} \)
11 \( 1 - \)\(13\!\cdots\!24\)\( T + \)\(35\!\cdots\!14\)\( p^{2} T^{2} - \)\(31\!\cdots\!08\)\( p^{4} T^{3} + \)\(36\!\cdots\!43\)\( p^{8} T^{4} - \)\(20\!\cdots\!20\)\( p^{12} T^{5} + \)\(16\!\cdots\!48\)\( p^{16} T^{6} - \)\(20\!\cdots\!20\)\( p^{81} T^{7} + \)\(36\!\cdots\!43\)\( p^{146} T^{8} - \)\(31\!\cdots\!08\)\( p^{211} T^{9} + \)\(35\!\cdots\!14\)\( p^{278} T^{10} - \)\(13\!\cdots\!24\)\( p^{345} T^{11} + p^{414} T^{12} \)
13 \( 1 - \)\(13\!\cdots\!32\)\( T + \)\(10\!\cdots\!30\)\( p T^{2} - \)\(21\!\cdots\!88\)\( p^{3} T^{3} + \)\(31\!\cdots\!35\)\( p^{6} T^{4} - \)\(29\!\cdots\!28\)\( p^{10} T^{5} + \)\(41\!\cdots\!64\)\( p^{14} T^{6} - \)\(29\!\cdots\!28\)\( p^{79} T^{7} + \)\(31\!\cdots\!35\)\( p^{144} T^{8} - \)\(21\!\cdots\!88\)\( p^{210} T^{9} + \)\(10\!\cdots\!30\)\( p^{277} T^{10} - \)\(13\!\cdots\!32\)\( p^{345} T^{11} + p^{414} T^{12} \)
17 \( 1 + \)\(27\!\cdots\!88\)\( T + \)\(16\!\cdots\!26\)\( p T^{2} + \)\(43\!\cdots\!40\)\( p^{3} T^{3} + \)\(27\!\cdots\!15\)\( p^{5} T^{4} - \)\(87\!\cdots\!52\)\( p^{8} T^{5} + \)\(31\!\cdots\!92\)\( p^{9} T^{6} - \)\(87\!\cdots\!52\)\( p^{77} T^{7} + \)\(27\!\cdots\!15\)\( p^{143} T^{8} + \)\(43\!\cdots\!40\)\( p^{210} T^{9} + \)\(16\!\cdots\!26\)\( p^{277} T^{10} + \)\(27\!\cdots\!88\)\( p^{345} T^{11} + p^{414} T^{12} \)
19 \( 1 - \)\(13\!\cdots\!12\)\( T + \)\(34\!\cdots\!98\)\( p T^{2} - \)\(21\!\cdots\!84\)\( p^{2} T^{3} + \)\(78\!\cdots\!37\)\( p^{5} T^{4} - \)\(22\!\cdots\!52\)\( p^{7} T^{5} + \)\(62\!\cdots\!48\)\( p^{10} T^{6} - \)\(22\!\cdots\!52\)\( p^{76} T^{7} + \)\(78\!\cdots\!37\)\( p^{143} T^{8} - \)\(21\!\cdots\!84\)\( p^{209} T^{9} + \)\(34\!\cdots\!98\)\( p^{277} T^{10} - \)\(13\!\cdots\!12\)\( p^{345} T^{11} + p^{414} T^{12} \)
23 \( 1 - \)\(94\!\cdots\!96\)\( p T + \)\(88\!\cdots\!14\)\( p^{2} T^{2} - \)\(40\!\cdots\!68\)\( p^{3} T^{3} + \)\(19\!\cdots\!91\)\( p^{4} T^{4} - \)\(46\!\cdots\!68\)\( p^{5} T^{5} + \)\(22\!\cdots\!48\)\( p^{6} T^{6} - \)\(46\!\cdots\!68\)\( p^{74} T^{7} + \)\(19\!\cdots\!91\)\( p^{142} T^{8} - \)\(40\!\cdots\!68\)\( p^{210} T^{9} + \)\(88\!\cdots\!14\)\( p^{278} T^{10} - \)\(94\!\cdots\!96\)\( p^{346} T^{11} + p^{414} T^{12} \)
29 \( 1 - \)\(38\!\cdots\!04\)\( T + \)\(81\!\cdots\!66\)\( T^{2} - \)\(12\!\cdots\!48\)\( p T^{3} + \)\(37\!\cdots\!83\)\( p^{3} T^{4} - \)\(13\!\cdots\!76\)\( p^{5} T^{5} + \)\(73\!\cdots\!08\)\( p^{7} T^{6} - \)\(13\!\cdots\!76\)\( p^{74} T^{7} + \)\(37\!\cdots\!83\)\( p^{141} T^{8} - \)\(12\!\cdots\!48\)\( p^{208} T^{9} + \)\(81\!\cdots\!66\)\( p^{276} T^{10} - \)\(38\!\cdots\!04\)\( p^{345} T^{11} + p^{414} T^{12} \)
31 \( 1 + \)\(40\!\cdots\!24\)\( T + \)\(64\!\cdots\!78\)\( p T^{2} + \)\(14\!\cdots\!12\)\( p^{2} T^{3} + \)\(32\!\cdots\!83\)\( p^{4} T^{4} + \)\(18\!\cdots\!16\)\( p^{6} T^{5} + \)\(31\!\cdots\!72\)\( p^{8} T^{6} + \)\(18\!\cdots\!16\)\( p^{75} T^{7} + \)\(32\!\cdots\!83\)\( p^{142} T^{8} + \)\(14\!\cdots\!12\)\( p^{209} T^{9} + \)\(64\!\cdots\!78\)\( p^{277} T^{10} + \)\(40\!\cdots\!24\)\( p^{345} T^{11} + p^{414} T^{12} \)
37 \( 1 + \)\(10\!\cdots\!24\)\( T + \)\(32\!\cdots\!94\)\( T^{2} + \)\(14\!\cdots\!92\)\( T^{3} + \)\(15\!\cdots\!79\)\( p^{2} T^{4} + \)\(38\!\cdots\!88\)\( p^{2} T^{5} + \)\(77\!\cdots\!56\)\( p^{3} T^{6} + \)\(38\!\cdots\!88\)\( p^{71} T^{7} + \)\(15\!\cdots\!79\)\( p^{140} T^{8} + \)\(14\!\cdots\!92\)\( p^{207} T^{9} + \)\(32\!\cdots\!94\)\( p^{276} T^{10} + \)\(10\!\cdots\!24\)\( p^{345} T^{11} + p^{414} T^{12} \)
41 \( 1 + \)\(75\!\cdots\!04\)\( T + \)\(41\!\cdots\!58\)\( T^{2} + \)\(66\!\cdots\!92\)\( p T^{3} + \)\(11\!\cdots\!03\)\( p^{2} T^{4} + \)\(11\!\cdots\!36\)\( p^{3} T^{5} + \)\(12\!\cdots\!52\)\( p^{4} T^{6} + \)\(11\!\cdots\!36\)\( p^{72} T^{7} + \)\(11\!\cdots\!03\)\( p^{140} T^{8} + \)\(66\!\cdots\!92\)\( p^{208} T^{9} + \)\(41\!\cdots\!58\)\( p^{276} T^{10} + \)\(75\!\cdots\!04\)\( p^{345} T^{11} + p^{414} T^{12} \)
43 \( 1 - \)\(51\!\cdots\!20\)\( T + \)\(32\!\cdots\!10\)\( T^{2} - \)\(26\!\cdots\!20\)\( p T^{3} + \)\(22\!\cdots\!03\)\( p^{2} T^{4} - \)\(13\!\cdots\!80\)\( p^{3} T^{5} + \)\(83\!\cdots\!80\)\( p^{4} T^{6} - \)\(13\!\cdots\!80\)\( p^{72} T^{7} + \)\(22\!\cdots\!03\)\( p^{140} T^{8} - \)\(26\!\cdots\!20\)\( p^{208} T^{9} + \)\(32\!\cdots\!10\)\( p^{276} T^{10} - \)\(51\!\cdots\!20\)\( p^{345} T^{11} + p^{414} T^{12} \)
47 \( 1 - \)\(24\!\cdots\!80\)\( T + \)\(80\!\cdots\!90\)\( p T^{2} - \)\(11\!\cdots\!40\)\( p^{2} T^{3} + \)\(10\!\cdots\!29\)\( p^{3} T^{4} - \)\(14\!\cdots\!40\)\( p^{4} T^{5} + \)\(16\!\cdots\!20\)\( p^{5} T^{6} - \)\(14\!\cdots\!40\)\( p^{73} T^{7} + \)\(10\!\cdots\!29\)\( p^{141} T^{8} - \)\(11\!\cdots\!40\)\( p^{209} T^{9} + \)\(80\!\cdots\!90\)\( p^{277} T^{10} - \)\(24\!\cdots\!80\)\( p^{345} T^{11} + p^{414} T^{12} \)
53 \( 1 - \)\(74\!\cdots\!88\)\( T + \)\(41\!\cdots\!02\)\( p T^{2} - \)\(51\!\cdots\!64\)\( p^{2} T^{3} + \)\(20\!\cdots\!63\)\( p^{3} T^{4} - \)\(16\!\cdots\!44\)\( p^{4} T^{5} + \)\(80\!\cdots\!04\)\( p^{5} T^{6} - \)\(16\!\cdots\!44\)\( p^{73} T^{7} + \)\(20\!\cdots\!63\)\( p^{141} T^{8} - \)\(51\!\cdots\!64\)\( p^{209} T^{9} + \)\(41\!\cdots\!02\)\( p^{277} T^{10} - \)\(74\!\cdots\!88\)\( p^{345} T^{11} + p^{414} T^{12} \)
59 \( 1 - \)\(19\!\cdots\!92\)\( p T + \)\(23\!\cdots\!02\)\( p^{2} T^{2} - \)\(39\!\cdots\!84\)\( p^{3} T^{3} + \)\(24\!\cdots\!03\)\( p^{4} T^{4} - \)\(33\!\cdots\!28\)\( p^{5} T^{5} + \)\(14\!\cdots\!88\)\( p^{6} T^{6} - \)\(33\!\cdots\!28\)\( p^{74} T^{7} + \)\(24\!\cdots\!03\)\( p^{142} T^{8} - \)\(39\!\cdots\!84\)\( p^{210} T^{9} + \)\(23\!\cdots\!02\)\( p^{278} T^{10} - \)\(19\!\cdots\!92\)\( p^{346} T^{11} + p^{414} T^{12} \)
61 \( 1 - \)\(33\!\cdots\!60\)\( T + \)\(51\!\cdots\!14\)\( p T^{2} - \)\(34\!\cdots\!20\)\( p^{2} T^{3} + \)\(30\!\cdots\!15\)\( p^{3} T^{4} - \)\(18\!\cdots\!20\)\( p^{4} T^{5} + \)\(13\!\cdots\!80\)\( p^{5} T^{6} - \)\(18\!\cdots\!20\)\( p^{73} T^{7} + \)\(30\!\cdots\!15\)\( p^{141} T^{8} - \)\(34\!\cdots\!20\)\( p^{209} T^{9} + \)\(51\!\cdots\!14\)\( p^{277} T^{10} - \)\(33\!\cdots\!60\)\( p^{345} T^{11} + p^{414} T^{12} \)
67 \( 1 - \)\(27\!\cdots\!96\)\( p T + \)\(10\!\cdots\!78\)\( p^{2} T^{2} - \)\(24\!\cdots\!60\)\( p^{3} T^{3} + \)\(55\!\cdots\!55\)\( p^{4} T^{4} - \)\(99\!\cdots\!76\)\( p^{5} T^{5} + \)\(16\!\cdots\!56\)\( p^{6} T^{6} - \)\(99\!\cdots\!76\)\( p^{74} T^{7} + \)\(55\!\cdots\!55\)\( p^{142} T^{8} - \)\(24\!\cdots\!60\)\( p^{210} T^{9} + \)\(10\!\cdots\!78\)\( p^{278} T^{10} - \)\(27\!\cdots\!96\)\( p^{346} T^{11} + p^{414} T^{12} \)
71 \( 1 + \)\(13\!\cdots\!48\)\( T + \)\(23\!\cdots\!46\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!95\)\( T^{4} + \)\(65\!\cdots\!08\)\( T^{5} + \)\(68\!\cdots\!04\)\( T^{6} + \)\(65\!\cdots\!08\)\( p^{69} T^{7} + \)\(16\!\cdots\!95\)\( p^{138} T^{8} + \)\(16\!\cdots\!80\)\( p^{207} T^{9} + \)\(23\!\cdots\!46\)\( p^{276} T^{10} + \)\(13\!\cdots\!48\)\( p^{345} T^{11} + p^{414} T^{12} \)
73 \( 1 + \)\(10\!\cdots\!12\)\( T + \)\(15\!\cdots\!26\)\( T^{2} + \)\(10\!\cdots\!44\)\( T^{3} + \)\(11\!\cdots\!31\)\( T^{4} + \)\(52\!\cdots\!76\)\( T^{5} + \)\(53\!\cdots\!12\)\( T^{6} + \)\(52\!\cdots\!76\)\( p^{69} T^{7} + \)\(11\!\cdots\!31\)\( p^{138} T^{8} + \)\(10\!\cdots\!44\)\( p^{207} T^{9} + \)\(15\!\cdots\!26\)\( p^{276} T^{10} + \)\(10\!\cdots\!12\)\( p^{345} T^{11} + p^{414} T^{12} \)
79 \( 1 - \)\(84\!\cdots\!00\)\( T + \)\(62\!\cdots\!14\)\( T^{2} - \)\(31\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!15\)\( T^{4} - \)\(51\!\cdots\!00\)\( T^{5} + \)\(16\!\cdots\!80\)\( T^{6} - \)\(51\!\cdots\!00\)\( p^{69} T^{7} + \)\(14\!\cdots\!15\)\( p^{138} T^{8} - \)\(31\!\cdots\!00\)\( p^{207} T^{9} + \)\(62\!\cdots\!14\)\( p^{276} T^{10} - \)\(84\!\cdots\!00\)\( p^{345} T^{11} + p^{414} T^{12} \)
83 \( 1 + \)\(14\!\cdots\!96\)\( T + \)\(61\!\cdots\!06\)\( T^{2} + \)\(59\!\cdots\!32\)\( T^{3} + \)\(20\!\cdots\!27\)\( T^{4} + \)\(29\!\cdots\!32\)\( T^{5} + \)\(56\!\cdots\!56\)\( T^{6} + \)\(29\!\cdots\!32\)\( p^{69} T^{7} + \)\(20\!\cdots\!27\)\( p^{138} T^{8} + \)\(59\!\cdots\!32\)\( p^{207} T^{9} + \)\(61\!\cdots\!06\)\( p^{276} T^{10} + \)\(14\!\cdots\!96\)\( p^{345} T^{11} + p^{414} T^{12} \)
89 \( 1 - \)\(63\!\cdots\!32\)\( T + \)\(24\!\cdots\!02\)\( T^{2} - \)\(61\!\cdots\!44\)\( T^{3} + \)\(11\!\cdots\!83\)\( T^{4} - \)\(16\!\cdots\!28\)\( T^{5} + \)\(27\!\cdots\!28\)\( T^{6} - \)\(16\!\cdots\!28\)\( p^{69} T^{7} + \)\(11\!\cdots\!83\)\( p^{138} T^{8} - \)\(61\!\cdots\!44\)\( p^{207} T^{9} + \)\(24\!\cdots\!02\)\( p^{276} T^{10} - \)\(63\!\cdots\!32\)\( p^{345} T^{11} + p^{414} T^{12} \)
97 \( 1 - \)\(94\!\cdots\!72\)\( T + \)\(84\!\cdots\!62\)\( T^{2} - \)\(45\!\cdots\!80\)\( T^{3} + \)\(24\!\cdots\!55\)\( T^{4} - \)\(94\!\cdots\!32\)\( T^{5} + \)\(37\!\cdots\!44\)\( T^{6} - \)\(94\!\cdots\!32\)\( p^{69} T^{7} + \)\(24\!\cdots\!55\)\( p^{138} T^{8} - \)\(45\!\cdots\!80\)\( p^{207} T^{9} + \)\(84\!\cdots\!62\)\( p^{276} T^{10} - \)\(94\!\cdots\!72\)\( p^{345} T^{11} + p^{414} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.64124729788136685077313067111, −5.09750189667010382912499392164, −5.07026891539230571603437127317, −4.73401290924350088938580149437, −4.64468702314166554005171153955, −4.54099060560191992326640139429, −4.35904568248055772257137544962, −3.68975742318588968766091396001, −3.63169908159322299569794928493, −3.58835037824610219319583641839, −3.48087728370306431655981435195, −3.45334893702170339260922784269, −2.95672281531647364674445972615, −2.69639645959778054968897692991, −2.51068315269915549079655997119, −2.26771819753389743440397912112, −1.89880723636355160823814394912, −1.81111600633578855391064666083, −1.80577698387341244129524003488, −1.51638036963420704103343299895, −1.02851438463494725479759278013, −0.954024580345667086974190381908, −0.78400810702407690047624641480, −0.56981536586595189697480263143, −0.36076670471800663170813780980, 0.36076670471800663170813780980, 0.56981536586595189697480263143, 0.78400810702407690047624641480, 0.954024580345667086974190381908, 1.02851438463494725479759278013, 1.51638036963420704103343299895, 1.80577698387341244129524003488, 1.81111600633578855391064666083, 1.89880723636355160823814394912, 2.26771819753389743440397912112, 2.51068315269915549079655997119, 2.69639645959778054968897692991, 2.95672281531647364674445972615, 3.45334893702170339260922784269, 3.48087728370306431655981435195, 3.58835037824610219319583641839, 3.63169908159322299569794928493, 3.68975742318588968766091396001, 4.35904568248055772257137544962, 4.54099060560191992326640139429, 4.64468702314166554005171153955, 4.73401290924350088938580149437, 5.07026891539230571603437127317, 5.09750189667010382912499392164, 5.64124729788136685077313067111

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

Plot not available for L-functions of degree greater than 10.