Properties

Degree 24
Conductor $ 3^{12} $
Sign $1$
Motivic weight 38
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.14e8·3-s + 7.99e11·4-s + 8.10e15·7-s − 2.05e17·9-s − 9.17e19·12-s + 1.06e21·13-s + 2.19e23·16-s − 4.60e24·19-s − 9.30e23·21-s + 1.53e27·25-s + 1.20e27·27-s + 6.48e27·28-s + 6.24e28·31-s − 1.64e29·36-s − 1.04e30·37-s − 1.22e29·39-s + 1.00e31·43-s − 2.52e31·48-s − 7.83e32·49-s + 8.54e32·52-s + 5.28e32·57-s + 1.92e34·61-s − 1.66e33·63-s + 7.76e33·64-s − 1.22e34·67-s + 9.04e35·73-s − 1.76e35·75-s + ⋯
L(s)  = 1  − 0.0987·3-s + 2.90·4-s + 0.711·7-s − 0.152·9-s − 0.287·12-s + 0.731·13-s + 2.90·16-s − 2.32·19-s − 0.0702·21-s + 4.21·25-s + 0.764·27-s + 2.06·28-s + 2.88·31-s − 0.442·36-s − 1.67·37-s − 0.0721·39-s + 0.923·43-s − 0.287·48-s − 6.03·49-s + 2.12·52-s + 0.229·57-s + 2.30·61-s − 0.108·63-s + 0.373·64-s − 0.247·67-s + 3.57·73-s − 0.416·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr =\mathstrut & \, \Lambda(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s+19)^{12} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(531441\)    =    \(3^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(38\)
character  :  induced by $\chi_{3} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((24,\ 531441,\ (\ :[19]^{12}),\ 1)\)
\(L(\frac{39}{2})\)  \(\approx\)  \(15.44311873\)
\(L(\frac12)\)  \(\approx\)  \(15.44311873\)
\(L(20)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 3$,\(F_p(T)\) is a polynomial of degree 24. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 23.
$p$$F_p(T)$
bad3 \( 1 + 12749156 p^{2} T + 100019426323094 p^{7} T^{2} - 26746876162508623872 p^{16} T^{3} + \)\(11\!\cdots\!08\)\( p^{27} T^{4} - \)\(55\!\cdots\!72\)\( p^{41} T^{5} + \)\(10\!\cdots\!60\)\( p^{59} T^{6} - \)\(55\!\cdots\!72\)\( p^{79} T^{7} + \)\(11\!\cdots\!08\)\( p^{103} T^{8} - 26746876162508623872 p^{130} T^{9} + 100019426323094 p^{159} T^{10} + 12749156 p^{192} T^{11} + p^{228} T^{12} \)
good2 \( 1 - 24988443045 p^{5} T^{2} + \)\(20\!\cdots\!43\)\( p^{11} T^{4} - \)\(19\!\cdots\!35\)\( p^{23} T^{6} + \)\(98\!\cdots\!65\)\( p^{39} T^{8} - \)\(41\!\cdots\!65\)\( p^{65} T^{10} + \)\(17\!\cdots\!55\)\( p^{81} T^{12} - \)\(41\!\cdots\!65\)\( p^{141} T^{14} + \)\(98\!\cdots\!65\)\( p^{191} T^{16} - \)\(19\!\cdots\!35\)\( p^{251} T^{18} + \)\(20\!\cdots\!43\)\( p^{315} T^{20} - 24988443045 p^{385} T^{22} + p^{456} T^{24} \)
5 \( 1 - \)\(30\!\cdots\!48\)\( p T^{2} + \)\(43\!\cdots\!02\)\( p^{5} T^{4} - \)\(18\!\cdots\!44\)\( p^{11} T^{6} + \)\(24\!\cdots\!91\)\( p^{19} T^{8} - \)\(11\!\cdots\!36\)\( p^{29} T^{10} + \)\(18\!\cdots\!76\)\( p^{41} T^{12} - \)\(11\!\cdots\!36\)\( p^{105} T^{14} + \)\(24\!\cdots\!91\)\( p^{171} T^{16} - \)\(18\!\cdots\!44\)\( p^{239} T^{18} + \)\(43\!\cdots\!02\)\( p^{309} T^{20} - \)\(30\!\cdots\!48\)\( p^{381} T^{22} + p^{456} T^{24} \)
7 \( ( 1 - 579133731181332 p T + \)\(59\!\cdots\!14\)\( p T^{2} - \)\(98\!\cdots\!32\)\( p^{3} T^{3} + \)\(76\!\cdots\!51\)\( p^{6} T^{4} + \)\(31\!\cdots\!36\)\( p^{11} T^{5} + \)\(13\!\cdots\!72\)\( p^{13} T^{6} + \)\(31\!\cdots\!36\)\( p^{49} T^{7} + \)\(76\!\cdots\!51\)\( p^{82} T^{8} - \)\(98\!\cdots\!32\)\( p^{117} T^{9} + \)\(59\!\cdots\!14\)\( p^{153} T^{10} - 579133731181332 p^{191} T^{11} + p^{228} T^{12} )^{2} \)
11 \( 1 - \)\(13\!\cdots\!32\)\( T^{2} + \)\(59\!\cdots\!06\)\( p^{2} T^{4} - \)\(86\!\cdots\!20\)\( p^{5} T^{6} - \)\(13\!\cdots\!55\)\( p^{9} T^{8} + \)\(71\!\cdots\!08\)\( p^{15} T^{10} - \)\(17\!\cdots\!76\)\( p^{21} T^{12} + \)\(71\!\cdots\!08\)\( p^{91} T^{14} - \)\(13\!\cdots\!55\)\( p^{161} T^{16} - \)\(86\!\cdots\!20\)\( p^{233} T^{18} + \)\(59\!\cdots\!06\)\( p^{306} T^{20} - \)\(13\!\cdots\!32\)\( p^{380} T^{22} + p^{456} T^{24} \)
13 \( ( 1 - 41119183589747365452 p T + \)\(45\!\cdots\!66\)\( p T^{2} - \)\(27\!\cdots\!16\)\( p^{2} T^{3} + \)\(12\!\cdots\!11\)\( p^{2} T^{4} - \)\(58\!\cdots\!88\)\( p^{4} T^{5} + \)\(11\!\cdots\!16\)\( p^{6} T^{6} - \)\(58\!\cdots\!88\)\( p^{42} T^{7} + \)\(12\!\cdots\!11\)\( p^{78} T^{8} - \)\(27\!\cdots\!16\)\( p^{116} T^{9} + \)\(45\!\cdots\!66\)\( p^{153} T^{10} - 41119183589747365452 p^{191} T^{11} + p^{228} T^{12} )^{2} \)
17 \( 1 - \)\(15\!\cdots\!80\)\( p T^{2} + \)\(91\!\cdots\!18\)\( p^{3} T^{4} - \)\(37\!\cdots\!60\)\( p^{5} T^{6} + \)\(43\!\cdots\!55\)\( p^{9} T^{8} - \)\(39\!\cdots\!60\)\( p^{13} T^{10} + \)\(30\!\cdots\!00\)\( p^{17} T^{12} - \)\(39\!\cdots\!60\)\( p^{89} T^{14} + \)\(43\!\cdots\!55\)\( p^{161} T^{16} - \)\(37\!\cdots\!60\)\( p^{233} T^{18} + \)\(91\!\cdots\!18\)\( p^{307} T^{20} - \)\(15\!\cdots\!80\)\( p^{381} T^{22} + p^{456} T^{24} \)
19 \( ( 1 + \)\(12\!\cdots\!20\)\( p T + \)\(11\!\cdots\!26\)\( p T^{2} + \)\(10\!\cdots\!60\)\( p^{2} T^{3} + \)\(52\!\cdots\!55\)\( p^{2} T^{4} + \)\(40\!\cdots\!40\)\( p^{3} T^{5} + \)\(14\!\cdots\!80\)\( p^{3} T^{6} + \)\(40\!\cdots\!40\)\( p^{41} T^{7} + \)\(52\!\cdots\!55\)\( p^{78} T^{8} + \)\(10\!\cdots\!60\)\( p^{116} T^{9} + \)\(11\!\cdots\!26\)\( p^{153} T^{10} + \)\(12\!\cdots\!20\)\( p^{191} T^{11} + p^{228} T^{12} )^{2} \)
23 \( 1 - \)\(21\!\cdots\!80\)\( p T^{2} + \)\(11\!\cdots\!14\)\( T^{4} - \)\(34\!\cdots\!20\)\( p^{2} T^{6} + \)\(70\!\cdots\!95\)\( p^{4} T^{8} - \)\(11\!\cdots\!20\)\( p^{6} T^{10} + \)\(13\!\cdots\!60\)\( p^{8} T^{12} - \)\(11\!\cdots\!20\)\( p^{82} T^{14} + \)\(70\!\cdots\!95\)\( p^{156} T^{16} - \)\(34\!\cdots\!20\)\( p^{230} T^{18} + \)\(11\!\cdots\!14\)\( p^{304} T^{20} - \)\(21\!\cdots\!80\)\( p^{381} T^{22} + p^{456} T^{24} \)
29 \( 1 - \)\(22\!\cdots\!72\)\( T^{2} + \)\(33\!\cdots\!46\)\( p^{2} T^{4} - \)\(34\!\cdots\!20\)\( p^{4} T^{6} + \)\(26\!\cdots\!95\)\( p^{6} T^{8} - \)\(15\!\cdots\!72\)\( p^{8} T^{10} + \)\(78\!\cdots\!44\)\( p^{10} T^{12} - \)\(15\!\cdots\!72\)\( p^{84} T^{14} + \)\(26\!\cdots\!95\)\( p^{158} T^{16} - \)\(34\!\cdots\!20\)\( p^{232} T^{18} + \)\(33\!\cdots\!46\)\( p^{306} T^{20} - \)\(22\!\cdots\!72\)\( p^{380} T^{22} + p^{456} T^{24} \)
31 \( ( 1 - \)\(10\!\cdots\!32\)\( p T + \)\(45\!\cdots\!26\)\( p T^{2} - \)\(17\!\cdots\!20\)\( p^{2} T^{3} + \)\(48\!\cdots\!95\)\( p^{4} T^{4} + \)\(36\!\cdots\!48\)\( p^{4} T^{5} + \)\(19\!\cdots\!84\)\( p^{5} T^{6} + \)\(36\!\cdots\!48\)\( p^{42} T^{7} + \)\(48\!\cdots\!95\)\( p^{80} T^{8} - \)\(17\!\cdots\!20\)\( p^{116} T^{9} + \)\(45\!\cdots\!26\)\( p^{153} T^{10} - \)\(10\!\cdots\!32\)\( p^{191} T^{11} + p^{228} T^{12} )^{2} \)
37 \( ( 1 + \)\(14\!\cdots\!48\)\( p T + \)\(16\!\cdots\!82\)\( p^{2} T^{2} + \)\(18\!\cdots\!68\)\( p^{3} T^{3} + \)\(11\!\cdots\!19\)\( p^{4} T^{4} + \)\(26\!\cdots\!52\)\( p^{6} T^{5} + \)\(41\!\cdots\!16\)\( p^{6} T^{6} + \)\(26\!\cdots\!52\)\( p^{44} T^{7} + \)\(11\!\cdots\!19\)\( p^{80} T^{8} + \)\(18\!\cdots\!68\)\( p^{117} T^{9} + \)\(16\!\cdots\!82\)\( p^{154} T^{10} + \)\(14\!\cdots\!48\)\( p^{191} T^{11} + p^{228} T^{12} )^{2} \)
41 \( 1 - \)\(69\!\cdots\!12\)\( T^{2} + \)\(25\!\cdots\!06\)\( T^{4} - \)\(62\!\cdots\!20\)\( T^{6} + \)\(13\!\cdots\!95\)\( T^{8} - \)\(30\!\cdots\!92\)\( T^{10} + \)\(66\!\cdots\!24\)\( T^{12} - \)\(30\!\cdots\!92\)\( p^{76} T^{14} + \)\(13\!\cdots\!95\)\( p^{152} T^{16} - \)\(62\!\cdots\!20\)\( p^{228} T^{18} + \)\(25\!\cdots\!06\)\( p^{304} T^{20} - \)\(69\!\cdots\!12\)\( p^{380} T^{22} + p^{456} T^{24} \)
43 \( ( 1 - \)\(50\!\cdots\!96\)\( T + \)\(42\!\cdots\!98\)\( T^{2} - \)\(15\!\cdots\!04\)\( T^{3} + \)\(97\!\cdots\!99\)\( T^{4} - \)\(27\!\cdots\!08\)\( T^{5} + \)\(13\!\cdots\!04\)\( T^{6} - \)\(27\!\cdots\!08\)\( p^{38} T^{7} + \)\(97\!\cdots\!99\)\( p^{76} T^{8} - \)\(15\!\cdots\!04\)\( p^{114} T^{9} + \)\(42\!\cdots\!98\)\( p^{152} T^{10} - \)\(50\!\cdots\!96\)\( p^{190} T^{11} + p^{228} T^{12} )^{2} \)
47 \( 1 - \)\(43\!\cdots\!80\)\( p T^{2} + \)\(22\!\cdots\!94\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!35\)\( T^{8} - \)\(46\!\cdots\!20\)\( T^{10} + \)\(17\!\cdots\!00\)\( T^{12} - \)\(46\!\cdots\!20\)\( p^{76} T^{14} + \)\(10\!\cdots\!35\)\( p^{152} T^{16} - \)\(17\!\cdots\!20\)\( p^{228} T^{18} + \)\(22\!\cdots\!94\)\( p^{304} T^{20} - \)\(43\!\cdots\!80\)\( p^{381} T^{22} + p^{456} T^{24} \)
53 \( 1 - \)\(21\!\cdots\!60\)\( T^{2} + \)\(21\!\cdots\!94\)\( T^{4} - \)\(14\!\cdots\!20\)\( T^{6} + \)\(66\!\cdots\!35\)\( T^{8} - \)\(26\!\cdots\!20\)\( T^{10} + \)\(90\!\cdots\!00\)\( T^{12} - \)\(26\!\cdots\!20\)\( p^{76} T^{14} + \)\(66\!\cdots\!35\)\( p^{152} T^{16} - \)\(14\!\cdots\!20\)\( p^{228} T^{18} + \)\(21\!\cdots\!94\)\( p^{304} T^{20} - \)\(21\!\cdots\!60\)\( p^{380} T^{22} + p^{456} T^{24} \)
59 \( 1 - \)\(11\!\cdots\!12\)\( T^{2} + \)\(61\!\cdots\!06\)\( T^{4} - \)\(24\!\cdots\!20\)\( T^{6} + \)\(72\!\cdots\!95\)\( T^{8} - \)\(18\!\cdots\!92\)\( T^{10} + \)\(38\!\cdots\!24\)\( T^{12} - \)\(18\!\cdots\!92\)\( p^{76} T^{14} + \)\(72\!\cdots\!95\)\( p^{152} T^{16} - \)\(24\!\cdots\!20\)\( p^{228} T^{18} + \)\(61\!\cdots\!06\)\( p^{304} T^{20} - \)\(11\!\cdots\!12\)\( p^{380} T^{22} + p^{456} T^{24} \)
61 \( ( 1 - \)\(96\!\cdots\!12\)\( T + \)\(25\!\cdots\!46\)\( T^{2} - \)\(16\!\cdots\!20\)\( T^{3} + \)\(28\!\cdots\!95\)\( T^{4} - \)\(14\!\cdots\!92\)\( T^{5} + \)\(21\!\cdots\!24\)\( T^{6} - \)\(14\!\cdots\!92\)\( p^{38} T^{7} + \)\(28\!\cdots\!95\)\( p^{76} T^{8} - \)\(16\!\cdots\!20\)\( p^{114} T^{9} + \)\(25\!\cdots\!46\)\( p^{152} T^{10} - \)\(96\!\cdots\!12\)\( p^{190} T^{11} + p^{228} T^{12} )^{2} \)
67 \( ( 1 + \)\(61\!\cdots\!76\)\( T + \)\(73\!\cdots\!18\)\( T^{2} - \)\(18\!\cdots\!16\)\( T^{3} + \)\(30\!\cdots\!19\)\( T^{4} - \)\(14\!\cdots\!12\)\( T^{5} + \)\(89\!\cdots\!84\)\( T^{6} - \)\(14\!\cdots\!12\)\( p^{38} T^{7} + \)\(30\!\cdots\!19\)\( p^{76} T^{8} - \)\(18\!\cdots\!16\)\( p^{114} T^{9} + \)\(73\!\cdots\!18\)\( p^{152} T^{10} + \)\(61\!\cdots\!76\)\( p^{190} T^{11} + p^{228} T^{12} )^{2} \)
71 \( 1 - \)\(13\!\cdots\!72\)\( T^{2} + \)\(88\!\cdots\!86\)\( T^{4} - \)\(35\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!95\)\( T^{8} - \)\(25\!\cdots\!92\)\( T^{10} + \)\(58\!\cdots\!44\)\( T^{12} - \)\(25\!\cdots\!92\)\( p^{76} T^{14} + \)\(10\!\cdots\!95\)\( p^{152} T^{16} - \)\(35\!\cdots\!20\)\( p^{228} T^{18} + \)\(88\!\cdots\!86\)\( p^{304} T^{20} - \)\(13\!\cdots\!72\)\( p^{380} T^{22} + p^{456} T^{24} \)
73 \( ( 1 - \)\(45\!\cdots\!36\)\( T + \)\(29\!\cdots\!38\)\( T^{2} - \)\(11\!\cdots\!84\)\( T^{3} + \)\(43\!\cdots\!39\)\( T^{4} - \)\(12\!\cdots\!08\)\( T^{5} + \)\(35\!\cdots\!64\)\( T^{6} - \)\(12\!\cdots\!08\)\( p^{38} T^{7} + \)\(43\!\cdots\!39\)\( p^{76} T^{8} - \)\(11\!\cdots\!84\)\( p^{114} T^{9} + \)\(29\!\cdots\!38\)\( p^{152} T^{10} - \)\(45\!\cdots\!36\)\( p^{190} T^{11} + p^{228} T^{12} )^{2} \)
79 \( ( 1 - \)\(16\!\cdots\!60\)\( T + \)\(60\!\cdots\!54\)\( T^{2} - \)\(81\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!35\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{5} + \)\(27\!\cdots\!00\)\( T^{6} - \)\(17\!\cdots\!20\)\( p^{38} T^{7} + \)\(16\!\cdots\!35\)\( p^{76} T^{8} - \)\(81\!\cdots\!20\)\( p^{114} T^{9} + \)\(60\!\cdots\!54\)\( p^{152} T^{10} - \)\(16\!\cdots\!60\)\( p^{190} T^{11} + p^{228} T^{12} )^{2} \)
83 \( 1 - \)\(52\!\cdots\!60\)\( T^{2} + \)\(14\!\cdots\!34\)\( T^{4} - \)\(26\!\cdots\!20\)\( T^{6} + \)\(37\!\cdots\!35\)\( T^{8} - \)\(42\!\cdots\!20\)\( T^{10} + \)\(38\!\cdots\!00\)\( T^{12} - \)\(42\!\cdots\!20\)\( p^{76} T^{14} + \)\(37\!\cdots\!35\)\( p^{152} T^{16} - \)\(26\!\cdots\!20\)\( p^{228} T^{18} + \)\(14\!\cdots\!34\)\( p^{304} T^{20} - \)\(52\!\cdots\!60\)\( p^{380} T^{22} + p^{456} T^{24} \)
89 \( 1 - \)\(10\!\cdots\!32\)\( T^{2} + \)\(50\!\cdots\!26\)\( T^{4} - \)\(16\!\cdots\!20\)\( T^{6} + \)\(36\!\cdots\!95\)\( T^{8} - \)\(80\!\cdots\!52\)\( p^{2} T^{10} + \)\(13\!\cdots\!04\)\( p^{4} T^{12} - \)\(80\!\cdots\!52\)\( p^{78} T^{14} + \)\(36\!\cdots\!95\)\( p^{152} T^{16} - \)\(16\!\cdots\!20\)\( p^{228} T^{18} + \)\(50\!\cdots\!26\)\( p^{304} T^{20} - \)\(10\!\cdots\!32\)\( p^{380} T^{22} + p^{456} T^{24} \)
97 \( ( 1 - \)\(12\!\cdots\!64\)\( T + \)\(14\!\cdots\!78\)\( T^{2} - \)\(10\!\cdots\!96\)\( T^{3} + \)\(79\!\cdots\!79\)\( T^{4} - \)\(46\!\cdots\!12\)\( T^{5} + \)\(30\!\cdots\!24\)\( T^{6} - \)\(46\!\cdots\!12\)\( p^{38} T^{7} + \)\(79\!\cdots\!79\)\( p^{76} T^{8} - \)\(10\!\cdots\!96\)\( p^{114} T^{9} + \)\(14\!\cdots\!78\)\( p^{152} T^{10} - \)\(12\!\cdots\!64\)\( p^{190} T^{11} + p^{228} T^{12} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.58642711543248984438273543662, −3.52676726717133813511835211618, −3.52380295569413181016982407799, −3.24471855439660385653570281682, −3.22418553489561592771511727198, −3.04357356215223300359996580863, −3.04062994775100672200881936735, −2.69440408607607329674589574666, −2.43063665718798610214224573223, −2.38944076023095451840585459871, −2.15414724657371109330246305388, −2.11971244430886432161087341781, −2.01777581639893428985006130400, −1.91127159309978392584790721650, −1.90772571939114440967485893683, −1.70241565015694465514505831689, −1.23373382131977621061706890140, −1.01850926545568037647024150277, −1.00036783132926991728984195178, −0.990355989494126016923096730746, −0.972892460934494412561330724251, −0.67961730440859792950145733451, −0.43003190831744765631227611582, −0.34560847179440494012829232001, −0.083869322673026564294775701764, 0.083869322673026564294775701764, 0.34560847179440494012829232001, 0.43003190831744765631227611582, 0.67961730440859792950145733451, 0.972892460934494412561330724251, 0.990355989494126016923096730746, 1.00036783132926991728984195178, 1.01850926545568037647024150277, 1.23373382131977621061706890140, 1.70241565015694465514505831689, 1.90772571939114440967485893683, 1.91127159309978392584790721650, 2.01777581639893428985006130400, 2.11971244430886432161087341781, 2.15414724657371109330246305388, 2.38944076023095451840585459871, 2.43063665718798610214224573223, 2.69440408607607329674589574666, 3.04062994775100672200881936735, 3.04357356215223300359996580863, 3.22418553489561592771511727198, 3.24471855439660385653570281682, 3.52380295569413181016982407799, 3.52676726717133813511835211618, 3.58642711543248984438273543662

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

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