Properties

Label 12-75e6-1.1-c21e6-0-3
Degree $12$
Conductor $177978515625$
Sign $1$
Analytic cond. $8.48100\times 10^{13}$
Root an. cond. $14.4778$
Motivic weight $21$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 92·2-s − 3.54e5·3-s − 4.09e6·4-s − 3.25e7·6-s + 1.32e9·7-s − 1.85e9·8-s + 7.32e10·9-s − 7.83e10·11-s + 1.44e12·12-s + 1.16e11·13-s + 1.22e11·14-s + 6.43e12·16-s + 5.38e12·17-s + 6.73e12·18-s + 6.96e12·19-s − 4.70e14·21-s − 7.20e12·22-s + 2.95e14·23-s + 6.58e14·24-s + 1.06e13·26-s − 1.15e16·27-s − 5.43e15·28-s − 3.89e15·29-s + 4.92e14·31-s + 6.27e15·32-s + 2.77e16·33-s + 4.95e14·34-s + ⋯
L(s)  = 1  + 0.0635·2-s − 3.46·3-s − 1.95·4-s − 0.220·6-s + 1.77·7-s − 0.611·8-s + 7·9-s − 0.910·11-s + 6.75·12-s + 0.233·13-s + 0.112·14-s + 1.46·16-s + 0.647·17-s + 0.444·18-s + 0.260·19-s − 6.15·21-s − 0.0578·22-s + 1.48·23-s + 2.11·24-s + 0.0148·26-s − 10.7·27-s − 3.46·28-s − 1.71·29-s + 0.107·31-s + 0.985·32-s + 3.15·33-s + 0.0411·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(22-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+21/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{12}\)
Sign: $1$
Analytic conductor: \(8.48100\times 10^{13}\)
Root analytic conductor: \(14.4778\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 3^{6} \cdot 5^{12} ,\ ( \ : [21/2]^{6} ),\ 1 )\)

Particular Values

\(L(11)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + p^{10} T )^{6} \)
5 \( 1 \)
good2 \( 1 - 23 p^{2} T + 256279 p^{4} T^{2} + 17261771 p^{6} T^{3} + 19674337071 p^{9} T^{4} + 168655853167 p^{15} T^{5} + 20861711149109 p^{20} T^{6} + 168655853167 p^{36} T^{7} + 19674337071 p^{51} T^{8} + 17261771 p^{69} T^{9} + 256279 p^{88} T^{10} - 23 p^{107} T^{11} + p^{126} T^{12} \)
7 \( 1 - 189591922 p T + 48831503728259361 p^{2} T^{2} - \)\(70\!\cdots\!54\)\( p^{3} T^{3} + \)\(10\!\cdots\!26\)\( p^{4} T^{4} - \)\(25\!\cdots\!74\)\( p^{7} T^{5} + \)\(14\!\cdots\!97\)\( p^{6} T^{6} - \)\(25\!\cdots\!74\)\( p^{28} T^{7} + \)\(10\!\cdots\!26\)\( p^{46} T^{8} - \)\(70\!\cdots\!54\)\( p^{66} T^{9} + 48831503728259361 p^{86} T^{10} - 189591922 p^{106} T^{11} + p^{126} T^{12} \)
11 \( 1 + 78310112516 T + \)\(28\!\cdots\!94\)\( p T^{2} + \)\(20\!\cdots\!32\)\( T^{3} + \)\(47\!\cdots\!03\)\( T^{4} + \)\(23\!\cdots\!20\)\( p T^{5} + \)\(36\!\cdots\!88\)\( p^{2} T^{6} + \)\(23\!\cdots\!20\)\( p^{22} T^{7} + \)\(47\!\cdots\!03\)\( p^{42} T^{8} + \)\(20\!\cdots\!32\)\( p^{63} T^{9} + \)\(28\!\cdots\!94\)\( p^{85} T^{10} + 78310112516 p^{105} T^{11} + p^{126} T^{12} \)
13 \( 1 - 116029746338 T + \)\(10\!\cdots\!65\)\( T^{2} - \)\(12\!\cdots\!38\)\( p T^{3} + \)\(29\!\cdots\!10\)\( p^{2} T^{4} - \)\(34\!\cdots\!74\)\( p^{3} T^{5} + \)\(53\!\cdots\!01\)\( p^{4} T^{6} - \)\(34\!\cdots\!74\)\( p^{24} T^{7} + \)\(29\!\cdots\!10\)\( p^{44} T^{8} - \)\(12\!\cdots\!38\)\( p^{64} T^{9} + \)\(10\!\cdots\!65\)\( p^{84} T^{10} - 116029746338 p^{105} T^{11} + p^{126} T^{12} \)
17 \( 1 - 5382900513068 T + \)\(26\!\cdots\!62\)\( T^{2} - \)\(81\!\cdots\!60\)\( p T^{3} + \)\(11\!\cdots\!95\)\( p^{2} T^{4} - \)\(33\!\cdots\!36\)\( p^{3} T^{5} + \)\(33\!\cdots\!84\)\( p^{4} T^{6} - \)\(33\!\cdots\!36\)\( p^{24} T^{7} + \)\(11\!\cdots\!95\)\( p^{44} T^{8} - \)\(81\!\cdots\!60\)\( p^{64} T^{9} + \)\(26\!\cdots\!62\)\( p^{84} T^{10} - 5382900513068 p^{105} T^{11} + p^{126} T^{12} \)
19 \( 1 - 366823105138 p T + \)\(73\!\cdots\!37\)\( p^{2} T^{2} + \)\(15\!\cdots\!74\)\( p^{3} T^{3} + \)\(20\!\cdots\!38\)\( p^{4} T^{4} + \)\(16\!\cdots\!78\)\( p^{5} T^{5} + \)\(39\!\cdots\!73\)\( p^{6} T^{6} + \)\(16\!\cdots\!78\)\( p^{26} T^{7} + \)\(20\!\cdots\!38\)\( p^{46} T^{8} + \)\(15\!\cdots\!74\)\( p^{66} T^{9} + \)\(73\!\cdots\!37\)\( p^{86} T^{10} - 366823105138 p^{106} T^{11} + p^{126} T^{12} \)
23 \( 1 - 295754895311052 T + \)\(21\!\cdots\!06\)\( T^{2} - \)\(47\!\cdots\!84\)\( T^{3} + \)\(19\!\cdots\!91\)\( T^{4} - \)\(32\!\cdots\!56\)\( T^{5} + \)\(96\!\cdots\!72\)\( T^{6} - \)\(32\!\cdots\!56\)\( p^{21} T^{7} + \)\(19\!\cdots\!91\)\( p^{42} T^{8} - \)\(47\!\cdots\!84\)\( p^{63} T^{9} + \)\(21\!\cdots\!06\)\( p^{84} T^{10} - 295754895311052 p^{105} T^{11} + p^{126} T^{12} \)
29 \( 1 + 3895624305077596 T + \)\(28\!\cdots\!26\)\( T^{2} + \)\(32\!\cdots\!52\)\( p T^{3} + \)\(34\!\cdots\!67\)\( T^{4} + \)\(92\!\cdots\!16\)\( T^{5} + \)\(23\!\cdots\!32\)\( T^{6} + \)\(92\!\cdots\!16\)\( p^{21} T^{7} + \)\(34\!\cdots\!67\)\( p^{42} T^{8} + \)\(32\!\cdots\!52\)\( p^{64} T^{9} + \)\(28\!\cdots\!26\)\( p^{84} T^{10} + 3895624305077596 p^{105} T^{11} + p^{126} T^{12} \)
31 \( 1 - 492586859482706 T + \)\(88\!\cdots\!73\)\( T^{2} - \)\(12\!\cdots\!38\)\( T^{3} + \)\(36\!\cdots\!78\)\( T^{4} - \)\(58\!\cdots\!54\)\( T^{5} + \)\(93\!\cdots\!37\)\( T^{6} - \)\(58\!\cdots\!54\)\( p^{21} T^{7} + \)\(36\!\cdots\!78\)\( p^{42} T^{8} - \)\(12\!\cdots\!38\)\( p^{63} T^{9} + \)\(88\!\cdots\!73\)\( p^{84} T^{10} - 492586859482706 p^{105} T^{11} + p^{126} T^{12} \)
37 \( 1 + 26101348744076316 T + \)\(26\!\cdots\!14\)\( T^{2} + \)\(56\!\cdots\!68\)\( T^{3} + \)\(29\!\cdots\!11\)\( T^{4} + \)\(54\!\cdots\!28\)\( T^{5} + \)\(23\!\cdots\!88\)\( T^{6} + \)\(54\!\cdots\!28\)\( p^{21} T^{7} + \)\(29\!\cdots\!11\)\( p^{42} T^{8} + \)\(56\!\cdots\!68\)\( p^{63} T^{9} + \)\(26\!\cdots\!14\)\( p^{84} T^{10} + 26101348744076316 p^{105} T^{11} + p^{126} T^{12} \)
41 \( 1 - 112662882011424056 T + \)\(28\!\cdots\!18\)\( T^{2} - \)\(27\!\cdots\!28\)\( T^{3} + \)\(37\!\cdots\!63\)\( T^{4} - \)\(31\!\cdots\!64\)\( T^{5} + \)\(31\!\cdots\!92\)\( T^{6} - \)\(31\!\cdots\!64\)\( p^{21} T^{7} + \)\(37\!\cdots\!63\)\( p^{42} T^{8} - \)\(27\!\cdots\!28\)\( p^{63} T^{9} + \)\(28\!\cdots\!18\)\( p^{84} T^{10} - 112662882011424056 p^{105} T^{11} + p^{126} T^{12} \)
43 \( 1 + 13377804412633810 T + \)\(82\!\cdots\!25\)\( T^{2} + \)\(15\!\cdots\!30\)\( T^{3} + \)\(76\!\cdots\!54\)\( p T^{4} + \)\(64\!\cdots\!30\)\( T^{5} + \)\(80\!\cdots\!25\)\( T^{6} + \)\(64\!\cdots\!30\)\( p^{21} T^{7} + \)\(76\!\cdots\!54\)\( p^{43} T^{8} + \)\(15\!\cdots\!30\)\( p^{63} T^{9} + \)\(82\!\cdots\!25\)\( p^{84} T^{10} + 13377804412633810 p^{105} T^{11} + p^{126} T^{12} \)
47 \( 1 + 243206274450163780 T + \)\(59\!\cdots\!30\)\( T^{2} + \)\(11\!\cdots\!60\)\( T^{3} + \)\(15\!\cdots\!27\)\( T^{4} + \)\(24\!\cdots\!40\)\( T^{5} + \)\(25\!\cdots\!40\)\( T^{6} + \)\(24\!\cdots\!40\)\( p^{21} T^{7} + \)\(15\!\cdots\!27\)\( p^{42} T^{8} + \)\(11\!\cdots\!60\)\( p^{63} T^{9} + \)\(59\!\cdots\!30\)\( p^{84} T^{10} + 243206274450163780 p^{105} T^{11} + p^{126} T^{12} \)
53 \( 1 - 554044170675798992 T + \)\(57\!\cdots\!26\)\( T^{2} - \)\(13\!\cdots\!84\)\( T^{3} + \)\(16\!\cdots\!51\)\( T^{4} - \)\(25\!\cdots\!16\)\( T^{5} + \)\(33\!\cdots\!92\)\( T^{6} - \)\(25\!\cdots\!16\)\( p^{21} T^{7} + \)\(16\!\cdots\!51\)\( p^{42} T^{8} - \)\(13\!\cdots\!84\)\( p^{63} T^{9} + \)\(57\!\cdots\!26\)\( p^{84} T^{10} - 554044170675798992 p^{105} T^{11} + p^{126} T^{12} \)
59 \( 1 + 7003944869710758212 T + \)\(64\!\cdots\!02\)\( T^{2} + \)\(19\!\cdots\!04\)\( T^{3} + \)\(87\!\cdots\!43\)\( T^{4} - \)\(92\!\cdots\!52\)\( T^{5} + \)\(38\!\cdots\!08\)\( T^{6} - \)\(92\!\cdots\!52\)\( p^{21} T^{7} + \)\(87\!\cdots\!43\)\( p^{42} T^{8} + \)\(19\!\cdots\!04\)\( p^{63} T^{9} + \)\(64\!\cdots\!02\)\( p^{84} T^{10} + 7003944869710758212 p^{105} T^{11} + p^{126} T^{12} \)
61 \( 1 + 10235053877143753790 T + \)\(13\!\cdots\!09\)\( T^{2} + \)\(91\!\cdots\!30\)\( T^{3} + \)\(82\!\cdots\!90\)\( T^{4} + \)\(75\!\cdots\!30\)\( p T^{5} + \)\(32\!\cdots\!05\)\( T^{6} + \)\(75\!\cdots\!30\)\( p^{22} T^{7} + \)\(82\!\cdots\!90\)\( p^{42} T^{8} + \)\(91\!\cdots\!30\)\( p^{63} T^{9} + \)\(13\!\cdots\!09\)\( p^{84} T^{10} + 10235053877143753790 p^{105} T^{11} + p^{126} T^{12} \)
67 \( 1 - 8975831852618567158 T + \)\(10\!\cdots\!37\)\( T^{2} - \)\(10\!\cdots\!70\)\( T^{3} + \)\(51\!\cdots\!30\)\( T^{4} - \)\(50\!\cdots\!18\)\( T^{5} + \)\(14\!\cdots\!09\)\( T^{6} - \)\(50\!\cdots\!18\)\( p^{21} T^{7} + \)\(51\!\cdots\!30\)\( p^{42} T^{8} - \)\(10\!\cdots\!70\)\( p^{63} T^{9} + \)\(10\!\cdots\!37\)\( p^{84} T^{10} - 8975831852618567158 p^{105} T^{11} + p^{126} T^{12} \)
71 \( 1 - 18009408229245489272 T + \)\(57\!\cdots\!86\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!95\)\( T^{4} - \)\(96\!\cdots\!92\)\( T^{5} + \)\(10\!\cdots\!44\)\( T^{6} - \)\(96\!\cdots\!92\)\( p^{21} T^{7} + \)\(19\!\cdots\!95\)\( p^{42} T^{8} + \)\(15\!\cdots\!80\)\( p^{63} T^{9} + \)\(57\!\cdots\!86\)\( p^{84} T^{10} - 18009408229245489272 p^{105} T^{11} + p^{126} T^{12} \)
73 \( 1 - 81172357549902895652 T + \)\(84\!\cdots\!86\)\( T^{2} - \)\(43\!\cdots\!24\)\( T^{3} + \)\(26\!\cdots\!31\)\( T^{4} - \)\(10\!\cdots\!16\)\( T^{5} + \)\(46\!\cdots\!72\)\( T^{6} - \)\(10\!\cdots\!16\)\( p^{21} T^{7} + \)\(26\!\cdots\!31\)\( p^{42} T^{8} - \)\(43\!\cdots\!24\)\( p^{63} T^{9} + \)\(84\!\cdots\!86\)\( p^{84} T^{10} - 81172357549902895652 p^{105} T^{11} + p^{126} T^{12} \)
79 \( 1 + 79149540279186452080 T + \)\(25\!\cdots\!74\)\( T^{2} + \)\(15\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!15\)\( T^{4} + \)\(17\!\cdots\!00\)\( T^{5} + \)\(29\!\cdots\!80\)\( T^{6} + \)\(17\!\cdots\!00\)\( p^{21} T^{7} + \)\(33\!\cdots\!15\)\( p^{42} T^{8} + \)\(15\!\cdots\!00\)\( p^{63} T^{9} + \)\(25\!\cdots\!74\)\( p^{84} T^{10} + 79149540279186452080 p^{105} T^{11} + p^{126} T^{12} \)
83 \( 1 - \)\(23\!\cdots\!56\)\( T + \)\(83\!\cdots\!06\)\( T^{2} - \)\(16\!\cdots\!32\)\( T^{3} + \)\(34\!\cdots\!47\)\( T^{4} - \)\(55\!\cdots\!92\)\( T^{5} + \)\(84\!\cdots\!76\)\( T^{6} - \)\(55\!\cdots\!92\)\( p^{21} T^{7} + \)\(34\!\cdots\!47\)\( p^{42} T^{8} - \)\(16\!\cdots\!32\)\( p^{63} T^{9} + \)\(83\!\cdots\!06\)\( p^{84} T^{10} - \)\(23\!\cdots\!56\)\( p^{105} T^{11} + p^{126} T^{12} \)
89 \( 1 + \)\(71\!\cdots\!08\)\( T + \)\(53\!\cdots\!62\)\( T^{2} + \)\(25\!\cdots\!96\)\( T^{3} + \)\(11\!\cdots\!23\)\( T^{4} + \)\(39\!\cdots\!92\)\( T^{5} + \)\(13\!\cdots\!28\)\( T^{6} + \)\(39\!\cdots\!92\)\( p^{21} T^{7} + \)\(11\!\cdots\!23\)\( p^{42} T^{8} + \)\(25\!\cdots\!96\)\( p^{63} T^{9} + \)\(53\!\cdots\!62\)\( p^{84} T^{10} + \)\(71\!\cdots\!08\)\( p^{105} T^{11} + p^{126} T^{12} \)
97 \( 1 - \)\(14\!\cdots\!78\)\( T + \)\(26\!\cdots\!17\)\( T^{2} - \)\(20\!\cdots\!70\)\( T^{3} + \)\(20\!\cdots\!30\)\( T^{4} - \)\(10\!\cdots\!18\)\( T^{5} + \)\(97\!\cdots\!69\)\( T^{6} - \)\(10\!\cdots\!18\)\( p^{21} T^{7} + \)\(20\!\cdots\!30\)\( p^{42} T^{8} - \)\(20\!\cdots\!70\)\( p^{63} T^{9} + \)\(26\!\cdots\!17\)\( p^{84} T^{10} - \)\(14\!\cdots\!78\)\( p^{105} T^{11} + p^{126} T^{12} \)
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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

−5.32269986807292509432800869227, −4.92382130144626333009008673417, −4.86962128316358781525345112834, −4.78059883177774675264258757892, −4.70288980015461378718692912918, −4.69962312363552939119804011810, −4.63834832277891184665773100896, −3.89389040332164542194620265297, −3.88900399668348773016807928200, −3.74203256674241779179783019709, −3.59260968897993316390598011360, −3.55777017047490422171724713475, −3.07458598180431227795612741114, −2.73551200462672116027122970767, −2.64280519250350795021789583125, −2.45359339868058502071877342299, −2.03477515415093804182944642053, −1.86347746446649611117619491840, −1.73914552788353847110953939966, −1.42424124332925028921236891201, −1.30994891143682443975852466345, −1.01595089010952372099462337105, −1.00621754014060383817051786073, −0.927648764139489564594197663028, −0.794658356326118763510337156067, 0, 0, 0, 0, 0, 0, 0.794658356326118763510337156067, 0.927648764139489564594197663028, 1.00621754014060383817051786073, 1.01595089010952372099462337105, 1.30994891143682443975852466345, 1.42424124332925028921236891201, 1.73914552788353847110953939966, 1.86347746446649611117619491840, 2.03477515415093804182944642053, 2.45359339868058502071877342299, 2.64280519250350795021789583125, 2.73551200462672116027122970767, 3.07458598180431227795612741114, 3.55777017047490422171724713475, 3.59260968897993316390598011360, 3.74203256674241779179783019709, 3.88900399668348773016807928200, 3.89389040332164542194620265297, 4.63834832277891184665773100896, 4.69962312363552939119804011810, 4.70288980015461378718692912918, 4.78059883177774675264258757892, 4.86962128316358781525345112834, 4.92382130144626333009008673417, 5.32269986807292509432800869227

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

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