Dirichlet series
| L(s) = 1 | − 84·3-s − 3.75e3·5-s + 1.44e4·7-s − 5.32e4·9-s − 2.98e4·11-s + 1.62e4·13-s + 3.15e5·15-s − 3.73e5·17-s + 4.26e5·19-s − 1.21e6·21-s − 6.95e5·23-s + 8.20e6·25-s + 4.10e6·27-s + 1.80e5·29-s − 2.76e6·31-s + 2.51e6·33-s − 5.40e7·35-s + 1.69e7·37-s − 1.36e6·39-s − 2.44e6·41-s + 4.53e7·43-s + 1.99e8·45-s − 6.90e6·47-s + 1.21e8·49-s + 3.13e7·51-s + 5.78e7·53-s + 1.12e8·55-s + ⋯ |
| L(s) = 1 | − 0.598·3-s − 2.68·5-s + 2.26·7-s − 2.70·9-s − 0.615·11-s + 0.158·13-s + 1.60·15-s − 1.08·17-s + 0.751·19-s − 1.35·21-s − 0.517·23-s + 21/5·25-s + 1.48·27-s + 0.0475·29-s − 0.538·31-s + 0.368·33-s − 6.08·35-s + 1.48·37-s − 0.0947·39-s − 0.134·41-s + 2.02·43-s + 7.25·45-s − 0.206·47-s + 3·49-s + 0.649·51-s + 1.00·53-s + 1.65·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{18} \cdot 5^{6} \cdot 7^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.99441\times 10^{12}\) |
| Root analytic conductor: | \(12.0087\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 2^{18} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [9/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + p^{4} T )^{6} \) | |
| 7 | \( ( 1 - p^{4} T )^{6} \) | |
| good | 3 | \( 1 + 28 p T + 60272 T^{2} + 1808320 p T^{3} + 71883848 p^{3} T^{4} + 6780668900 p^{3} T^{5} + 545280228742 p^{4} T^{6} + 6780668900 p^{12} T^{7} + 71883848 p^{21} T^{8} + 1808320 p^{28} T^{9} + 60272 p^{36} T^{10} + 28 p^{46} T^{11} + p^{54} T^{12} \) |
| 11 | \( 1 + 29892 T + 6111066568 T^{2} + 169017564912376 T^{3} + 21581459145182019536 T^{4} + \)\(45\!\cdots\!44\)\( T^{5} + \)\(60\!\cdots\!38\)\( T^{6} + \)\(45\!\cdots\!44\)\( p^{9} T^{7} + 21581459145182019536 p^{18} T^{8} + 169017564912376 p^{27} T^{9} + 6111066568 p^{36} T^{10} + 29892 p^{45} T^{11} + p^{54} T^{12} \) | |
| 13 | \( 1 - 16288 T + 48562815056 T^{2} - 524102114144964 T^{3} + \)\(11\!\cdots\!48\)\( T^{4} - \)\(91\!\cdots\!84\)\( T^{5} + \)\(14\!\cdots\!38\)\( T^{6} - \)\(91\!\cdots\!84\)\( p^{9} T^{7} + \)\(11\!\cdots\!48\)\( p^{18} T^{8} - 524102114144964 p^{27} T^{9} + 48562815056 p^{36} T^{10} - 16288 p^{45} T^{11} + p^{54} T^{12} \) | |
| 17 | \( 1 + 373400 T + 480258256056 T^{2} + 122446828586139036 T^{3} + \)\(92\!\cdots\!64\)\( T^{4} + \)\(17\!\cdots\!44\)\( T^{5} + \)\(11\!\cdots\!34\)\( T^{6} + \)\(17\!\cdots\!44\)\( p^{9} T^{7} + \)\(92\!\cdots\!64\)\( p^{18} T^{8} + 122446828586139036 p^{27} T^{9} + 480258256056 p^{36} T^{10} + 373400 p^{45} T^{11} + p^{54} T^{12} \) | |
| 19 | \( 1 - 22464 p T + 1139121288298 T^{2} - 190328405548195072 T^{3} + \)\(56\!\cdots\!43\)\( T^{4} - \)\(26\!\cdots\!76\)\( T^{5} + \)\(20\!\cdots\!52\)\( T^{6} - \)\(26\!\cdots\!76\)\( p^{9} T^{7} + \)\(56\!\cdots\!43\)\( p^{18} T^{8} - 190328405548195072 p^{27} T^{9} + 1139121288298 p^{36} T^{10} - 22464 p^{46} T^{11} + p^{54} T^{12} \) | |
| 23 | \( 1 + 695088 T + 7172597876482 T^{2} + 4492323029184542800 T^{3} + \)\(25\!\cdots\!03\)\( T^{4} + \)\(13\!\cdots\!84\)\( T^{5} + \)\(58\!\cdots\!64\)\( T^{6} + \)\(13\!\cdots\!84\)\( p^{9} T^{7} + \)\(25\!\cdots\!03\)\( p^{18} T^{8} + 4492323029184542800 p^{27} T^{9} + 7172597876482 p^{36} T^{10} + 695088 p^{45} T^{11} + p^{54} T^{12} \) | |
| 29 | \( 1 - 180920 T + 47767409117880 T^{2} + 3089837988600784980 p T^{3} + \)\(99\!\cdots\!04\)\( T^{4} + \)\(34\!\cdots\!88\)\( T^{5} + \)\(15\!\cdots\!22\)\( T^{6} + \)\(34\!\cdots\!88\)\( p^{9} T^{7} + \)\(99\!\cdots\!04\)\( p^{18} T^{8} + 3089837988600784980 p^{28} T^{9} + 47767409117880 p^{36} T^{10} - 180920 p^{45} T^{11} + p^{54} T^{12} \) | |
| 31 | \( 1 + 2769080 T + 68527483782122 T^{2} + 98188035906598075240 T^{3} + \)\(28\!\cdots\!83\)\( T^{4} + \)\(29\!\cdots\!20\)\( T^{5} + \)\(86\!\cdots\!04\)\( T^{6} + \)\(29\!\cdots\!20\)\( p^{9} T^{7} + \)\(28\!\cdots\!83\)\( p^{18} T^{8} + 98188035906598075240 p^{27} T^{9} + 68527483782122 p^{36} T^{10} + 2769080 p^{45} T^{11} + p^{54} T^{12} \) | |
| 37 | \( 1 - 16932844 T + 11299181492306 p T^{2} - \)\(35\!\cdots\!60\)\( T^{3} + \)\(52\!\cdots\!15\)\( T^{4} - \)\(25\!\cdots\!96\)\( T^{5} + \)\(51\!\cdots\!44\)\( T^{6} - \)\(25\!\cdots\!96\)\( p^{9} T^{7} + \)\(52\!\cdots\!15\)\( p^{18} T^{8} - \)\(35\!\cdots\!60\)\( p^{27} T^{9} + 11299181492306 p^{37} T^{10} - 16932844 p^{45} T^{11} + p^{54} T^{12} \) | |
| 41 | \( 1 + 2441028 T + 1511459974545930 T^{2} + \)\(59\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!75\)\( T^{4} - \)\(57\!\cdots\!04\)\( T^{5} + \)\(41\!\cdots\!88\)\( T^{6} - \)\(57\!\cdots\!04\)\( p^{9} T^{7} + \)\(10\!\cdots\!75\)\( p^{18} T^{8} + \)\(59\!\cdots\!36\)\( p^{27} T^{9} + 1511459974545930 p^{36} T^{10} + 2441028 p^{45} T^{11} + p^{54} T^{12} \) | |
| 43 | \( 1 - 45387448 T + 2492081743949130 T^{2} - \)\(78\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!07\)\( T^{4} - \)\(65\!\cdots\!52\)\( T^{5} + \)\(16\!\cdots\!40\)\( T^{6} - \)\(65\!\cdots\!52\)\( p^{9} T^{7} + \)\(26\!\cdots\!07\)\( p^{18} T^{8} - \)\(78\!\cdots\!20\)\( p^{27} T^{9} + 2492081743949130 p^{36} T^{10} - 45387448 p^{45} T^{11} + p^{54} T^{12} \) | |
| 47 | \( 1 + 6904132 T + 2014218733610568 T^{2} + \)\(19\!\cdots\!48\)\( T^{3} + \)\(41\!\cdots\!00\)\( T^{4} + \)\(69\!\cdots\!88\)\( p T^{5} + \)\(50\!\cdots\!70\)\( T^{6} + \)\(69\!\cdots\!88\)\( p^{10} T^{7} + \)\(41\!\cdots\!00\)\( p^{18} T^{8} + \)\(19\!\cdots\!48\)\( p^{27} T^{9} + 2014218733610568 p^{36} T^{10} + 6904132 p^{45} T^{11} + p^{54} T^{12} \) | |
| 53 | \( 1 - 57875148 T + 11718330329393698 T^{2} - \)\(58\!\cdots\!80\)\( T^{3} + \)\(68\!\cdots\!43\)\( T^{4} - \)\(29\!\cdots\!56\)\( T^{5} + \)\(26\!\cdots\!00\)\( T^{6} - \)\(29\!\cdots\!56\)\( p^{9} T^{7} + \)\(68\!\cdots\!43\)\( p^{18} T^{8} - \)\(58\!\cdots\!80\)\( p^{27} T^{9} + 11718330329393698 p^{36} T^{10} - 57875148 p^{45} T^{11} + p^{54} T^{12} \) | |
| 59 | \( 1 + 384608 T + 26722865554688034 T^{2} + \)\(96\!\cdots\!88\)\( T^{3} + \)\(30\!\cdots\!47\)\( T^{4} + \)\(23\!\cdots\!20\)\( T^{5} + \)\(26\!\cdots\!92\)\( T^{6} + \)\(23\!\cdots\!20\)\( p^{9} T^{7} + \)\(30\!\cdots\!47\)\( p^{18} T^{8} + \)\(96\!\cdots\!88\)\( p^{27} T^{9} + 26722865554688034 p^{36} T^{10} + 384608 p^{45} T^{11} + p^{54} T^{12} \) | |
| 61 | \( 1 - 152779164 T + 59717062425231874 T^{2} - \)\(74\!\cdots\!36\)\( T^{3} + \)\(15\!\cdots\!27\)\( T^{4} - \)\(16\!\cdots\!72\)\( T^{5} + \)\(23\!\cdots\!48\)\( T^{6} - \)\(16\!\cdots\!72\)\( p^{9} T^{7} + \)\(15\!\cdots\!27\)\( p^{18} T^{8} - \)\(74\!\cdots\!36\)\( p^{27} T^{9} + 59717062425231874 p^{36} T^{10} - 152779164 p^{45} T^{11} + p^{54} T^{12} \) | |
| 67 | \( 1 + 175266152 T + 87723420173799746 T^{2} + \)\(14\!\cdots\!12\)\( T^{3} + \)\(38\!\cdots\!11\)\( T^{4} + \)\(68\!\cdots\!28\)\( T^{5} + \)\(12\!\cdots\!88\)\( T^{6} + \)\(68\!\cdots\!28\)\( p^{9} T^{7} + \)\(38\!\cdots\!11\)\( p^{18} T^{8} + \)\(14\!\cdots\!12\)\( p^{27} T^{9} + 87723420173799746 p^{36} T^{10} + 175266152 p^{45} T^{11} + p^{54} T^{12} \) | |
| 71 | \( 1 + 341384512 T + 233127935086967082 T^{2} + \)\(70\!\cdots\!28\)\( T^{3} + \)\(24\!\cdots\!67\)\( T^{4} + \)\(61\!\cdots\!84\)\( T^{5} + \)\(14\!\cdots\!48\)\( T^{6} + \)\(61\!\cdots\!84\)\( p^{9} T^{7} + \)\(24\!\cdots\!67\)\( p^{18} T^{8} + \)\(70\!\cdots\!28\)\( p^{27} T^{9} + 233127935086967082 p^{36} T^{10} + 341384512 p^{45} T^{11} + p^{54} T^{12} \) | |
| 73 | \( 1 - 350356156 T + 189619003746551826 T^{2} - \)\(53\!\cdots\!92\)\( T^{3} + \)\(14\!\cdots\!99\)\( T^{4} - \)\(43\!\cdots\!16\)\( T^{5} + \)\(88\!\cdots\!80\)\( T^{6} - \)\(43\!\cdots\!16\)\( p^{9} T^{7} + \)\(14\!\cdots\!99\)\( p^{18} T^{8} - \)\(53\!\cdots\!92\)\( p^{27} T^{9} + 189619003746551826 p^{36} T^{10} - 350356156 p^{45} T^{11} + p^{54} T^{12} \) | |
| 79 | \( 1 + 394960660 T + 578587858223504512 T^{2} + \)\(16\!\cdots\!12\)\( T^{3} + \)\(14\!\cdots\!24\)\( T^{4} + \)\(32\!\cdots\!84\)\( T^{5} + \)\(21\!\cdots\!54\)\( T^{6} + \)\(32\!\cdots\!84\)\( p^{9} T^{7} + \)\(14\!\cdots\!24\)\( p^{18} T^{8} + \)\(16\!\cdots\!12\)\( p^{27} T^{9} + 578587858223504512 p^{36} T^{10} + 394960660 p^{45} T^{11} + p^{54} T^{12} \) | |
| 83 | \( 1 + 1201830720 T + 645086872070323730 T^{2} + \)\(19\!\cdots\!40\)\( T^{3} + \)\(80\!\cdots\!99\)\( T^{4} + \)\(62\!\cdots\!00\)\( T^{5} + \)\(36\!\cdots\!12\)\( T^{6} + \)\(62\!\cdots\!00\)\( p^{9} T^{7} + \)\(80\!\cdots\!99\)\( p^{18} T^{8} + \)\(19\!\cdots\!40\)\( p^{27} T^{9} + 645086872070323730 p^{36} T^{10} + 1201830720 p^{45} T^{11} + p^{54} T^{12} \) | |
| 89 | \( 1 + 84565172 T + 1417678585647687562 T^{2} + \)\(15\!\cdots\!96\)\( T^{3} + \)\(95\!\cdots\!75\)\( T^{4} + \)\(11\!\cdots\!64\)\( T^{5} + \)\(40\!\cdots\!68\)\( T^{6} + \)\(11\!\cdots\!64\)\( p^{9} T^{7} + \)\(95\!\cdots\!75\)\( p^{18} T^{8} + \)\(15\!\cdots\!96\)\( p^{27} T^{9} + 1417678585647687562 p^{36} T^{10} + 84565172 p^{45} T^{11} + p^{54} T^{12} \) | |
| 97 | \( 1 + 1933233544 T + 5344538294692433448 T^{2} + \)\(70\!\cdots\!16\)\( T^{3} + \)\(10\!\cdots\!04\)\( T^{4} + \)\(10\!\cdots\!16\)\( T^{5} + \)\(11\!\cdots\!30\)\( T^{6} + \)\(10\!\cdots\!16\)\( p^{9} T^{7} + \)\(10\!\cdots\!04\)\( p^{18} T^{8} + \)\(70\!\cdots\!16\)\( p^{27} T^{9} + 5344538294692433448 p^{36} T^{10} + 1933233544 p^{45} T^{11} + p^{54} T^{12} \) | |
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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
−5.63448670062635522194268648865, −5.06092689114510165785469116682, −4.90463004842734306866389012607, −4.86058911135300050122241704209, −4.69435537565843170536281964688, −4.60624557975204076352214933339, −4.44610205901894749020592650930, −4.10650394713657760481815669507, −3.81512770286379136579094806920, −3.72074443968499508997911211911, −3.69605432082374688613144458560, −3.49658464334327882748368083043, −3.38677172512467128737740802069, −2.68525135939489879493425020718, −2.64675347777135884011263435639, −2.50497405464325648092799182265, −2.49611564391781744926073593898, −2.45867887114359448961682495008, −2.27971363628863246117659360991, −1.49856959067265986936932412708, −1.44093043305659129425549500308, −1.19143267991024585489674114768, −1.11354034945675402909219165906, −0.982255513024838762416242347523, −0.876227984993091406999710577385, 0, 0, 0, 0, 0, 0, 0.876227984993091406999710577385, 0.982255513024838762416242347523, 1.11354034945675402909219165906, 1.19143267991024585489674114768, 1.44093043305659129425549500308, 1.49856959067265986936932412708, 2.27971363628863246117659360991, 2.45867887114359448961682495008, 2.49611564391781744926073593898, 2.50497405464325648092799182265, 2.64675347777135884011263435639, 2.68525135939489879493425020718, 3.38677172512467128737740802069, 3.49658464334327882748368083043, 3.69605432082374688613144458560, 3.72074443968499508997911211911, 3.81512770286379136579094806920, 4.10650394713657760481815669507, 4.44610205901894749020592650930, 4.60624557975204076352214933339, 4.69435537565843170536281964688, 4.86058911135300050122241704209, 4.90463004842734306866389012607, 5.06092689114510165785469116682, 5.63448670062635522194268648865
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.