Dirichlet series
| L(s) = 1 | + 68·3-s − 176·5-s + 9.66e3·7-s − 4.32e4·9-s + 1.59e4·11-s + 1.71e5·13-s − 1.19e4·15-s + 2.15e5·17-s + 5.19e5·19-s + 6.57e5·21-s + 6.91e5·23-s − 5.94e6·25-s − 2.96e6·27-s − 5.46e5·29-s + 5.35e5·31-s + 1.08e6·33-s − 1.70e6·35-s − 1.87e4·37-s + 1.16e7·39-s − 8.89e6·41-s + 2.13e7·43-s + 7.61e6·45-s + 6.03e7·47-s − 1.12e8·49-s + 1.46e7·51-s − 9.51e7·53-s − 2.81e6·55-s + ⋯ |
| L(s) = 1 | + 0.484·3-s − 0.125·5-s + 1.52·7-s − 2.19·9-s + 0.329·11-s + 1.66·13-s − 0.0610·15-s + 0.624·17-s + 0.915·19-s + 0.737·21-s + 0.515·23-s − 3.04·25-s − 1.07·27-s − 0.143·29-s + 0.104·31-s + 0.159·33-s − 0.191·35-s − 0.00164·37-s + 0.806·39-s − 0.491·41-s + 0.953·43-s + 0.276·45-s + 1.80·47-s − 2.79·49-s + 0.302·51-s − 1.65·53-s − 0.0414·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{24} \cdot 13^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.51148\times 10^{12}\) |
| Root analytic conductor: | \(10.3502\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{24} \cdot 13^{6} ,\ ( \ : [9/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(24.95670722\) |
| \(L(\frac12)\) | \(\approx\) | \(24.95670722\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 13 | \( ( 1 - p^{4} T )^{6} \) | |
| good | 3 | \( 1 - 68 T + 15964 p T^{2} - 359624 p^{2} T^{3} + 57208420 p^{3} T^{4} - 1246127204 p^{4} T^{5} + 154492204066 p^{5} T^{6} - 1246127204 p^{13} T^{7} + 57208420 p^{21} T^{8} - 359624 p^{29} T^{9} + 15964 p^{37} T^{10} - 68 p^{45} T^{11} + p^{54} T^{12} \) |
| 5 | \( 1 + 176 T + 5972904 T^{2} - 990619836 T^{3} + 4104867114072 p T^{4} - 111516379165376 p^{2} T^{5} + 80430587420758142 p^{4} T^{6} - 111516379165376 p^{11} T^{7} + 4104867114072 p^{19} T^{8} - 990619836 p^{27} T^{9} + 5972904 p^{36} T^{10} + 176 p^{45} T^{11} + p^{54} T^{12} \) | |
| 7 | \( 1 - 9664 T + 205982460 T^{2} - 207381974452 p T^{3} + 371517987365316 p^{2} T^{4} - 291954749447357040 p^{3} T^{5} + \)\(38\!\cdots\!02\)\( p^{4} T^{6} - 291954749447357040 p^{12} T^{7} + 371517987365316 p^{20} T^{8} - 207381974452 p^{28} T^{9} + 205982460 p^{36} T^{10} - 9664 p^{45} T^{11} + p^{54} T^{12} \) | |
| 11 | \( 1 - 15980 T + 4075723226 T^{2} - 349167177107028 T^{3} + 1365036502310206453 p T^{4} - \)\(88\!\cdots\!72\)\( T^{5} + \)\(64\!\cdots\!40\)\( T^{6} - \)\(88\!\cdots\!72\)\( p^{9} T^{7} + 1365036502310206453 p^{19} T^{8} - 349167177107028 p^{27} T^{9} + 4075723226 p^{36} T^{10} - 15980 p^{45} T^{11} + p^{54} T^{12} \) | |
| 17 | \( 1 - 215048 T + 411854330672 T^{2} - 54622194681783332 T^{3} + \)\(84\!\cdots\!52\)\( T^{4} - \)\(69\!\cdots\!96\)\( T^{5} + \)\(11\!\cdots\!86\)\( T^{6} - \)\(69\!\cdots\!96\)\( p^{9} T^{7} + \)\(84\!\cdots\!52\)\( p^{18} T^{8} - 54622194681783332 p^{27} T^{9} + 411854330672 p^{36} T^{10} - 215048 p^{45} T^{11} + p^{54} T^{12} \) | |
| 19 | \( 1 - 519868 T + 1144019768314 T^{2} - 683575449521647044 T^{3} + \)\(69\!\cdots\!43\)\( T^{4} - \)\(39\!\cdots\!84\)\( T^{5} + \)\(27\!\cdots\!08\)\( T^{6} - \)\(39\!\cdots\!84\)\( p^{9} T^{7} + \)\(69\!\cdots\!43\)\( p^{18} T^{8} - 683575449521647044 p^{27} T^{9} + 1144019768314 p^{36} T^{10} - 519868 p^{45} T^{11} + p^{54} T^{12} \) | |
| 23 | \( 1 - 691224 T + 7110282037706 T^{2} - 5713036781735609032 T^{3} + \)\(23\!\cdots\!07\)\( T^{4} - \)\(19\!\cdots\!76\)\( T^{5} + \)\(49\!\cdots\!92\)\( T^{6} - \)\(19\!\cdots\!76\)\( p^{9} T^{7} + \)\(23\!\cdots\!07\)\( p^{18} T^{8} - 5713036781735609032 p^{27} T^{9} + 7110282037706 p^{36} T^{10} - 691224 p^{45} T^{11} + p^{54} T^{12} \) | |
| 29 | \( 1 + 546932 T + 55030694948218 T^{2} + 15518941127778012948 T^{3} + \)\(14\!\cdots\!51\)\( T^{4} + \)\(23\!\cdots\!56\)\( T^{5} + \)\(26\!\cdots\!28\)\( T^{6} + \)\(23\!\cdots\!56\)\( p^{9} T^{7} + \)\(14\!\cdots\!51\)\( p^{18} T^{8} + 15518941127778012948 p^{27} T^{9} + 55030694948218 p^{36} T^{10} + 546932 p^{45} T^{11} + p^{54} T^{12} \) | |
| 31 | \( 1 - 535300 T + 141886964252506 T^{2} - 36813181696591465708 T^{3} + \)\(87\!\cdots\!03\)\( T^{4} - \)\(10\!\cdots\!52\)\( T^{5} + \)\(29\!\cdots\!00\)\( T^{6} - \)\(10\!\cdots\!52\)\( p^{9} T^{7} + \)\(87\!\cdots\!03\)\( p^{18} T^{8} - 36813181696591465708 p^{27} T^{9} + 141886964252506 p^{36} T^{10} - 535300 p^{45} T^{11} + p^{54} T^{12} \) | |
| 37 | \( 1 + 18760 T + 344824755411344 T^{2} + \)\(18\!\cdots\!36\)\( T^{3} + \)\(59\!\cdots\!52\)\( T^{4} + \)\(55\!\cdots\!64\)\( T^{5} + \)\(78\!\cdots\!22\)\( T^{6} + \)\(55\!\cdots\!64\)\( p^{9} T^{7} + \)\(59\!\cdots\!52\)\( p^{18} T^{8} + \)\(18\!\cdots\!36\)\( p^{27} T^{9} + 344824755411344 p^{36} T^{10} + 18760 p^{45} T^{11} + p^{54} T^{12} \) | |
| 41 | \( 1 + 8895252 T + 1723518102702778 T^{2} + \)\(14\!\cdots\!96\)\( T^{3} + \)\(12\!\cdots\!91\)\( T^{4} + \)\(98\!\cdots\!08\)\( T^{5} + \)\(13\!\cdots\!68\)\( p T^{6} + \)\(98\!\cdots\!08\)\( p^{9} T^{7} + \)\(12\!\cdots\!91\)\( p^{18} T^{8} + \)\(14\!\cdots\!96\)\( p^{27} T^{9} + 1723518102702778 p^{36} T^{10} + 8895252 p^{45} T^{11} + p^{54} T^{12} \) | |
| 43 | \( 1 - 21376108 T + 2412221622373052 T^{2} - \)\(44\!\cdots\!40\)\( T^{3} + \)\(27\!\cdots\!08\)\( T^{4} - \)\(40\!\cdots\!72\)\( T^{5} + \)\(17\!\cdots\!46\)\( T^{6} - \)\(40\!\cdots\!72\)\( p^{9} T^{7} + \)\(27\!\cdots\!08\)\( p^{18} T^{8} - \)\(44\!\cdots\!40\)\( p^{27} T^{9} + 2412221622373052 p^{36} T^{10} - 21376108 p^{45} T^{11} + p^{54} T^{12} \) | |
| 47 | \( 1 - 60317048 T + 5344345906182148 T^{2} - \)\(27\!\cdots\!56\)\( T^{3} + \)\(12\!\cdots\!52\)\( T^{4} - \)\(54\!\cdots\!76\)\( T^{5} + \)\(18\!\cdots\!10\)\( T^{6} - \)\(54\!\cdots\!76\)\( p^{9} T^{7} + \)\(12\!\cdots\!52\)\( p^{18} T^{8} - \)\(27\!\cdots\!56\)\( p^{27} T^{9} + 5344345906182148 p^{36} T^{10} - 60317048 p^{45} T^{11} + p^{54} T^{12} \) | |
| 53 | \( 1 + 95149004 T + 13435212476775554 T^{2} + \)\(10\!\cdots\!88\)\( T^{3} + \)\(90\!\cdots\!55\)\( T^{4} + \)\(54\!\cdots\!04\)\( T^{5} + \)\(36\!\cdots\!36\)\( T^{6} + \)\(54\!\cdots\!04\)\( p^{9} T^{7} + \)\(90\!\cdots\!55\)\( p^{18} T^{8} + \)\(10\!\cdots\!88\)\( p^{27} T^{9} + 13435212476775554 p^{36} T^{10} + 95149004 p^{45} T^{11} + p^{54} T^{12} \) | |
| 59 | \( 1 - 181069068 T + 43715287396438714 T^{2} - \)\(38\!\cdots\!56\)\( T^{3} + \)\(50\!\cdots\!47\)\( T^{4} - \)\(20\!\cdots\!44\)\( T^{5} + \)\(33\!\cdots\!12\)\( T^{6} - \)\(20\!\cdots\!44\)\( p^{9} T^{7} + \)\(50\!\cdots\!47\)\( p^{18} T^{8} - \)\(38\!\cdots\!56\)\( p^{27} T^{9} + 43715287396438714 p^{36} T^{10} - 181069068 p^{45} T^{11} + p^{54} T^{12} \) | |
| 61 | \( 1 + 169024572 T + 36982766927158418 T^{2} + \)\(40\!\cdots\!28\)\( T^{3} + \)\(64\!\cdots\!67\)\( T^{4} + \)\(64\!\cdots\!76\)\( T^{5} + \)\(92\!\cdots\!96\)\( T^{6} + \)\(64\!\cdots\!76\)\( p^{9} T^{7} + \)\(64\!\cdots\!67\)\( p^{18} T^{8} + \)\(40\!\cdots\!28\)\( p^{27} T^{9} + 36982766927158418 p^{36} T^{10} + 169024572 p^{45} T^{11} + p^{54} T^{12} \) | |
| 67 | \( 1 - 356012572 T + 186576639901864954 T^{2} - \)\(47\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!83\)\( T^{4} - \)\(25\!\cdots\!88\)\( T^{5} + \)\(49\!\cdots\!56\)\( T^{6} - \)\(25\!\cdots\!88\)\( p^{9} T^{7} + \)\(13\!\cdots\!83\)\( p^{18} T^{8} - \)\(47\!\cdots\!40\)\( p^{27} T^{9} + 186576639901864954 p^{36} T^{10} - 356012572 p^{45} T^{11} + p^{54} T^{12} \) | |
| 71 | \( 1 - 661588848 T + 323584227026634540 T^{2} - \)\(10\!\cdots\!12\)\( T^{3} + \)\(30\!\cdots\!32\)\( T^{4} - \)\(73\!\cdots\!76\)\( T^{5} + \)\(16\!\cdots\!86\)\( T^{6} - \)\(73\!\cdots\!76\)\( p^{9} T^{7} + \)\(30\!\cdots\!32\)\( p^{18} T^{8} - \)\(10\!\cdots\!12\)\( p^{27} T^{9} + 323584227026634540 p^{36} T^{10} - 661588848 p^{45} T^{11} + p^{54} T^{12} \) | |
| 73 | \( 1 + 315676732 T + 170082902385264226 T^{2} + \)\(31\!\cdots\!72\)\( T^{3} + \)\(13\!\cdots\!63\)\( T^{4} + \)\(18\!\cdots\!28\)\( T^{5} + \)\(77\!\cdots\!44\)\( T^{6} + \)\(18\!\cdots\!28\)\( p^{9} T^{7} + \)\(13\!\cdots\!63\)\( p^{18} T^{8} + \)\(31\!\cdots\!72\)\( p^{27} T^{9} + 170082902385264226 p^{36} T^{10} + 315676732 p^{45} T^{11} + p^{54} T^{12} \) | |
| 79 | \( 1 - 1268146056 T + 1273792919977288922 T^{2} - \)\(85\!\cdots\!08\)\( T^{3} + \)\(48\!\cdots\!11\)\( T^{4} - \)\(21\!\cdots\!20\)\( T^{5} + \)\(83\!\cdots\!12\)\( T^{6} - \)\(21\!\cdots\!20\)\( p^{9} T^{7} + \)\(48\!\cdots\!11\)\( p^{18} T^{8} - \)\(85\!\cdots\!08\)\( p^{27} T^{9} + 1273792919977288922 p^{36} T^{10} - 1268146056 p^{45} T^{11} + p^{54} T^{12} \) | |
| 83 | \( 1 - 1026540484 T + 829742469726860786 T^{2} - \)\(36\!\cdots\!88\)\( T^{3} + \)\(14\!\cdots\!71\)\( T^{4} - \)\(31\!\cdots\!88\)\( T^{5} + \)\(12\!\cdots\!36\)\( T^{6} - \)\(31\!\cdots\!88\)\( p^{9} T^{7} + \)\(14\!\cdots\!71\)\( p^{18} T^{8} - \)\(36\!\cdots\!88\)\( p^{27} T^{9} + 829742469726860786 p^{36} T^{10} - 1026540484 p^{45} T^{11} + p^{54} T^{12} \) | |
| 89 | \( 1 + 201055172 T + 1257673486253043058 T^{2} + \)\(42\!\cdots\!08\)\( T^{3} + \)\(81\!\cdots\!99\)\( T^{4} + \)\(27\!\cdots\!24\)\( T^{5} + \)\(35\!\cdots\!32\)\( T^{6} + \)\(27\!\cdots\!24\)\( p^{9} T^{7} + \)\(81\!\cdots\!99\)\( p^{18} T^{8} + \)\(42\!\cdots\!08\)\( p^{27} T^{9} + 1257673486253043058 p^{36} T^{10} + 201055172 p^{45} T^{11} + p^{54} T^{12} \) | |
| 97 | \( 1 - 566920580 T + 2715826450591846706 T^{2} - \)\(79\!\cdots\!84\)\( T^{3} + \)\(31\!\cdots\!63\)\( T^{4} - \)\(22\!\cdots\!64\)\( T^{5} + \)\(24\!\cdots\!00\)\( T^{6} - \)\(22\!\cdots\!64\)\( p^{9} T^{7} + \)\(31\!\cdots\!63\)\( p^{18} T^{8} - \)\(79\!\cdots\!84\)\( p^{27} T^{9} + 2715826450591846706 p^{36} T^{10} - 566920580 p^{45} T^{11} + p^{54} T^{12} \) | |
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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
−5.30697270469542945496903686328, −4.91664717032111069568895420522, −4.82212654502427116303100701149, −4.51965123768610050127595582910, −4.21351755944862401377473298911, −4.16428074661086271034689089494, −4.12467356963057697661068473374, −3.52000697403989347823087760672, −3.39290122965665658035819272086, −3.31469075667027151811112709151, −3.29675951499351319721446572679, −3.10730659873043153609009149819, −2.94567911298969885345145408264, −2.23662766701315276891055236001, −2.18701674705708496072915996131, −2.07701649844886886980470916968, −2.03375664544225356500983203018, −1.70435454649767689119748687023, −1.63871270240083400496281611999, −1.17078673870739115676480532859, −0.892444440575439117318069898817, −0.77822010880140179245153000561, −0.56084688168555401057950476491, −0.49601746325539122621097519508, −0.28512917802586920224904874020, 0.28512917802586920224904874020, 0.49601746325539122621097519508, 0.56084688168555401057950476491, 0.77822010880140179245153000561, 0.892444440575439117318069898817, 1.17078673870739115676480532859, 1.63871270240083400496281611999, 1.70435454649767689119748687023, 2.03375664544225356500983203018, 2.07701649844886886980470916968, 2.18701674705708496072915996131, 2.23662766701315276891055236001, 2.94567911298969885345145408264, 3.10730659873043153609009149819, 3.29675951499351319721446572679, 3.31469075667027151811112709151, 3.39290122965665658035819272086, 3.52000697403989347823087760672, 4.12467356963057697661068473374, 4.16428074661086271034689089494, 4.21351755944862401377473298911, 4.51965123768610050127595582910, 4.82212654502427116303100701149, 4.91664717032111069568895420522, 5.30697270469542945496903686328
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.