Dirichlet series
| L(s) = 1 | − 4.23e3·2-s + 7.21e4·3-s − 4.83e7·4-s − 1.10e8·5-s − 3.04e8·6-s + 1.49e11·8-s − 1.90e12·9-s + 4.65e11·10-s − 1.67e12·11-s − 3.48e12·12-s − 5.28e12·13-s − 7.94e12·15-s + 1.08e15·16-s + 3.30e15·17-s + 8.05e15·18-s − 2.31e16·19-s + 5.33e15·20-s + 7.08e15·22-s − 1.40e15·23-s + 1.08e16·24-s − 1.06e18·25-s + 2.23e16·26-s + 1.08e18·27-s + 6.77e17·29-s + 3.35e16·30-s + 1.49e19·31-s − 5.19e18·32-s + ⋯ |
| L(s) = 1 | − 0.730·2-s + 0.0783·3-s − 1.44·4-s − 0.201·5-s − 0.0572·6-s + 0.771·8-s − 2.24·9-s + 0.147·10-s − 0.161·11-s − 0.112·12-s − 0.0629·13-s − 0.0158·15-s + 0.961·16-s + 1.37·17-s + 1.64·18-s − 2.39·19-s + 0.290·20-s + 0.117·22-s − 0.0133·23-s + 0.0604·24-s − 3.58·25-s + 0.0459·26-s + 1.39·27-s + 0.355·29-s + 0.0115·30-s + 3.41·31-s − 0.796·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(26-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12}\right)^{s/2} \, \Gamma_{\C}(s+25/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.33735\times 10^{13}\) |
| Root analytic conductor: | \(13.9297\) |
| Motivic weight: | \(25\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 7^{12} ,\ ( \ : [25/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(13)\) | \(\approx\) | \(0.4655876741\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4655876741\) |
| \(L(\frac{27}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 2115 p T + 8285073 p^{3} T^{2} + 163634841 p^{11} T^{3} + 1419822351915 p^{11} T^{4} + 73224717827355 p^{18} T^{5} + 1872901485996441 p^{26} T^{6} + 73224717827355 p^{43} T^{7} + 1419822351915 p^{61} T^{8} + 163634841 p^{86} T^{9} + 8285073 p^{103} T^{10} + 2115 p^{126} T^{11} + p^{150} T^{12} \) |
| 3 | \( 1 - 72104 T + 636872612974 p T^{2} - 16834365904987000 p^{4} T^{3} + \)\(81\!\cdots\!17\)\( p^{7} T^{4} - \)\(47\!\cdots\!24\)\( p^{10} T^{5} + \)\(27\!\cdots\!56\)\( p^{16} T^{6} - \)\(47\!\cdots\!24\)\( p^{35} T^{7} + \)\(81\!\cdots\!17\)\( p^{57} T^{8} - 16834365904987000 p^{79} T^{9} + 636872612974 p^{101} T^{10} - 72104 p^{125} T^{11} + p^{150} T^{12} \) | |
| 5 | \( 1 + 110161332 T + 1079576927761484706 T^{2} - \)\(50\!\cdots\!32\)\( p^{2} T^{3} + \)\(16\!\cdots\!47\)\( p^{5} T^{4} - \)\(88\!\cdots\!48\)\( p^{6} T^{5} + \)\(43\!\cdots\!64\)\( p^{8} T^{6} - \)\(88\!\cdots\!48\)\( p^{31} T^{7} + \)\(16\!\cdots\!47\)\( p^{55} T^{8} - \)\(50\!\cdots\!32\)\( p^{77} T^{9} + 1079576927761484706 p^{100} T^{10} + 110161332 p^{125} T^{11} + p^{150} T^{12} \) | |
| 11 | \( 1 + 1675999103976 T + \)\(41\!\cdots\!74\)\( p T^{2} + \)\(11\!\cdots\!76\)\( p^{2} T^{3} + \)\(69\!\cdots\!19\)\( p^{4} T^{4} + \)\(17\!\cdots\!60\)\( p^{6} T^{5} + \)\(64\!\cdots\!08\)\( p^{8} T^{6} + \)\(17\!\cdots\!60\)\( p^{31} T^{7} + \)\(69\!\cdots\!19\)\( p^{54} T^{8} + \)\(11\!\cdots\!76\)\( p^{77} T^{9} + \)\(41\!\cdots\!74\)\( p^{101} T^{10} + 1675999103976 p^{125} T^{11} + p^{150} T^{12} \) | |
| 13 | \( 1 + 5288670743748 T + \)\(19\!\cdots\!06\)\( p T^{2} + \)\(95\!\cdots\!72\)\( T^{3} + \)\(29\!\cdots\!31\)\( T^{4} + \)\(13\!\cdots\!80\)\( p T^{5} + \)\(13\!\cdots\!84\)\( p^{2} T^{6} + \)\(13\!\cdots\!80\)\( p^{26} T^{7} + \)\(29\!\cdots\!31\)\( p^{50} T^{8} + \)\(95\!\cdots\!72\)\( p^{75} T^{9} + \)\(19\!\cdots\!06\)\( p^{101} T^{10} + 5288670743748 p^{125} T^{11} + p^{150} T^{12} \) | |
| 17 | \( 1 - 3300624486895572 T + \)\(20\!\cdots\!98\)\( T^{2} - \)\(35\!\cdots\!92\)\( T^{3} + \)\(58\!\cdots\!23\)\( p^{2} T^{4} - \)\(42\!\cdots\!16\)\( p^{3} T^{5} + \)\(21\!\cdots\!84\)\( p^{3} T^{6} - \)\(42\!\cdots\!16\)\( p^{28} T^{7} + \)\(58\!\cdots\!23\)\( p^{52} T^{8} - \)\(35\!\cdots\!92\)\( p^{75} T^{9} + \)\(20\!\cdots\!98\)\( p^{100} T^{10} - 3300624486895572 p^{125} T^{11} + p^{150} T^{12} \) | |
| 19 | \( 1 + 23126727727224696 T + \)\(54\!\cdots\!94\)\( T^{2} + \)\(40\!\cdots\!16\)\( p T^{3} + \)\(16\!\cdots\!85\)\( p^{3} T^{4} + \)\(18\!\cdots\!32\)\( p^{3} T^{5} + \)\(10\!\cdots\!80\)\( p^{4} T^{6} + \)\(18\!\cdots\!32\)\( p^{28} T^{7} + \)\(16\!\cdots\!85\)\( p^{53} T^{8} + \)\(40\!\cdots\!16\)\( p^{76} T^{9} + \)\(54\!\cdots\!94\)\( p^{100} T^{10} + 23126727727224696 p^{125} T^{11} + p^{150} T^{12} \) | |
| 23 | \( 1 + 61075609011936 p T + \)\(54\!\cdots\!30\)\( p^{2} T^{2} - \)\(30\!\cdots\!04\)\( p^{3} T^{3} + \)\(17\!\cdots\!91\)\( p^{4} T^{4} + \)\(12\!\cdots\!68\)\( p^{5} T^{5} + \)\(42\!\cdots\!36\)\( p^{6} T^{6} + \)\(12\!\cdots\!68\)\( p^{30} T^{7} + \)\(17\!\cdots\!91\)\( p^{54} T^{8} - \)\(30\!\cdots\!04\)\( p^{78} T^{9} + \)\(54\!\cdots\!30\)\( p^{102} T^{10} + 61075609011936 p^{126} T^{11} + p^{150} T^{12} \) | |
| 29 | \( 1 - 677678534303781204 T + \)\(67\!\cdots\!34\)\( T^{2} - \)\(12\!\cdots\!56\)\( T^{3} + \)\(43\!\cdots\!55\)\( T^{4} - \)\(68\!\cdots\!12\)\( T^{5} + \)\(17\!\cdots\!20\)\( T^{6} - \)\(68\!\cdots\!12\)\( p^{25} T^{7} + \)\(43\!\cdots\!55\)\( p^{50} T^{8} - \)\(12\!\cdots\!56\)\( p^{75} T^{9} + \)\(67\!\cdots\!34\)\( p^{100} T^{10} - 677678534303781204 p^{125} T^{11} + p^{150} T^{12} \) | |
| 31 | \( 1 - 14968503768754072128 T + \)\(19\!\cdots\!34\)\( T^{2} - \)\(15\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!35\)\( T^{4} - \)\(62\!\cdots\!80\)\( T^{5} + \)\(30\!\cdots\!04\)\( T^{6} - \)\(62\!\cdots\!80\)\( p^{25} T^{7} + \)\(11\!\cdots\!35\)\( p^{50} T^{8} - \)\(15\!\cdots\!20\)\( p^{75} T^{9} + \)\(19\!\cdots\!34\)\( p^{100} T^{10} - 14968503768754072128 p^{125} T^{11} + p^{150} T^{12} \) | |
| 37 | \( 1 + \)\(12\!\cdots\!44\)\( T + \)\(10\!\cdots\!86\)\( T^{2} + \)\(63\!\cdots\!32\)\( T^{3} + \)\(35\!\cdots\!99\)\( T^{4} + \)\(17\!\cdots\!40\)\( T^{5} + \)\(75\!\cdots\!08\)\( T^{6} + \)\(17\!\cdots\!40\)\( p^{25} T^{7} + \)\(35\!\cdots\!99\)\( p^{50} T^{8} + \)\(63\!\cdots\!32\)\( p^{75} T^{9} + \)\(10\!\cdots\!86\)\( p^{100} T^{10} + \)\(12\!\cdots\!44\)\( p^{125} T^{11} + p^{150} T^{12} \) | |
| 41 | \( 1 - 91172275091591492484 T + \)\(82\!\cdots\!38\)\( T^{2} - \)\(64\!\cdots\!40\)\( T^{3} + \)\(33\!\cdots\!55\)\( T^{4} - \)\(23\!\cdots\!56\)\( T^{5} + \)\(87\!\cdots\!52\)\( T^{6} - \)\(23\!\cdots\!56\)\( p^{25} T^{7} + \)\(33\!\cdots\!55\)\( p^{50} T^{8} - \)\(64\!\cdots\!40\)\( p^{75} T^{9} + \)\(82\!\cdots\!38\)\( p^{100} T^{10} - 91172275091591492484 p^{125} T^{11} + p^{150} T^{12} \) | |
| 43 | \( 1 - \)\(25\!\cdots\!28\)\( T + \)\(31\!\cdots\!34\)\( T^{2} - \)\(60\!\cdots\!20\)\( T^{3} + \)\(43\!\cdots\!51\)\( T^{4} - \)\(65\!\cdots\!08\)\( T^{5} + \)\(36\!\cdots\!88\)\( T^{6} - \)\(65\!\cdots\!08\)\( p^{25} T^{7} + \)\(43\!\cdots\!51\)\( p^{50} T^{8} - \)\(60\!\cdots\!20\)\( p^{75} T^{9} + \)\(31\!\cdots\!34\)\( p^{100} T^{10} - \)\(25\!\cdots\!28\)\( p^{125} T^{11} + p^{150} T^{12} \) | |
| 47 | \( 1 + \)\(13\!\cdots\!92\)\( T + \)\(33\!\cdots\!18\)\( T^{2} + \)\(31\!\cdots\!36\)\( T^{3} + \)\(47\!\cdots\!15\)\( T^{4} + \)\(35\!\cdots\!96\)\( T^{5} + \)\(39\!\cdots\!76\)\( T^{6} + \)\(35\!\cdots\!96\)\( p^{25} T^{7} + \)\(47\!\cdots\!15\)\( p^{50} T^{8} + \)\(31\!\cdots\!36\)\( p^{75} T^{9} + \)\(33\!\cdots\!18\)\( p^{100} T^{10} + \)\(13\!\cdots\!92\)\( p^{125} T^{11} + p^{150} T^{12} \) | |
| 53 | \( 1 - \)\(24\!\cdots\!44\)\( T + \)\(78\!\cdots\!54\)\( p T^{2} - \)\(12\!\cdots\!60\)\( T^{3} + \)\(87\!\cdots\!79\)\( T^{4} - \)\(29\!\cdots\!04\)\( T^{5} + \)\(12\!\cdots\!84\)\( T^{6} - \)\(29\!\cdots\!04\)\( p^{25} T^{7} + \)\(87\!\cdots\!79\)\( p^{50} T^{8} - \)\(12\!\cdots\!60\)\( p^{75} T^{9} + \)\(78\!\cdots\!54\)\( p^{101} T^{10} - \)\(24\!\cdots\!44\)\( p^{125} T^{11} + p^{150} T^{12} \) | |
| 59 | \( 1 - \)\(52\!\cdots\!08\)\( T + \)\(13\!\cdots\!10\)\( T^{2} - \)\(21\!\cdots\!40\)\( T^{3} + \)\(21\!\cdots\!71\)\( T^{4} - \)\(12\!\cdots\!40\)\( T^{5} + \)\(80\!\cdots\!36\)\( T^{6} - \)\(12\!\cdots\!40\)\( p^{25} T^{7} + \)\(21\!\cdots\!71\)\( p^{50} T^{8} - \)\(21\!\cdots\!40\)\( p^{75} T^{9} + \)\(13\!\cdots\!10\)\( p^{100} T^{10} - \)\(52\!\cdots\!08\)\( p^{125} T^{11} + p^{150} T^{12} \) | |
| 61 | \( 1 - \)\(61\!\cdots\!20\)\( T + \)\(40\!\cdots\!06\)\( T^{2} - \)\(14\!\cdots\!12\)\( T^{3} + \)\(54\!\cdots\!63\)\( T^{4} - \)\(13\!\cdots\!48\)\( T^{5} + \)\(33\!\cdots\!00\)\( T^{6} - \)\(13\!\cdots\!48\)\( p^{25} T^{7} + \)\(54\!\cdots\!63\)\( p^{50} T^{8} - \)\(14\!\cdots\!12\)\( p^{75} T^{9} + \)\(40\!\cdots\!06\)\( p^{100} T^{10} - \)\(61\!\cdots\!20\)\( p^{125} T^{11} + p^{150} T^{12} \) | |
| 67 | \( 1 - \)\(11\!\cdots\!64\)\( T + \)\(13\!\cdots\!74\)\( T^{2} - \)\(39\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!83\)\( T^{4} + \)\(30\!\cdots\!96\)\( T^{5} - \)\(13\!\cdots\!72\)\( T^{6} + \)\(30\!\cdots\!96\)\( p^{25} T^{7} + \)\(14\!\cdots\!83\)\( p^{50} T^{8} - \)\(39\!\cdots\!20\)\( p^{75} T^{9} + \)\(13\!\cdots\!74\)\( p^{100} T^{10} - \)\(11\!\cdots\!64\)\( p^{125} T^{11} + p^{150} T^{12} \) | |
| 71 | \( 1 + \)\(80\!\cdots\!08\)\( T + \)\(42\!\cdots\!38\)\( T^{2} - \)\(10\!\cdots\!44\)\( T^{3} + \)\(53\!\cdots\!39\)\( T^{4} - \)\(65\!\cdots\!64\)\( T^{5} + \)\(55\!\cdots\!44\)\( T^{6} - \)\(65\!\cdots\!64\)\( p^{25} T^{7} + \)\(53\!\cdots\!39\)\( p^{50} T^{8} - \)\(10\!\cdots\!44\)\( p^{75} T^{9} + \)\(42\!\cdots\!38\)\( p^{100} T^{10} + \)\(80\!\cdots\!08\)\( p^{125} T^{11} + p^{150} T^{12} \) | |
| 73 | \( 1 + \)\(53\!\cdots\!68\)\( T + \)\(12\!\cdots\!50\)\( T^{2} + \)\(44\!\cdots\!32\)\( T^{3} + \)\(72\!\cdots\!43\)\( T^{4} - \)\(23\!\cdots\!96\)\( T^{5} + \)\(31\!\cdots\!76\)\( T^{6} - \)\(23\!\cdots\!96\)\( p^{25} T^{7} + \)\(72\!\cdots\!43\)\( p^{50} T^{8} + \)\(44\!\cdots\!32\)\( p^{75} T^{9} + \)\(12\!\cdots\!50\)\( p^{100} T^{10} + \)\(53\!\cdots\!68\)\( p^{125} T^{11} + p^{150} T^{12} \) | |
| 79 | \( 1 - \)\(47\!\cdots\!00\)\( T + \)\(10\!\cdots\!54\)\( T^{2} - \)\(58\!\cdots\!04\)\( T^{3} + \)\(62\!\cdots\!15\)\( T^{4} - \)\(28\!\cdots\!88\)\( T^{5} + \)\(22\!\cdots\!60\)\( T^{6} - \)\(28\!\cdots\!88\)\( p^{25} T^{7} + \)\(62\!\cdots\!15\)\( p^{50} T^{8} - \)\(58\!\cdots\!04\)\( p^{75} T^{9} + \)\(10\!\cdots\!54\)\( p^{100} T^{10} - \)\(47\!\cdots\!00\)\( p^{125} T^{11} + p^{150} T^{12} \) | |
| 83 | \( 1 - \)\(11\!\cdots\!56\)\( T + \)\(27\!\cdots\!94\)\( T^{2} - \)\(12\!\cdots\!08\)\( T^{3} + \)\(18\!\cdots\!67\)\( T^{4} + \)\(95\!\cdots\!12\)\( T^{5} + \)\(24\!\cdots\!12\)\( T^{6} + \)\(95\!\cdots\!12\)\( p^{25} T^{7} + \)\(18\!\cdots\!67\)\( p^{50} T^{8} - \)\(12\!\cdots\!08\)\( p^{75} T^{9} + \)\(27\!\cdots\!94\)\( p^{100} T^{10} - \)\(11\!\cdots\!56\)\( p^{125} T^{11} + p^{150} T^{12} \) | |
| 89 | \( 1 - \)\(13\!\cdots\!96\)\( T + \)\(29\!\cdots\!94\)\( T^{2} - \)\(33\!\cdots\!44\)\( T^{3} + \)\(37\!\cdots\!35\)\( T^{4} - \)\(34\!\cdots\!08\)\( T^{5} + \)\(26\!\cdots\!40\)\( T^{6} - \)\(34\!\cdots\!08\)\( p^{25} T^{7} + \)\(37\!\cdots\!35\)\( p^{50} T^{8} - \)\(33\!\cdots\!44\)\( p^{75} T^{9} + \)\(29\!\cdots\!94\)\( p^{100} T^{10} - \)\(13\!\cdots\!96\)\( p^{125} T^{11} + p^{150} T^{12} \) | |
| 97 | \( 1 - \)\(14\!\cdots\!72\)\( T + \)\(16\!\cdots\!46\)\( T^{2} - \)\(17\!\cdots\!64\)\( T^{3} + \)\(15\!\cdots\!47\)\( T^{4} - \)\(12\!\cdots\!76\)\( T^{5} + \)\(88\!\cdots\!08\)\( T^{6} - \)\(12\!\cdots\!76\)\( p^{25} T^{7} + \)\(15\!\cdots\!47\)\( p^{50} T^{8} - \)\(17\!\cdots\!64\)\( p^{75} T^{9} + \)\(16\!\cdots\!46\)\( p^{100} T^{10} - \)\(14\!\cdots\!72\)\( p^{125} T^{11} + p^{150} T^{12} \) | |
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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
−4.87920877520666544297281060908, −4.67750370569884890395621090618, −4.40919487627425726042163906883, −4.11099599292289029167117769235, −3.97854135443586848932235746995, −3.84419094313783797254039467644, −3.81068386419693569097884925171, −3.44067939010728960453387228985, −3.22632256702280174786690715067, −3.19951161252474950900036731261, −2.89559269583759734154881309306, −2.50914229508642696090542377402, −2.39276682751047639157152613419, −2.28662642876879031217750703029, −2.22234610505284009970056640522, −1.90825498174236848535965464576, −1.80517302990990728795237051742, −1.34974905724012633373561967006, −1.17725861477407649930859916511, −0.861895632425082874490137327875, −0.68363839448233650155091006545, −0.66590511843101454324118095669, −0.47509420173513610976278303296, −0.22558844903335827426187997272, −0.12527844413031739941923098733, 0.12527844413031739941923098733, 0.22558844903335827426187997272, 0.47509420173513610976278303296, 0.66590511843101454324118095669, 0.68363839448233650155091006545, 0.861895632425082874490137327875, 1.17725861477407649930859916511, 1.34974905724012633373561967006, 1.80517302990990728795237051742, 1.90825498174236848535965464576, 2.22234610505284009970056640522, 2.28662642876879031217750703029, 2.39276682751047639157152613419, 2.50914229508642696090542377402, 2.89559269583759734154881309306, 3.19951161252474950900036731261, 3.22632256702280174786690715067, 3.44067939010728960453387228985, 3.81068386419693569097884925171, 3.84419094313783797254039467644, 3.97854135443586848932235746995, 4.11099599292289029167117769235, 4.40919487627425726042163906883, 4.67750370569884890395621090618, 4.87920877520666544297281060908
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.