Dirichlet series
| L(s) = 1 | + 96·2-s + 3.07e3·4-s − 1.04e3·5-s + 4.57e4·7-s − 6.55e4·8-s − 1.00e5·10-s − 1.81e5·11-s + 1.18e6·13-s + 4.39e6·14-s − 9.43e6·16-s + 7.01e5·17-s − 7.89e6·19-s − 3.21e6·20-s − 1.74e7·22-s − 5.92e7·23-s + 1.36e8·25-s + 1.13e8·26-s + 1.40e8·28-s + 3.00e7·29-s − 2.99e8·31-s − 3.01e8·32-s + 6.73e7·34-s − 4.77e7·35-s − 3.52e8·37-s − 7.57e8·38-s + 6.84e7·40-s − 1.19e9·41-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s − 0.149·5-s + 1.02·7-s − 0.707·8-s − 0.317·10-s − 0.339·11-s + 0.886·13-s + 2.18·14-s − 9/4·16-s + 0.119·17-s − 0.731·19-s − 0.224·20-s − 0.721·22-s − 1.91·23-s + 2.78·25-s + 1.87·26-s + 1.54·28-s + 0.271·29-s − 1.87·31-s − 1.59·32-s + 0.254·34-s − 0.153·35-s − 0.835·37-s − 1.55·38-s + 0.105·40-s − 1.61·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 3^{12} \cdot 7^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.23293\times 10^{11}\) |
| Root analytic conductor: | \(9.83927\) |
| Motivic weight: | \(11\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{6} \cdot 3^{12} \cdot 7^{6} ,\ ( \ : [11/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(6)\) | \(\approx\) | \(0.0007553027068\) |
| \(L(\frac12)\) | \(\approx\) | \(0.0007553027068\) |
| \(L(\frac{13}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{5} T + p^{10} T^{2} )^{3} \) |
| 3 | \( 1 \) | |
| 7 | \( 1 - 6533 p T - 45532 p^{5} T^{2} + 3434983 p^{9} T^{3} - 45532 p^{16} T^{4} - 6533 p^{23} T^{5} + p^{33} T^{6} \) | |
| good | 5 | \( 1 + 209 p T - 5399558 p^{2} T^{2} - 610578661 p^{3} T^{3} + 18814836107792 p^{4} T^{4} + 1111355088495509 p^{5} T^{5} - 42370253381372678864 p^{6} T^{6} + 1111355088495509 p^{16} T^{7} + 18814836107792 p^{26} T^{8} - 610578661 p^{36} T^{9} - 5399558 p^{46} T^{10} + 209 p^{56} T^{11} + p^{66} T^{12} \) |
| 11 | \( 1 + 181565 T - 628806735368 T^{2} - 44476528761771415 T^{3} + \)\(23\!\cdots\!76\)\( T^{4} + \)\(69\!\cdots\!65\)\( T^{5} - \)\(74\!\cdots\!02\)\( T^{6} + \)\(69\!\cdots\!65\)\( p^{11} T^{7} + \)\(23\!\cdots\!76\)\( p^{22} T^{8} - 44476528761771415 p^{33} T^{9} - 628806735368 p^{44} T^{10} + 181565 p^{55} T^{11} + p^{66} T^{12} \) | |
| 13 | \( ( 1 - 593182 T + 2697743671096 T^{2} - 1728838193778432488 T^{3} + 2697743671096 p^{11} T^{4} - 593182 p^{22} T^{5} + p^{33} T^{6} )^{2} \) | |
| 17 | \( 1 - 701848 T - 32234003410835 T^{2} - \)\(27\!\cdots\!44\)\( T^{3} + \)\(38\!\cdots\!62\)\( T^{4} + \)\(49\!\cdots\!84\)\( T^{5} + \)\(59\!\cdots\!21\)\( T^{6} + \)\(49\!\cdots\!84\)\( p^{11} T^{7} + \)\(38\!\cdots\!62\)\( p^{22} T^{8} - \)\(27\!\cdots\!44\)\( p^{33} T^{9} - 32234003410835 p^{44} T^{10} - 701848 p^{55} T^{11} + p^{66} T^{12} \) | |
| 19 | \( 1 + 7893102 T - 236749686164814 T^{2} - \)\(99\!\cdots\!80\)\( T^{3} + \)\(41\!\cdots\!66\)\( T^{4} + \)\(59\!\cdots\!98\)\( T^{5} - \)\(52\!\cdots\!18\)\( T^{6} + \)\(59\!\cdots\!98\)\( p^{11} T^{7} + \)\(41\!\cdots\!66\)\( p^{22} T^{8} - \)\(99\!\cdots\!80\)\( p^{33} T^{9} - 236749686164814 p^{44} T^{10} + 7893102 p^{55} T^{11} + p^{66} T^{12} \) | |
| 23 | \( 1 + 59247728 T + 2098614481805803 T^{2} + \)\(29\!\cdots\!12\)\( T^{3} - \)\(94\!\cdots\!10\)\( T^{4} - \)\(67\!\cdots\!92\)\( T^{5} - \)\(26\!\cdots\!33\)\( T^{6} - \)\(67\!\cdots\!92\)\( p^{11} T^{7} - \)\(94\!\cdots\!10\)\( p^{22} T^{8} + \)\(29\!\cdots\!12\)\( p^{33} T^{9} + 2098614481805803 p^{44} T^{10} + 59247728 p^{55} T^{11} + p^{66} T^{12} \) | |
| 29 | \( ( 1 - 15020035 T + 32141555328672327 T^{2} - \)\(41\!\cdots\!30\)\( T^{3} + 32141555328672327 p^{11} T^{4} - 15020035 p^{22} T^{5} + p^{33} T^{6} )^{2} \) | |
| 31 | \( 1 + 299540305 T + 25034090138478309 T^{2} - \)\(38\!\cdots\!22\)\( T^{3} - \)\(93\!\cdots\!45\)\( T^{4} - \)\(42\!\cdots\!99\)\( p^{2} T^{5} + \)\(33\!\cdots\!82\)\( p^{4} T^{6} - \)\(42\!\cdots\!99\)\( p^{13} T^{7} - \)\(93\!\cdots\!45\)\( p^{22} T^{8} - \)\(38\!\cdots\!22\)\( p^{33} T^{9} + 25034090138478309 p^{44} T^{10} + 299540305 p^{55} T^{11} + p^{66} T^{12} \) | |
| 37 | \( 1 + 352325094 T - 203476169283109692 T^{2} - \)\(15\!\cdots\!36\)\( T^{3} + \)\(17\!\cdots\!24\)\( T^{4} + \)\(14\!\cdots\!06\)\( T^{5} + \)\(62\!\cdots\!74\)\( T^{6} + \)\(14\!\cdots\!06\)\( p^{11} T^{7} + \)\(17\!\cdots\!24\)\( p^{22} T^{8} - \)\(15\!\cdots\!36\)\( p^{33} T^{9} - 203476169283109692 p^{44} T^{10} + 352325094 p^{55} T^{11} + p^{66} T^{12} \) | |
| 41 | \( ( 1 + 599977236 T + 225477063483943983 T^{2} + \)\(27\!\cdots\!92\)\( T^{3} + 225477063483943983 p^{11} T^{4} + 599977236 p^{22} T^{5} + p^{33} T^{6} )^{2} \) | |
| 43 | \( ( 1 + 2149422288 T + 1119208578495351942 T^{2} - \)\(14\!\cdots\!38\)\( T^{3} + 1119208578495351942 p^{11} T^{4} + 2149422288 p^{22} T^{5} + p^{33} T^{6} )^{2} \) | |
| 47 | \( 1 - 1179123942 T - 2301335671962298965 T^{2} + \)\(68\!\cdots\!14\)\( T^{3} - \)\(20\!\cdots\!18\)\( T^{4} - \)\(62\!\cdots\!34\)\( T^{5} + \)\(19\!\cdots\!11\)\( T^{6} - \)\(62\!\cdots\!34\)\( p^{11} T^{7} - \)\(20\!\cdots\!18\)\( p^{22} T^{8} + \)\(68\!\cdots\!14\)\( p^{33} T^{9} - 2301335671962298965 p^{44} T^{10} - 1179123942 p^{55} T^{11} + p^{66} T^{12} \) | |
| 53 | \( 1 - 67875849 T - 6928544352228691182 T^{2} + \)\(28\!\cdots\!69\)\( T^{3} - \)\(17\!\cdots\!44\)\( T^{4} - \)\(98\!\cdots\!37\)\( T^{5} + \)\(13\!\cdots\!28\)\( T^{6} - \)\(98\!\cdots\!37\)\( p^{11} T^{7} - \)\(17\!\cdots\!44\)\( p^{22} T^{8} + \)\(28\!\cdots\!69\)\( p^{33} T^{9} - 6928544352228691182 p^{44} T^{10} - 67875849 p^{55} T^{11} + p^{66} T^{12} \) | |
| 59 | \( 1 - 2911039823 T - 69160345804932375776 T^{2} + \)\(10\!\cdots\!25\)\( T^{3} + \)\(32\!\cdots\!20\)\( T^{4} - \)\(16\!\cdots\!43\)\( T^{5} - \)\(10\!\cdots\!66\)\( T^{6} - \)\(16\!\cdots\!43\)\( p^{11} T^{7} + \)\(32\!\cdots\!20\)\( p^{22} T^{8} + \)\(10\!\cdots\!25\)\( p^{33} T^{9} - 69160345804932375776 p^{44} T^{10} - 2911039823 p^{55} T^{11} + p^{66} T^{12} \) | |
| 61 | \( 1 + 10148733466 T - 42264532243202499207 T^{2} - \)\(30\!\cdots\!82\)\( T^{3} + \)\(57\!\cdots\!06\)\( T^{4} + \)\(19\!\cdots\!70\)\( T^{5} - \)\(15\!\cdots\!23\)\( T^{6} + \)\(19\!\cdots\!70\)\( p^{11} T^{7} + \)\(57\!\cdots\!06\)\( p^{22} T^{8} - \)\(30\!\cdots\!82\)\( p^{33} T^{9} - 42264532243202499207 p^{44} T^{10} + 10148733466 p^{55} T^{11} + p^{66} T^{12} \) | |
| 67 | \( 1 + 5535975380 T - \)\(16\!\cdots\!86\)\( T^{2} - \)\(10\!\cdots\!56\)\( T^{3} + \)\(71\!\cdots\!18\)\( T^{4} + \)\(28\!\cdots\!88\)\( T^{5} - \)\(10\!\cdots\!70\)\( T^{6} + \)\(28\!\cdots\!88\)\( p^{11} T^{7} + \)\(71\!\cdots\!18\)\( p^{22} T^{8} - \)\(10\!\cdots\!56\)\( p^{33} T^{9} - \)\(16\!\cdots\!86\)\( p^{44} T^{10} + 5535975380 p^{55} T^{11} + p^{66} T^{12} \) | |
| 71 | \( ( 1 - 10279098938 T + 9446153607445971879 p T^{2} - \)\(44\!\cdots\!44\)\( T^{3} + 9446153607445971879 p^{12} T^{4} - 10279098938 p^{22} T^{5} + p^{33} T^{6} )^{2} \) | |
| 73 | \( 1 - 29230853572 T - \)\(13\!\cdots\!72\)\( T^{2} + \)\(22\!\cdots\!44\)\( T^{3} + \)\(20\!\cdots\!88\)\( T^{4} - \)\(10\!\cdots\!88\)\( T^{5} - \)\(86\!\cdots\!98\)\( T^{6} - \)\(10\!\cdots\!88\)\( p^{11} T^{7} + \)\(20\!\cdots\!88\)\( p^{22} T^{8} + \)\(22\!\cdots\!44\)\( p^{33} T^{9} - \)\(13\!\cdots\!72\)\( p^{44} T^{10} - 29230853572 p^{55} T^{11} + p^{66} T^{12} \) | |
| 79 | \( 1 - 58886709107 T + \)\(11\!\cdots\!17\)\( T^{2} - \)\(13\!\cdots\!54\)\( T^{3} + \)\(15\!\cdots\!51\)\( T^{4} + \)\(28\!\cdots\!85\)\( T^{5} - \)\(13\!\cdots\!02\)\( T^{6} + \)\(28\!\cdots\!85\)\( p^{11} T^{7} + \)\(15\!\cdots\!51\)\( p^{22} T^{8} - \)\(13\!\cdots\!54\)\( p^{33} T^{9} + \)\(11\!\cdots\!17\)\( p^{44} T^{10} - 58886709107 p^{55} T^{11} + p^{66} T^{12} \) | |
| 83 | \( ( 1 + 13000856863 T + \)\(30\!\cdots\!17\)\( T^{2} + \)\(38\!\cdots\!22\)\( T^{3} + \)\(30\!\cdots\!17\)\( p^{11} T^{4} + 13000856863 p^{22} T^{5} + p^{33} T^{6} )^{2} \) | |
| 89 | \( 1 - 32237877702 T - \)\(67\!\cdots\!19\)\( T^{2} + \)\(87\!\cdots\!46\)\( T^{3} + \)\(34\!\cdots\!90\)\( T^{4} - \)\(18\!\cdots\!62\)\( T^{5} - \)\(10\!\cdots\!11\)\( T^{6} - \)\(18\!\cdots\!62\)\( p^{11} T^{7} + \)\(34\!\cdots\!90\)\( p^{22} T^{8} + \)\(87\!\cdots\!46\)\( p^{33} T^{9} - \)\(67\!\cdots\!19\)\( p^{44} T^{10} - 32237877702 p^{55} T^{11} + p^{66} T^{12} \) | |
| 97 | \( ( 1 - 87723438467 T + \)\(19\!\cdots\!87\)\( T^{2} - \)\(12\!\cdots\!06\)\( T^{3} + \)\(19\!\cdots\!87\)\( p^{11} T^{4} - 87723438467 p^{22} T^{5} + p^{33} T^{6} )^{2} \) | |
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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
−5.27045181332902757435932284128, −5.04560663148275439325087674369, −4.77010194556534455475205456769, −4.75070318057499180663551931260, −4.63466096381608203794954813648, −4.52599971386766965079465445872, −3.96453694381847746434424638577, −3.78733381866688219883538309580, −3.69788529566892797757754132088, −3.68958251172123835977012851837, −3.38372441779726583352840305050, −3.10362598617126381307187106159, −3.02299879124219761460487160326, −2.80924220213366517693794338947, −2.39041546414007856634844736709, −2.02636455583095651333419134114, −1.92675985060955629109711426815, −1.92314788838172332360216880237, −1.77206345431418180397143153171, −1.16093136677184723105317584364, −1.14763495402390804317285670899, −0.881325910465644355221311248214, −0.61667156089895694408651362413, −0.24446416174055826914943448962, −0.00175527016652961083875922858, 0.00175527016652961083875922858, 0.24446416174055826914943448962, 0.61667156089895694408651362413, 0.881325910465644355221311248214, 1.14763495402390804317285670899, 1.16093136677184723105317584364, 1.77206345431418180397143153171, 1.92314788838172332360216880237, 1.92675985060955629109711426815, 2.02636455583095651333419134114, 2.39041546414007856634844736709, 2.80924220213366517693794338947, 3.02299879124219761460487160326, 3.10362598617126381307187106159, 3.38372441779726583352840305050, 3.68958251172123835977012851837, 3.69788529566892797757754132088, 3.78733381866688219883538309580, 3.96453694381847746434424638577, 4.52599971386766965079465445872, 4.63466096381608203794954813648, 4.75070318057499180663551931260, 4.77010194556534455475205456769, 5.04560663148275439325087674369, 5.27045181332902757435932284128
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.