Dirichlet series
| L(s) = 1 | − 2·2-s − 19·4-s − 100·5-s + 100·8-s + 200·10-s − 604·11-s + 1.35e3·13-s + 5·16-s − 3.02e3·17-s + 1.72e3·19-s + 1.90e3·20-s + 1.20e3·22-s + 4.48e3·23-s − 1.97e3·25-s − 2.70e3·26-s + 5.32e3·29-s + 3.97e3·31-s + 5.30e3·32-s + 6.05e3·34-s + 2.26e4·37-s − 3.45e3·38-s − 1.00e4·40-s − 2.87e4·41-s − 6.76e3·43-s + 1.14e4·44-s − 8.96e3·46-s − 5.15e4·47-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 0.593·4-s − 1.78·5-s + 0.552·8-s + 0.632·10-s − 1.50·11-s + 2.21·13-s + 0.00488·16-s − 2.54·17-s + 1.09·19-s + 1.06·20-s + 0.532·22-s + 1.76·23-s − 0.631·25-s − 0.784·26-s + 1.17·29-s + 0.743·31-s + 0.915·32-s + 0.898·34-s + 2.72·37-s − 0.388·38-s − 0.988·40-s − 2.67·41-s − 0.558·43-s + 0.893·44-s − 0.624·46-s − 3.40·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(3^{12} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.25197\times 10^{11}\) |
| Root analytic conductor: | \(8.41006\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 3^{12} \cdot 7^{12} ,\ ( \ : [5/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(4.720114677\) |
| \(L(\frac12)\) | \(\approx\) | \(4.720114677\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) | |
| good | 2 | \( 1 + p T + 23 T^{2} - p^{4} T^{3} + 25 p^{3} T^{4} - 235 p^{5} T^{5} - 1241 p^{4} T^{6} - 235 p^{10} T^{7} + 25 p^{13} T^{8} - p^{19} T^{9} + 23 p^{20} T^{10} + p^{26} T^{11} + p^{30} T^{12} \) |
| 5 | \( 1 + 4 p^{2} T + 11972 T^{2} + 911188 T^{3} + 71428667 T^{4} + 4618786664 T^{5} + 283606162312 T^{6} + 4618786664 p^{5} T^{7} + 71428667 p^{10} T^{8} + 911188 p^{15} T^{9} + 11972 p^{20} T^{10} + 4 p^{27} T^{11} + p^{30} T^{12} \) | |
| 11 | \( 1 + 604 T + 465470 T^{2} + 126422564 T^{3} + 54553413495 T^{4} - 2126286797512 T^{5} + 1724800346353156 T^{6} - 2126286797512 p^{5} T^{7} + 54553413495 p^{10} T^{8} + 126422564 p^{15} T^{9} + 465470 p^{20} T^{10} + 604 p^{25} T^{11} + p^{30} T^{12} \) | |
| 13 | \( 1 - 8 p^{2} T + 2101160 T^{2} - 1701654952 T^{3} + 1567459550619 T^{4} - 957961480510864 T^{5} + 695838444947896272 T^{6} - 957961480510864 p^{5} T^{7} + 1567459550619 p^{10} T^{8} - 1701654952 p^{15} T^{9} + 2101160 p^{20} T^{10} - 8 p^{27} T^{11} + p^{30} T^{12} \) | |
| 17 | \( 1 + 3028 T + 9480044 T^{2} + 16573942004 T^{3} + 30101485361331 T^{4} + 38257820797073480 T^{5} + 3111964669826641688 p T^{6} + 38257820797073480 p^{5} T^{7} + 30101485361331 p^{10} T^{8} + 16573942004 p^{15} T^{9} + 9480044 p^{20} T^{10} + 3028 p^{25} T^{11} + p^{30} T^{12} \) | |
| 19 | \( 1 - 1728 T + 362526 p T^{2} - 10046617920 T^{3} + 29965558163367 T^{4} - 38539774127461248 T^{5} + 90560842902848100108 T^{6} - 38539774127461248 p^{5} T^{7} + 29965558163367 p^{10} T^{8} - 10046617920 p^{15} T^{9} + 362526 p^{21} T^{10} - 1728 p^{25} T^{11} + p^{30} T^{12} \) | |
| 23 | \( 1 - 4484 T + 18866246 T^{2} - 54000132140 T^{3} + 162349820698943 T^{4} - 365036667648018376 T^{5} + \)\(10\!\cdots\!56\)\( T^{6} - 365036667648018376 p^{5} T^{7} + 162349820698943 p^{10} T^{8} - 54000132140 p^{15} T^{9} + 18866246 p^{20} T^{10} - 4484 p^{25} T^{11} + p^{30} T^{12} \) | |
| 29 | \( 1 - 5320 T + 81920486 T^{2} - 415516942408 T^{3} + 3329157696001607 T^{4} - 14790454945150175696 T^{5} + \)\(84\!\cdots\!28\)\( T^{6} - 14790454945150175696 p^{5} T^{7} + 3329157696001607 p^{10} T^{8} - 415516942408 p^{15} T^{9} + 81920486 p^{20} T^{10} - 5320 p^{25} T^{11} + p^{30} T^{12} \) | |
| 31 | \( 1 - 3976 T + 83286866 T^{2} - 428560488920 T^{3} + 3788786084033967 T^{4} - 19749839011446525392 T^{5} + \)\(40\!\cdots\!00\)\( p T^{6} - 19749839011446525392 p^{5} T^{7} + 3788786084033967 p^{10} T^{8} - 428560488920 p^{15} T^{9} + 83286866 p^{20} T^{10} - 3976 p^{25} T^{11} + p^{30} T^{12} \) | |
| 37 | \( 1 - 22680 T + 503329518 T^{2} - 6259800940888 T^{3} + 78591894954668055 T^{4} - \)\(69\!\cdots\!28\)\( T^{5} + \)\(66\!\cdots\!96\)\( T^{6} - \)\(69\!\cdots\!28\)\( p^{5} T^{7} + 78591894954668055 p^{10} T^{8} - 6259800940888 p^{15} T^{9} + 503329518 p^{20} T^{10} - 22680 p^{25} T^{11} + p^{30} T^{12} \) | |
| 41 | \( 1 + 28756 T + 750193916 T^{2} + 13026738107732 T^{3} + 211306160121247683 T^{4} + \)\(26\!\cdots\!00\)\( T^{5} + \)\(31\!\cdots\!92\)\( T^{6} + \)\(26\!\cdots\!00\)\( p^{5} T^{7} + 211306160121247683 p^{10} T^{8} + 13026738107732 p^{15} T^{9} + 750193916 p^{20} T^{10} + 28756 p^{25} T^{11} + p^{30} T^{12} \) | |
| 43 | \( 1 + 6768 T + 494671554 T^{2} + 2742913738512 T^{3} + 119159445546447735 T^{4} + \)\(53\!\cdots\!92\)\( T^{5} + \)\(19\!\cdots\!64\)\( T^{6} + \)\(53\!\cdots\!92\)\( p^{5} T^{7} + 119159445546447735 p^{10} T^{8} + 2742913738512 p^{15} T^{9} + 494671554 p^{20} T^{10} + 6768 p^{25} T^{11} + p^{30} T^{12} \) | |
| 47 | \( 1 + 51552 T + 2111840514 T^{2} + 61246957962528 T^{3} + 1452388942293427215 T^{4} + \)\(28\!\cdots\!36\)\( T^{5} + \)\(46\!\cdots\!52\)\( T^{6} + \)\(28\!\cdots\!36\)\( p^{5} T^{7} + 1452388942293427215 p^{10} T^{8} + 61246957962528 p^{15} T^{9} + 2111840514 p^{20} T^{10} + 51552 p^{25} T^{11} + p^{30} T^{12} \) | |
| 53 | \( 1 + 80884 T + 4007879066 T^{2} + 141303056465588 T^{3} + 4187639515951405527 T^{4} + \)\(10\!\cdots\!44\)\( T^{5} + \)\(23\!\cdots\!56\)\( T^{6} + \)\(10\!\cdots\!44\)\( p^{5} T^{7} + 4187639515951405527 p^{10} T^{8} + 141303056465588 p^{15} T^{9} + 4007879066 p^{20} T^{10} + 80884 p^{25} T^{11} + p^{30} T^{12} \) | |
| 59 | \( 1 + 8872 T + 3182645786 T^{2} + 29070200620696 T^{3} + 4679842080106833143 T^{4} + \)\(40\!\cdots\!84\)\( T^{5} + \)\(41\!\cdots\!88\)\( T^{6} + \)\(40\!\cdots\!84\)\( p^{5} T^{7} + 4679842080106833143 p^{10} T^{8} + 29070200620696 p^{15} T^{9} + 3182645786 p^{20} T^{10} + 8872 p^{25} T^{11} + p^{30} T^{12} \) | |
| 61 | \( 1 - 50896 T + 4845455240 T^{2} - 185639096405136 T^{3} + 9992630357119665211 T^{4} - \)\(29\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!08\)\( T^{6} - \)\(29\!\cdots\!68\)\( p^{5} T^{7} + 9992630357119665211 p^{10} T^{8} - 185639096405136 p^{15} T^{9} + 4845455240 p^{20} T^{10} - 50896 p^{25} T^{11} + p^{30} T^{12} \) | |
| 67 | \( 1 - 6480 T + 5099780754 T^{2} - 27953333833392 T^{3} + 13041442652014676199 T^{4} - \)\(58\!\cdots\!88\)\( T^{5} + \)\(21\!\cdots\!48\)\( T^{6} - \)\(58\!\cdots\!88\)\( p^{5} T^{7} + 13041442652014676199 p^{10} T^{8} - 27953333833392 p^{15} T^{9} + 5099780754 p^{20} T^{10} - 6480 p^{25} T^{11} + p^{30} T^{12} \) | |
| 71 | \( 1 - 110852 T + 11683518662 T^{2} - 634743040311404 T^{3} + 35407535175826294175 T^{4} - \)\(12\!\cdots\!64\)\( T^{5} + \)\(60\!\cdots\!80\)\( T^{6} - \)\(12\!\cdots\!64\)\( p^{5} T^{7} + 35407535175826294175 p^{10} T^{8} - 634743040311404 p^{15} T^{9} + 11683518662 p^{20} T^{10} - 110852 p^{25} T^{11} + p^{30} T^{12} \) | |
| 73 | \( 1 - 64232 T + 7052410688 T^{2} - 368948613770536 T^{3} + 322469435420857371 p T^{4} - \)\(12\!\cdots\!16\)\( T^{5} + \)\(58\!\cdots\!12\)\( T^{6} - \)\(12\!\cdots\!16\)\( p^{5} T^{7} + 322469435420857371 p^{11} T^{8} - 368948613770536 p^{15} T^{9} + 7052410688 p^{20} T^{10} - 64232 p^{25} T^{11} + p^{30} T^{12} \) | |
| 79 | \( 1 - 111696 T + 16876475322 T^{2} - 1210556899938160 T^{3} + \)\(10\!\cdots\!15\)\( T^{4} - \)\(56\!\cdots\!24\)\( T^{5} + \)\(37\!\cdots\!56\)\( T^{6} - \)\(56\!\cdots\!24\)\( p^{5} T^{7} + \)\(10\!\cdots\!15\)\( p^{10} T^{8} - 1210556899938160 p^{15} T^{9} + 16876475322 p^{20} T^{10} - 111696 p^{25} T^{11} + p^{30} T^{12} \) | |
| 83 | \( 1 + 101128 T + 21476446850 T^{2} + 1732192887980152 T^{3} + \)\(20\!\cdots\!55\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!76\)\( T^{6} + \)\(12\!\cdots\!00\)\( p^{5} T^{7} + \)\(20\!\cdots\!55\)\( p^{10} T^{8} + 1732192887980152 p^{15} T^{9} + 21476446850 p^{20} T^{10} + 101128 p^{25} T^{11} + p^{30} T^{12} \) | |
| 89 | \( 1 - 35012 T + 15153790364 T^{2} - 1129254702966916 T^{3} + \)\(14\!\cdots\!47\)\( T^{4} - \)\(92\!\cdots\!96\)\( T^{5} + \)\(11\!\cdots\!48\)\( T^{6} - \)\(92\!\cdots\!96\)\( p^{5} T^{7} + \)\(14\!\cdots\!47\)\( p^{10} T^{8} - 1129254702966916 p^{15} T^{9} + 15153790364 p^{20} T^{10} - 35012 p^{25} T^{11} + p^{30} T^{12} \) | |
| 97 | \( 1 - 70952 T + 38177362064 T^{2} - 2926886015275816 T^{3} + \)\(65\!\cdots\!71\)\( T^{4} - \)\(49\!\cdots\!92\)\( T^{5} + \)\(69\!\cdots\!20\)\( T^{6} - \)\(49\!\cdots\!92\)\( p^{5} T^{7} + \)\(65\!\cdots\!71\)\( p^{10} T^{8} - 2926886015275816 p^{15} T^{9} + 38177362064 p^{20} T^{10} - 70952 p^{25} T^{11} + p^{30} T^{12} \) | |
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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
−5.08648441306597981462623978581, −4.72061184994483414669157201079, −4.57067178682265318762104381353, −4.56967433566146887009955407425, −4.51350319740305998498649929404, −4.48054948826344086378089647401, −4.17611291045419306133614161158, −3.58650631837099791146819413697, −3.56626318911878645799061855458, −3.56433683213133392981291043245, −3.23451796605010854838555101776, −3.21519642319761477829202733349, −3.07588185089115723203885900992, −2.66057772223384130906927997967, −2.36407322668277769993734927910, −2.33245959779544026687990431913, −1.86122559167696053741780709712, −1.70618293776182515758503806292, −1.56890949167602548222081931281, −1.39031295884614459618954061233, −0.75530530917433973799639384177, −0.69577313422658594818532222450, −0.54328570081352305606004310810, −0.48765561596250828757499328245, −0.27646577229502036129393587035, 0.27646577229502036129393587035, 0.48765561596250828757499328245, 0.54328570081352305606004310810, 0.69577313422658594818532222450, 0.75530530917433973799639384177, 1.39031295884614459618954061233, 1.56890949167602548222081931281, 1.70618293776182515758503806292, 1.86122559167696053741780709712, 2.33245959779544026687990431913, 2.36407322668277769993734927910, 2.66057772223384130906927997967, 3.07588185089115723203885900992, 3.21519642319761477829202733349, 3.23451796605010854838555101776, 3.56433683213133392981291043245, 3.56626318911878645799061855458, 3.58650631837099791146819413697, 4.17611291045419306133614161158, 4.48054948826344086378089647401, 4.51350319740305998498649929404, 4.56967433566146887009955407425, 4.57067178682265318762104381353, 4.72061184994483414669157201079, 5.08648441306597981462623978581
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.