Dirichlet series
| L(s) = 1 | − 3·2-s − 3·3-s − 12·4-s − 12·5-s + 9·6-s + 12·7-s + 45·8-s + 54·9-s + 36·10-s + 483·11-s + 36·12-s − 6·13-s − 36·14-s + 36·15-s + 129·16-s − 162·18-s − 258·19-s + 144·20-s − 36·21-s − 1.44e3·22-s − 282·23-s − 135·24-s − 1.00e3·25-s + 18·26-s − 135·27-s − 144·28-s − 1.05e3·29-s + ⋯ |
| L(s) = 1 | − 3/4·2-s − 1/3·3-s − 3/4·4-s − 0.479·5-s + 1/4·6-s + 0.244·7-s + 0.703·8-s + 2/3·9-s + 9/25·10-s + 3.99·11-s + 1/4·12-s − 0.0355·13-s − 0.183·14-s + 4/25·15-s + 0.503·16-s − 1/2·18-s − 0.714·19-s + 9/25·20-s − 0.0816·21-s − 2.99·22-s − 0.533·23-s − 0.234·24-s − 1.60·25-s + 0.0266·26-s − 0.185·27-s − 0.183·28-s − 1.25·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s+2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(531441\) = \(3^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.648367\) |
| Root analytic conductor: | \(0.964535\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 531441,\ (\ :[2]^{6}),\ 1)\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(0.5430324473\) |
| \(L(\frac12)\) | \(\approx\) | \(0.5430324473\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 + p T - 5 p^{2} T^{2} - 2 p^{4} T^{3} - 5 p^{6} T^{4} + p^{9} T^{5} + p^{12} T^{6} \) |
| good | 2 | \( 1 + 3 T + 21 T^{2} + 27 p T^{3} + 75 p T^{4} + 321 p^{2} T^{5} + 139 p^{2} T^{6} + 321 p^{6} T^{7} + 75 p^{9} T^{8} + 27 p^{13} T^{9} + 21 p^{16} T^{10} + 3 p^{20} T^{11} + p^{24} T^{12} \) |
| 5 | \( 1 + 12 T + 1146 T^{2} + 13176 T^{3} + 624786 T^{4} + 24253584 T^{5} + 478052638 T^{6} + 24253584 p^{4} T^{7} + 624786 p^{8} T^{8} + 13176 p^{12} T^{9} + 1146 p^{16} T^{10} + 12 p^{20} T^{11} + p^{24} T^{12} \) | |
| 7 | \( 1 - 12 T - 2598 T^{2} - 158692 T^{3} + 1689750 T^{4} + 261282744 T^{5} + 12756026130 T^{6} + 261282744 p^{4} T^{7} + 1689750 p^{8} T^{8} - 158692 p^{12} T^{9} - 2598 p^{16} T^{10} - 12 p^{20} T^{11} + p^{24} T^{12} \) | |
| 11 | \( 1 - 483 T + 146469 T^{2} - 3016818 p T^{3} + 6185106393 T^{4} - 86431460865 p T^{5} + 124654869741274 T^{6} - 86431460865 p^{5} T^{7} + 6185106393 p^{8} T^{8} - 3016818 p^{13} T^{9} + 146469 p^{16} T^{10} - 483 p^{20} T^{11} + p^{24} T^{12} \) | |
| 13 | \( 1 + 6 T - 68352 T^{2} - 1958440 T^{3} + 2716582572 T^{4} + 62453281302 T^{5} - 85607901136722 T^{6} + 62453281302 p^{4} T^{7} + 2716582572 p^{8} T^{8} - 1958440 p^{12} T^{9} - 68352 p^{16} T^{10} + 6 p^{20} T^{11} + p^{24} T^{12} \) | |
| 17 | \( 1 - 346011 T^{2} + 53617620939 T^{4} - 5294214329064626 T^{6} + 53617620939 p^{8} T^{8} - 346011 p^{16} T^{10} + p^{24} T^{12} \) | |
| 19 | \( ( 1 + 129 T + 372939 T^{2} + 32427790 T^{3} + 372939 p^{4} T^{4} + 129 p^{8} T^{5} + p^{12} T^{6} )^{2} \) | |
| 23 | \( 1 + 282 T + 28212 p T^{2} + 175507776 T^{3} + 227497773480 T^{4} + 72727641623526 T^{5} + 70224828144985186 T^{6} + 72727641623526 p^{4} T^{7} + 227497773480 p^{8} T^{8} + 175507776 p^{12} T^{9} + 28212 p^{17} T^{10} + 282 p^{20} T^{11} + p^{24} T^{12} \) | |
| 29 | \( 1 + 1056 T + 1968402 T^{2} + 1686104640 T^{3} + 1760514009810 T^{4} + 1015827714042684 T^{5} + 1221310573219134286 T^{6} + 1015827714042684 p^{4} T^{7} + 1760514009810 p^{8} T^{8} + 1686104640 p^{12} T^{9} + 1968402 p^{16} T^{10} + 1056 p^{20} T^{11} + p^{24} T^{12} \) | |
| 31 | \( 1 - 1290 T - 570606 T^{2} + 44436716 p T^{3} + 182213722890 T^{4} - 535899775714218 T^{5} + 81606055489213290 T^{6} - 535899775714218 p^{4} T^{7} + 182213722890 p^{8} T^{8} + 44436716 p^{13} T^{9} - 570606 p^{16} T^{10} - 1290 p^{20} T^{11} + p^{24} T^{12} \) | |
| 37 | \( ( 1 - 6 T + 3399531 T^{2} + 1254253444 T^{3} + 3399531 p^{4} T^{4} - 6 p^{8} T^{5} + p^{12} T^{6} )^{2} \) | |
| 41 | \( 1 - 7629 T + 33279051 T^{2} - 105879107016 T^{3} + 271025936843037 T^{4} - 581054677368191211 T^{5} + \)\(10\!\cdots\!02\)\( T^{6} - 581054677368191211 p^{4} T^{7} + 271025936843037 p^{8} T^{8} - 105879107016 p^{12} T^{9} + 33279051 p^{16} T^{10} - 7629 p^{20} T^{11} + p^{24} T^{12} \) | |
| 43 | \( 1 + 285 T - 4416855 T^{2} + 6696709682 T^{3} + 5626286140005 T^{4} - 17612350767962595 T^{5} + 17128129577980595658 T^{6} - 17612350767962595 p^{4} T^{7} + 5626286140005 p^{8} T^{8} + 6696709682 p^{12} T^{9} - 4416855 p^{16} T^{10} + 285 p^{20} T^{11} + p^{24} T^{12} \) | |
| 47 | \( 1 + 9642 T + 52651236 T^{2} + 208863538416 T^{3} + 680262116291880 T^{4} + 1905117042381356886 T^{5} + \)\(45\!\cdots\!66\)\( T^{6} + 1905117042381356886 p^{4} T^{7} + 680262116291880 p^{8} T^{8} + 208863538416 p^{12} T^{9} + 52651236 p^{16} T^{10} + 9642 p^{20} T^{11} + p^{24} T^{12} \) | |
| 53 | \( 1 - 20649822 T^{2} + 267970470920223 T^{4} - \)\(22\!\cdots\!72\)\( T^{6} + 267970470920223 p^{8} T^{8} - 20649822 p^{16} T^{10} + p^{24} T^{12} \) | |
| 59 | \( 1 - 6225 T + 34368069 T^{2} - 133533682650 T^{3} + 415161196812177 T^{4} - 1308007205289880689 T^{5} + \)\(36\!\cdots\!46\)\( T^{6} - 1308007205289880689 p^{4} T^{7} + 415161196812177 p^{8} T^{8} - 133533682650 p^{12} T^{9} + 34368069 p^{16} T^{10} - 6225 p^{20} T^{11} + p^{24} T^{12} \) | |
| 61 | \( 1 - 3630 T - 31672896 T^{2} + 38334701264 T^{3} + 1025452935692820 T^{4} - 698091809532688782 T^{5} - \)\(14\!\cdots\!30\)\( T^{6} - 698091809532688782 p^{4} T^{7} + 1025452935692820 p^{8} T^{8} + 38334701264 p^{12} T^{9} - 31672896 p^{16} T^{10} - 3630 p^{20} T^{11} + p^{24} T^{12} \) | |
| 67 | \( 1 + 5055 T - 13352055 T^{2} + 5710009118 T^{3} + 246985082323965 T^{4} - 2158812454985978265 T^{5} - \)\(15\!\cdots\!22\)\( T^{6} - 2158812454985978265 p^{4} T^{7} + 246985082323965 p^{8} T^{8} + 5710009118 p^{12} T^{9} - 13352055 p^{16} T^{10} + 5055 p^{20} T^{11} + p^{24} T^{12} \) | |
| 71 | \( 1 - 87967842 T^{2} + 4070274316914543 T^{4} - \)\(12\!\cdots\!12\)\( T^{6} + 4070274316914543 p^{8} T^{8} - 87967842 p^{16} T^{10} + p^{24} T^{12} \) | |
| 73 | \( ( 1 + 7311 T + 93929019 T^{2} + 399497247430 T^{3} + 93929019 p^{4} T^{4} + 7311 p^{8} T^{5} + p^{12} T^{6} )^{2} \) | |
| 79 | \( 1 - 4764 T - 88341198 T^{2} + 209973394004 T^{3} + 6244109393857182 T^{4} - 7045428820224395952 T^{5} - \)\(25\!\cdots\!90\)\( T^{6} - 7045428820224395952 p^{4} T^{7} + 6244109393857182 p^{8} T^{8} + 209973394004 p^{12} T^{9} - 88341198 p^{16} T^{10} - 4764 p^{20} T^{11} + p^{24} T^{12} \) | |
| 83 | \( 1 + 1866 T + 10594740 T^{2} + 17604008208 T^{3} + 123292807619064 T^{4} + 3108136851733394166 T^{5} - \)\(11\!\cdots\!18\)\( T^{6} + 3108136851733394166 p^{4} T^{7} + 123292807619064 p^{8} T^{8} + 17604008208 p^{12} T^{9} + 10594740 p^{16} T^{10} + 1866 p^{20} T^{11} + p^{24} T^{12} \) | |
| 89 | \( 1 - 92525118 T^{2} + 4141804686688959 T^{4} - \)\(27\!\cdots\!72\)\( T^{6} + 4141804686688959 p^{8} T^{8} - 92525118 p^{16} T^{10} + p^{24} T^{12} \) | |
| 97 | \( 1 + 28959 T + 315999867 T^{2} + 3434149167476 T^{3} + 57450550006007937 T^{4} + \)\(60\!\cdots\!37\)\( T^{5} + \)\(48\!\cdots\!90\)\( T^{6} + \)\(60\!\cdots\!37\)\( p^{4} T^{7} + 57450550006007937 p^{8} T^{8} + 3434149167476 p^{12} T^{9} + 315999867 p^{16} T^{10} + 28959 p^{20} T^{11} + p^{24} T^{12} \) | |
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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
−12.20779491062379161780508694691, −12.15948300402956413529349975776, −11.98078364337353344745302622226, −11.52581995196533337180556170483, −11.24982781826406031449774784490, −11.20679484717477736322310650472, −10.77997602641107249984782359224, −10.08940977307677130487187661268, −9.721693755352397029136819496706, −9.657344127998523603231432929950, −9.535298948836360342899739804578, −8.935528831918611667198271517288, −8.767143711347760277045197583229, −8.497175320800506517831948013721, −7.922935610310642112929559617236, −7.53365314695087897131605859684, −7.25259808378858088326653754619, −6.46389251052961450853473964807, −6.43733661612261843547882195491, −5.97151479953988768215279134759, −5.25087573306371985183922040991, −4.16441993780772380695722002942, −4.14130148445312395828322151129, −3.88336784903434659782190843136, −1.43416954902843804324915323752, 1.43416954902843804324915323752, 3.88336784903434659782190843136, 4.14130148445312395828322151129, 4.16441993780772380695722002942, 5.25087573306371985183922040991, 5.97151479953988768215279134759, 6.43733661612261843547882195491, 6.46389251052961450853473964807, 7.25259808378858088326653754619, 7.53365314695087897131605859684, 7.922935610310642112929559617236, 8.497175320800506517831948013721, 8.767143711347760277045197583229, 8.935528831918611667198271517288, 9.535298948836360342899739804578, 9.657344127998523603231432929950, 9.721693755352397029136819496706, 10.08940977307677130487187661268, 10.77997602641107249984782359224, 11.20679484717477736322310650472, 11.24982781826406031449774784490, 11.52581995196533337180556170483, 11.98078364337353344745302622226, 12.15948300402956413529349975776, 12.20779491062379161780508694691
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.