Dirichlet series
| L(s) = 1 | + 12·5-s − 12·7-s + 483·11-s − 6·13-s + 258·19-s − 282·23-s − 1.00e3·25-s + 1.05e3·29-s − 1.29e3·31-s − 144·35-s + 12·37-s − 7.62e3·41-s + 285·43-s − 9.64e3·47-s + 2.74e3·49-s + 5.79e3·55-s + 6.22e3·59-s + 3.63e3·61-s − 72·65-s + 5.05e3·67-s − 1.46e4·73-s − 5.79e3·77-s − 4.76e3·79-s − 1.86e3·83-s + 72·91-s + 3.09e3·95-s − 2.89e4·97-s + ⋯ |
| L(s) = 1 | + 0.479·5-s − 0.244·7-s + 3.99·11-s − 0.0355·13-s + 0.714·19-s − 0.533·23-s − 1.60·25-s + 1.25·29-s − 1.34·31-s − 0.117·35-s + 0.00876·37-s − 4.53·41-s + 0.154·43-s − 4.36·47-s + 1.14·49-s + 1.91·55-s + 1.78·59-s + 0.975·61-s − 0.0170·65-s + 1.12·67-s − 2.74·73-s − 0.977·77-s − 0.763·79-s − 0.270·83-s + 0.00869·91-s + 0.343·95-s − 3.07·97-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s+2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{24} \cdot 3^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.92991\times 10^{9}\) |
| Root analytic conductor: | \(6.68250\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{24} \cdot 3^{18} ,\ ( \ : [2]^{6} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(4.113013340\) |
| \(L(\frac12)\) | \(\approx\) | \(4.113013340\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| good | 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
−5.29525964015205456597876149744, −5.22453833131791004997527145261, −5.05406485801981234217053278207, −5.04836757670623302342689203102, −4.55131900843807546026152354479, −4.23406501550041045821548460402, −4.05965427139136134928181504878, −4.04145142391815889290103342853, −3.97089601923909681003815777548, −3.80542473445015004122662195264, −3.41186480317275410666095085895, −3.25855060715554136871548589289, −3.19245284984097001401622472318, −2.98827793941466387366618101358, −2.62894392058290167232033875492, −2.22849137330113375143855167790, −2.17134329620023564310517929179, −1.69404503994132849894140350248, −1.63190899588932867241226381574, −1.36517461131896586975565159246, −1.34526628240335677781941263842, −1.32821729724130489703699679479, −0.63818186234179573923019687882, −0.37738300889631770602925106737, −0.18987112340304472185578731080, 0.18987112340304472185578731080, 0.37738300889631770602925106737, 0.63818186234179573923019687882, 1.32821729724130489703699679479, 1.34526628240335677781941263842, 1.36517461131896586975565159246, 1.63190899588932867241226381574, 1.69404503994132849894140350248, 2.17134329620023564310517929179, 2.22849137330113375143855167790, 2.62894392058290167232033875492, 2.98827793941466387366618101358, 3.19245284984097001401622472318, 3.25855060715554136871548589289, 3.41186480317275410666095085895, 3.80542473445015004122662195264, 3.97089601923909681003815777548, 4.04145142391815889290103342853, 4.05965427139136134928181504878, 4.23406501550041045821548460402, 4.55131900843807546026152354479, 5.04836757670623302342689203102, 5.05406485801981234217053278207, 5.22453833131791004997527145261, 5.29525964015205456597876149744
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.