Dirichlet series
| L(s) = 1 | + 54·5-s + 132·7-s − 315·11-s − 744·13-s − 2.89e3·17-s − 2.26e3·19-s − 3.16e3·23-s + 4.70e3·25-s + 5.14e3·29-s + 8.61e3·31-s + 7.12e3·35-s + 3.99e4·37-s − 5.04e3·41-s + 3.13e4·43-s + 1.29e4·47-s + 7.49e3·49-s + 9.60e4·53-s − 1.70e4·55-s + 6.29e4·59-s − 7.59e4·61-s − 4.01e4·65-s + 3.29e4·67-s − 1.29e5·71-s − 8.46e3·73-s − 4.15e4·77-s − 8.92e4·79-s + 3.26e4·83-s + ⋯ |
| L(s) = 1 | + 0.965·5-s + 1.01·7-s − 0.784·11-s − 1.22·13-s − 2.43·17-s − 1.43·19-s − 1.24·23-s + 1.50·25-s + 1.13·29-s + 1.60·31-s + 0.983·35-s + 4.79·37-s − 0.469·41-s + 2.58·43-s + 0.853·47-s + 0.445·49-s + 4.69·53-s − 0.758·55-s + 2.35·59-s − 2.61·61-s − 1.17·65-s + 0.897·67-s − 3.05·71-s − 0.185·73-s − 0.799·77-s − 1.60·79-s + 0.519·83-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{24} \cdot 3^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.10628\times 10^{11}\) |
| Root analytic conductor: | \(8.32380\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{24} \cdot 3^{18} ,\ ( \ : [5/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(0.7080688353\) |
| \(L(\frac12)\) | \(\approx\) | \(0.7080688353\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 - 54 T - 1788 T^{2} - 10584 T^{3} + 854952 T^{4} + 487704186 T^{5} - 21775421594 T^{6} + 487704186 p^{5} T^{7} + 854952 p^{10} T^{8} - 10584 p^{15} T^{9} - 1788 p^{20} T^{10} - 54 p^{25} T^{11} + p^{30} T^{12} \) |
| 7 | \( 1 - 132 T + 9930 T^{2} - 2383268 T^{3} + 22216338 T^{4} + 1188047664 T^{5} + 4115243377158 T^{6} + 1188047664 p^{5} T^{7} + 22216338 p^{10} T^{8} - 2383268 p^{15} T^{9} + 9930 p^{20} T^{10} - 132 p^{25} T^{11} + p^{30} T^{12} \) | |
| 11 | \( 1 + 315 T - 222063 T^{2} - 18182214 T^{3} + 32678625651 T^{4} - 477210982239 p T^{5} - 7280422055915978 T^{6} - 477210982239 p^{6} T^{7} + 32678625651 p^{10} T^{8} - 18182214 p^{15} T^{9} - 222063 p^{20} T^{10} + 315 p^{25} T^{11} + p^{30} T^{12} \) | |
| 13 | \( 1 + 744 T + 110316 T^{2} - 5563444 T^{3} - 92859229188 T^{4} - 67995162383184 T^{5} - 18766393745055906 T^{6} - 67995162383184 p^{5} T^{7} - 92859229188 p^{10} T^{8} - 5563444 p^{15} T^{9} + 110316 p^{20} T^{10} + 744 p^{25} T^{11} + p^{30} T^{12} \) | |
| 17 | \( ( 1 + 1449 T + 3759531 T^{2} + 3184553142 T^{3} + 3759531 p^{5} T^{4} + 1449 p^{10} T^{5} + p^{15} T^{6} )^{2} \) | |
| 19 | \( ( 1 + 1131 T + 5928369 T^{2} + 5953391858 T^{3} + 5928369 p^{5} T^{4} + 1131 p^{10} T^{5} + p^{15} T^{6} )^{2} \) | |
| 23 | \( 1 + 3168 T - 11449086 T^{2} - 14347077444 T^{3} + 192790258568754 T^{4} + 140383346504289372 T^{5} - \)\(11\!\cdots\!18\)\( T^{6} + 140383346504289372 p^{5} T^{7} + 192790258568754 p^{10} T^{8} - 14347077444 p^{15} T^{9} - 11449086 p^{20} T^{10} + 3168 p^{25} T^{11} + p^{30} T^{12} \) | |
| 29 | \( 1 - 5148 T - 22104564 T^{2} + 185150934252 T^{3} + 244953744801108 T^{4} - 2482021182847364820 T^{5} + \)\(50\!\cdots\!82\)\( T^{6} - 2482021182847364820 p^{5} T^{7} + 244953744801108 p^{10} T^{8} + 185150934252 p^{15} T^{9} - 22104564 p^{20} T^{10} - 5148 p^{25} T^{11} + p^{30} T^{12} \) | |
| 31 | \( 1 - 8610 T - 10689342 T^{2} + 137042221612 T^{3} + 1032348577619862 T^{4} - 2025683261171583642 T^{5} - \)\(31\!\cdots\!06\)\( T^{6} - 2025683261171583642 p^{5} T^{7} + 1032348577619862 p^{10} T^{8} + 137042221612 p^{15} T^{9} - 10689342 p^{20} T^{10} - 8610 p^{25} T^{11} + p^{30} T^{12} \) | |
| 37 | \( ( 1 - 19968 T + 327446979 T^{2} - 2957339330768 T^{3} + 327446979 p^{5} T^{4} - 19968 p^{10} T^{5} + p^{15} T^{6} )^{2} \) | |
| 41 | \( 1 + 5049 T - 49911369 T^{2} + 2355608603160 T^{3} + 1935356544736665 T^{4} - 94902326907305939361 T^{5} + \)\(30\!\cdots\!26\)\( T^{6} - 94902326907305939361 p^{5} T^{7} + 1935356544736665 p^{10} T^{8} + 2355608603160 p^{15} T^{9} - 49911369 p^{20} T^{10} + 5049 p^{25} T^{11} + p^{30} T^{12} \) | |
| 43 | \( 1 - 31389 T + 371875485 T^{2} - 42403467926 p T^{3} + 12515413892662971 T^{4} - \)\(36\!\cdots\!65\)\( T^{5} + \)\(57\!\cdots\!06\)\( T^{6} - \)\(36\!\cdots\!65\)\( p^{5} T^{7} + 12515413892662971 p^{10} T^{8} - 42403467926 p^{16} T^{9} + 371875485 p^{20} T^{10} - 31389 p^{25} T^{11} + p^{30} T^{12} \) | |
| 47 | \( 1 - 12924 T - 448629342 T^{2} + 4068633746340 T^{3} + 160129591294416234 T^{4} - \)\(73\!\cdots\!76\)\( T^{5} - \)\(35\!\cdots\!10\)\( T^{6} - \)\(73\!\cdots\!76\)\( p^{5} T^{7} + 160129591294416234 p^{10} T^{8} + 4068633746340 p^{15} T^{9} - 448629342 p^{20} T^{10} - 12924 p^{25} T^{11} + p^{30} T^{12} \) | |
| 53 | \( ( 1 - 48024 T + 1811738067 T^{2} - 41931353529216 T^{3} + 1811738067 p^{5} T^{4} - 48024 p^{10} T^{5} + p^{15} T^{6} )^{2} \) | |
| 59 | \( 1 - 62955 T + 1128633393 T^{2} - 9982647295866 T^{3} + 705724247444789571 T^{4} - \)\(15\!\cdots\!79\)\( T^{5} + \)\(56\!\cdots\!06\)\( T^{6} - \)\(15\!\cdots\!79\)\( p^{5} T^{7} + 705724247444789571 p^{10} T^{8} - 9982647295866 p^{15} T^{9} + 1128633393 p^{20} T^{10} - 62955 p^{25} T^{11} + p^{30} T^{12} \) | |
| 61 | \( 1 + 75966 T + 1651965492 T^{2} + 45532183560440 T^{3} + 3331090328104364760 T^{4} + \)\(76\!\cdots\!06\)\( T^{5} + \)\(63\!\cdots\!14\)\( T^{6} + \)\(76\!\cdots\!06\)\( p^{5} T^{7} + 3331090328104364760 p^{10} T^{8} + 45532183560440 p^{15} T^{9} + 1651965492 p^{20} T^{10} + 75966 p^{25} T^{11} + p^{30} T^{12} \) | |
| 67 | \( 1 - 32991 T - 1801789179 T^{2} + 76747890991738 T^{3} + 1612255572283172571 T^{4} - \)\(51\!\cdots\!59\)\( T^{5} - \)\(93\!\cdots\!82\)\( T^{6} - \)\(51\!\cdots\!59\)\( p^{5} T^{7} + 1612255572283172571 p^{10} T^{8} + 76747890991738 p^{15} T^{9} - 1801789179 p^{20} T^{10} - 32991 p^{25} T^{11} + p^{30} T^{12} \) | |
| 71 | \( ( 1 + 64836 T + 5330360517 T^{2} + 233818077065976 T^{3} + 5330360517 p^{5} T^{4} + 64836 p^{10} T^{5} + p^{15} T^{6} )^{2} \) | |
| 73 | \( ( 1 + 4233 T + 1481704827 T^{2} + 31872982860070 T^{3} + 1481704827 p^{5} T^{4} + 4233 p^{10} T^{5} + p^{15} T^{6} )^{2} \) | |
| 79 | \( 1 + 89202 T - 946471758 T^{2} - 73104980305676 T^{3} + 13192286849737226406 T^{4} - \)\(18\!\cdots\!50\)\( T^{5} - \)\(72\!\cdots\!62\)\( T^{6} - \)\(18\!\cdots\!50\)\( p^{5} T^{7} + 13192286849737226406 p^{10} T^{8} - 73104980305676 p^{15} T^{9} - 946471758 p^{20} T^{10} + 89202 p^{25} T^{11} + p^{30} T^{12} \) | |
| 83 | \( 1 - 32634 T - 7761884658 T^{2} + 19287559006164 T^{3} + 38129210867128845714 T^{4} + \)\(40\!\cdots\!90\)\( T^{5} - \)\(17\!\cdots\!58\)\( T^{6} + \)\(40\!\cdots\!90\)\( p^{5} T^{7} + 38129210867128845714 p^{10} T^{8} + 19287559006164 p^{15} T^{9} - 7761884658 p^{20} T^{10} - 32634 p^{25} T^{11} + p^{30} T^{12} \) | |
| 89 | \( ( 1 + 33066 T + 16131261399 T^{2} + 360180435327660 T^{3} + 16131261399 p^{5} T^{4} + 33066 p^{10} T^{5} + p^{15} T^{6} )^{2} \) | |
| 97 | \( 1 - 46245 T - 17067186177 T^{2} + 116084978368376 T^{3} + \)\(18\!\cdots\!05\)\( T^{4} + \)\(22\!\cdots\!89\)\( T^{5} - \)\(18\!\cdots\!78\)\( T^{6} + \)\(22\!\cdots\!89\)\( p^{5} T^{7} + \)\(18\!\cdots\!05\)\( p^{10} T^{8} + 116084978368376 p^{15} T^{9} - 17067186177 p^{20} T^{10} - 46245 p^{25} T^{11} + p^{30} T^{12} \) | |
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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
−4.94380818217684192319474319925, −4.81020479758843359276122605921, −4.72507438870421166561144032576, −4.65250037708897745202128830289, −4.54223309721499718662224313658, −4.26795831277447493427659176952, −4.14533567142126161209665845783, −4.06739176506705177437443430399, −3.82615732640285928815240145330, −3.57768335753677543583466105826, −3.04278717962441169952691306322, −2.77619950384084585590654925099, −2.70611758592658411058592287441, −2.56208549149007301377412313268, −2.42736747354408584243923560268, −2.36487502427848206386709121250, −2.19389548430297005113077299167, −1.83173223063805881320396583616, −1.75646416191428584492567597251, −1.12740245461596870478985412935, −1.00808775377977555384626012042, −0.930786149268317625249634598731, −0.799715271963220810157954565059, −0.35016754861972618053073765999, −0.06299316940170122638926302094, 0.06299316940170122638926302094, 0.35016754861972618053073765999, 0.799715271963220810157954565059, 0.930786149268317625249634598731, 1.00808775377977555384626012042, 1.12740245461596870478985412935, 1.75646416191428584492567597251, 1.83173223063805881320396583616, 2.19389548430297005113077299167, 2.36487502427848206386709121250, 2.42736747354408584243923560268, 2.56208549149007301377412313268, 2.70611758592658411058592287441, 2.77619950384084585590654925099, 3.04278717962441169952691306322, 3.57768335753677543583466105826, 3.82615732640285928815240145330, 4.06739176506705177437443430399, 4.14533567142126161209665845783, 4.26795831277447493427659176952, 4.54223309721499718662224313658, 4.65250037708897745202128830289, 4.72507438870421166561144032576, 4.81020479758843359276122605921, 4.94380818217684192319474319925
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.