Dirichlet series
| L(s) = 1 | − 2-s + 12·3-s − 7·4-s − 14·5-s − 12·6-s − 5·7-s + 8·8-s + 78·9-s + 14·10-s − 84·12-s − 7·13-s + 5·14-s − 168·15-s + 22·16-s + 12·17-s − 78·18-s + 3·19-s + 98·20-s − 60·21-s − 39·23-s + 96·24-s + 72·25-s + 7·26-s + 364·27-s + 35·28-s + 10·29-s + 168·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 6.92·3-s − 7/2·4-s − 6.26·5-s − 4.89·6-s − 1.88·7-s + 2.82·8-s + 26·9-s + 4.42·10-s − 24.2·12-s − 1.94·13-s + 1.33·14-s − 43.3·15-s + 11/2·16-s + 2.91·17-s − 18.3·18-s + 0.688·19-s + 21.9·20-s − 13.0·21-s − 8.13·23-s + 19.5·24-s + 72/5·25-s + 1.37·26-s + 70.0·27-s + 6.61·28-s + 1.85·29-s + 30.6·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{24} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{24} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{12} \cdot 11^{24} \cdot 17^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.04923\times 10^{20}\) |
| Root analytic conductor: | \(7.01966\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 3^{12} \cdot 11^{24} \cdot 17^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{12} \) |
| 11 | \( 1 \) | |
| 17 | \( ( 1 - T )^{12} \) | |
| good | 2 | \( 1 + T + p^{3} T^{2} + 7 T^{3} + 33 T^{4} + 7 p^{2} T^{5} + 103 T^{6} + 95 T^{7} + 275 T^{8} + 137 p T^{9} + 639 T^{10} + 639 T^{11} + 1331 T^{12} + 639 p T^{13} + 639 p^{2} T^{14} + 137 p^{4} T^{15} + 275 p^{4} T^{16} + 95 p^{5} T^{17} + 103 p^{6} T^{18} + 7 p^{9} T^{19} + 33 p^{8} T^{20} + 7 p^{9} T^{21} + p^{13} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 + 14 T + 124 T^{2} + 819 T^{3} + 4422 T^{4} + 20347 T^{5} + 16427 p T^{6} + 295954 T^{7} + 964572 T^{8} + 2868024 T^{9} + 1565981 p T^{10} + 19704031 T^{11} + 45820451 T^{12} + 19704031 p T^{13} + 1565981 p^{3} T^{14} + 2868024 p^{3} T^{15} + 964572 p^{4} T^{16} + 295954 p^{5} T^{17} + 16427 p^{7} T^{18} + 20347 p^{7} T^{19} + 4422 p^{8} T^{20} + 819 p^{9} T^{21} + 124 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 + 5 T + 62 T^{2} + 268 T^{3} + 1847 T^{4} + 6997 T^{5} + 5031 p T^{6} + 117926 T^{7} + 479909 T^{8} + 1426654 T^{9} + 4927242 T^{10} + 13000734 T^{11} + 39088624 T^{12} + 13000734 p T^{13} + 4927242 p^{2} T^{14} + 1426654 p^{3} T^{15} + 479909 p^{4} T^{16} + 117926 p^{5} T^{17} + 5031 p^{7} T^{18} + 6997 p^{7} T^{19} + 1847 p^{8} T^{20} + 268 p^{9} T^{21} + 62 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 7 T + 113 T^{2} + 561 T^{3} + 5352 T^{4} + 1554 p T^{5} + 151498 T^{6} + 450849 T^{7} + 3037955 T^{8} + 7336227 T^{9} + 48279809 T^{10} + 100208578 T^{11} + 662671583 T^{12} + 100208578 p T^{13} + 48279809 p^{2} T^{14} + 7336227 p^{3} T^{15} + 3037955 p^{4} T^{16} + 450849 p^{5} T^{17} + 151498 p^{6} T^{18} + 1554 p^{8} T^{19} + 5352 p^{8} T^{20} + 561 p^{9} T^{21} + 113 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 3 T + 124 T^{2} - 189 T^{3} + 7549 T^{4} - 4977 T^{5} + 320272 T^{6} - 60657 T^{7} + 10463728 T^{8} + 32712 p T^{9} + 271765970 T^{10} + 43751659 T^{11} + 5718355747 T^{12} + 43751659 p T^{13} + 271765970 p^{2} T^{14} + 32712 p^{4} T^{15} + 10463728 p^{4} T^{16} - 60657 p^{5} T^{17} + 320272 p^{6} T^{18} - 4977 p^{7} T^{19} + 7549 p^{8} T^{20} - 189 p^{9} T^{21} + 124 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 39 T + 871 T^{2} + 13981 T^{3} + 178602 T^{4} + 1909633 T^{5} + 17649620 T^{6} + 143885386 T^{7} + 1049529401 T^{8} + 6915158131 T^{9} + 41435897648 T^{10} + 226749291542 T^{11} + 1136083563063 T^{12} + 226749291542 p T^{13} + 41435897648 p^{2} T^{14} + 6915158131 p^{3} T^{15} + 1049529401 p^{4} T^{16} + 143885386 p^{5} T^{17} + 17649620 p^{6} T^{18} + 1909633 p^{7} T^{19} + 178602 p^{8} T^{20} + 13981 p^{9} T^{21} + 871 p^{10} T^{22} + 39 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 10 T + 218 T^{2} - 1984 T^{3} + 24721 T^{4} - 196226 T^{5} + 1844933 T^{6} - 12819840 T^{7} + 99842696 T^{8} - 614253984 T^{9} + 4132121123 T^{10} - 22661211024 T^{11} + 134518110435 T^{12} - 22661211024 p T^{13} + 4132121123 p^{2} T^{14} - 614253984 p^{3} T^{15} + 99842696 p^{4} T^{16} - 12819840 p^{5} T^{17} + 1844933 p^{6} T^{18} - 196226 p^{7} T^{19} + 24721 p^{8} T^{20} - 1984 p^{9} T^{21} + 218 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 25 T + 15 p T^{2} + 6432 T^{3} + 76170 T^{4} + 777385 T^{5} + 7137976 T^{6} + 59195375 T^{7} + 14591980 p T^{8} + 3189829362 T^{9} + 20990251540 T^{10} + 128770071785 T^{11} + 741393454424 T^{12} + 128770071785 p T^{13} + 20990251540 p^{2} T^{14} + 3189829362 p^{3} T^{15} + 14591980 p^{5} T^{16} + 59195375 p^{5} T^{17} + 7137976 p^{6} T^{18} + 777385 p^{7} T^{19} + 76170 p^{8} T^{20} + 6432 p^{9} T^{21} + 15 p^{11} T^{22} + 25 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 10 T + 291 T^{2} + 2098 T^{3} + 36532 T^{4} + 189856 T^{5} + 2666908 T^{6} + 9080202 T^{7} + 128574875 T^{8} + 205540201 T^{9} + 4654735776 T^{10} + 215998239 T^{11} + 160872796370 T^{12} + 215998239 p T^{13} + 4654735776 p^{2} T^{14} + 205540201 p^{3} T^{15} + 128574875 p^{4} T^{16} + 9080202 p^{5} T^{17} + 2666908 p^{6} T^{18} + 189856 p^{7} T^{19} + 36532 p^{8} T^{20} + 2098 p^{9} T^{21} + 291 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 17 T + 360 T^{2} - 4332 T^{3} + 55847 T^{4} - 549997 T^{5} + 133602 p T^{6} - 46857678 T^{7} + 392445985 T^{8} - 2997158100 T^{9} + 536268255 p T^{10} - 151665241292 T^{11} + 996463022012 T^{12} - 151665241292 p T^{13} + 536268255 p^{3} T^{14} - 2997158100 p^{3} T^{15} + 392445985 p^{4} T^{16} - 46857678 p^{5} T^{17} + 133602 p^{7} T^{18} - 549997 p^{7} T^{19} + 55847 p^{8} T^{20} - 4332 p^{9} T^{21} + 360 p^{10} T^{22} - 17 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 18 T + 447 T^{2} + 5789 T^{3} + 85857 T^{4} + 892646 T^{5} + 10050507 T^{6} + 88536172 T^{7} + 825800192 T^{8} + 6345665920 T^{9} + 51128013034 T^{10} + 347888912495 T^{11} + 2473087263700 T^{12} + 347888912495 p T^{13} + 51128013034 p^{2} T^{14} + 6345665920 p^{3} T^{15} + 825800192 p^{4} T^{16} + 88536172 p^{5} T^{17} + 10050507 p^{6} T^{18} + 892646 p^{7} T^{19} + 85857 p^{8} T^{20} + 5789 p^{9} T^{21} + 447 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 38 T + 947 T^{2} + 17196 T^{3} + 258857 T^{4} + 3317426 T^{5} + 37746239 T^{6} + 385633907 T^{7} + 3610768569 T^{8} + 31160614349 T^{9} + 250810821145 T^{10} + 40140175834 p T^{11} + 13348708571272 T^{12} + 40140175834 p^{2} T^{13} + 250810821145 p^{2} T^{14} + 31160614349 p^{3} T^{15} + 3610768569 p^{4} T^{16} + 385633907 p^{5} T^{17} + 37746239 p^{6} T^{18} + 3317426 p^{7} T^{19} + 258857 p^{8} T^{20} + 17196 p^{9} T^{21} + 947 p^{10} T^{22} + 38 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 40 T + 1154 T^{2} + 24329 T^{3} + 431820 T^{4} + 6504842 T^{5} + 86799831 T^{6} + 1030751927 T^{7} + 11099738186 T^{8} + 108599551189 T^{9} + 974827529555 T^{10} + 8025803040565 T^{11} + 60924534382698 T^{12} + 8025803040565 p T^{13} + 974827529555 p^{2} T^{14} + 108599551189 p^{3} T^{15} + 11099738186 p^{4} T^{16} + 1030751927 p^{5} T^{17} + 86799831 p^{6} T^{18} + 6504842 p^{7} T^{19} + 431820 p^{8} T^{20} + 24329 p^{9} T^{21} + 1154 p^{10} T^{22} + 40 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 18 T + 485 T^{2} + 6805 T^{3} + 112127 T^{4} + 1310047 T^{5} + 16739874 T^{6} + 169239223 T^{7} + 1818852520 T^{8} + 16247188657 T^{9} + 152233341576 T^{10} + 1212297003894 T^{11} + 10068453262990 T^{12} + 1212297003894 p T^{13} + 152233341576 p^{2} T^{14} + 16247188657 p^{3} T^{15} + 1818852520 p^{4} T^{16} + 169239223 p^{5} T^{17} + 16739874 p^{6} T^{18} + 1310047 p^{7} T^{19} + 112127 p^{8} T^{20} + 6805 p^{9} T^{21} + 485 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 22 T + 707 T^{2} + 11691 T^{3} + 219598 T^{4} + 2932867 T^{5} + 41127430 T^{6} + 461586237 T^{7} + 5274029971 T^{8} + 50854922219 T^{9} + 492756013564 T^{10} + 4126116834974 T^{11} + 34547370207762 T^{12} + 4126116834974 p T^{13} + 492756013564 p^{2} T^{14} + 50854922219 p^{3} T^{15} + 5274029971 p^{4} T^{16} + 461586237 p^{5} T^{17} + 41127430 p^{6} T^{18} + 2932867 p^{7} T^{19} + 219598 p^{8} T^{20} + 11691 p^{9} T^{21} + 707 p^{10} T^{22} + 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 9 T + 414 T^{2} + 4341 T^{3} + 94861 T^{4} + 994901 T^{5} + 15161596 T^{6} + 150744515 T^{7} + 1814584016 T^{8} + 16803822864 T^{9} + 169996647767 T^{10} + 1436937497672 T^{11} + 12720901443518 T^{12} + 1436937497672 p T^{13} + 169996647767 p^{2} T^{14} + 16803822864 p^{3} T^{15} + 1814584016 p^{4} T^{16} + 150744515 p^{5} T^{17} + 15161596 p^{6} T^{18} + 994901 p^{7} T^{19} + 94861 p^{8} T^{20} + 4341 p^{9} T^{21} + 414 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 24 T + 687 T^{2} + 11180 T^{3} + 196240 T^{4} + 2486703 T^{5} + 33455168 T^{6} + 353054767 T^{7} + 3983003005 T^{8} + 36631865911 T^{9} + 366564951540 T^{10} + 3052872090328 T^{11} + 28092631098963 T^{12} + 3052872090328 p T^{13} + 366564951540 p^{2} T^{14} + 36631865911 p^{3} T^{15} + 3983003005 p^{4} T^{16} + 353054767 p^{5} T^{17} + 33455168 p^{6} T^{18} + 2486703 p^{7} T^{19} + 196240 p^{8} T^{20} + 11180 p^{9} T^{21} + 687 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - T + 472 T^{2} - 361 T^{3} + 117861 T^{4} - 98690 T^{5} + 20061705 T^{6} - 18363253 T^{7} + 2563579257 T^{8} - 2443481645 T^{9} + 258051058240 T^{10} - 238531529954 T^{11} + 20917215676996 T^{12} - 238531529954 p T^{13} + 258051058240 p^{2} T^{14} - 2443481645 p^{3} T^{15} + 2563579257 p^{4} T^{16} - 18363253 p^{5} T^{17} + 20061705 p^{6} T^{18} - 98690 p^{7} T^{19} + 117861 p^{8} T^{20} - 361 p^{9} T^{21} + 472 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 3 T + 245 T^{2} - 1881 T^{3} + 27941 T^{4} - 483850 T^{5} + 5422970 T^{6} - 53973414 T^{7} + 819100546 T^{8} - 6401899409 T^{9} + 77157907825 T^{10} - 8984520903 p T^{11} + 6167968344544 T^{12} - 8984520903 p^{2} T^{13} + 77157907825 p^{2} T^{14} - 6401899409 p^{3} T^{15} + 819100546 p^{4} T^{16} - 53973414 p^{5} T^{17} + 5422970 p^{6} T^{18} - 483850 p^{7} T^{19} + 27941 p^{8} T^{20} - 1881 p^{9} T^{21} + 245 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 4 T + 611 T^{2} - 2172 T^{3} + 182091 T^{4} - 610804 T^{5} + 35449479 T^{6} - 117295359 T^{7} + 5090250253 T^{8} - 16758564735 T^{9} + 574566492543 T^{10} - 1822054375866 T^{11} + 52660032985592 T^{12} - 1822054375866 p T^{13} + 574566492543 p^{2} T^{14} - 16758564735 p^{3} T^{15} + 5090250253 p^{4} T^{16} - 117295359 p^{5} T^{17} + 35449479 p^{6} T^{18} - 610804 p^{7} T^{19} + 182091 p^{8} T^{20} - 2172 p^{9} T^{21} + 611 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 62 T + 2424 T^{2} + 68588 T^{3} + 1571877 T^{4} + 30292836 T^{5} + 509478952 T^{6} + 7611176678 T^{7} + 102849838382 T^{8} + 1268938278976 T^{9} + 14432679369101 T^{10} + 151938872362836 T^{11} + 1486834075205326 T^{12} + 151938872362836 p T^{13} + 14432679369101 p^{2} T^{14} + 1268938278976 p^{3} T^{15} + 102849838382 p^{4} T^{16} + 7611176678 p^{5} T^{17} + 509478952 p^{6} T^{18} + 30292836 p^{7} T^{19} + 1571877 p^{8} T^{20} + 68588 p^{9} T^{21} + 2424 p^{10} T^{22} + 62 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 5 T + 673 T^{2} + 4064 T^{3} + 220022 T^{4} + 1530532 T^{5} + 47929764 T^{6} + 356349238 T^{7} + 7949810559 T^{8} + 58180993890 T^{9} + 1058254694070 T^{10} + 7178571301925 T^{11} + 114126170122726 T^{12} + 7178571301925 p T^{13} + 1058254694070 p^{2} T^{14} + 58180993890 p^{3} T^{15} + 7949810559 p^{4} T^{16} + 356349238 p^{5} T^{17} + 47929764 p^{6} T^{18} + 1530532 p^{7} T^{19} + 220022 p^{8} T^{20} + 4064 p^{9} T^{21} + 673 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.10242656701675296504885219058, −2.89619234879211889561300683218, −2.72994419209824229082769172358, −2.72714320875040103344217880430, −2.65597309781283640523391005694, −2.51439608492355699517108894232, −2.34035698214974575320674519410, −2.31972894576000714757123748669, −2.29945048511787238187050154150, −2.06292068996911688832947815866, −2.01594046550860401293166150827, −1.89864609277754449591693428977, −1.86624768419348607197380586582, −1.83405290224567076612171745758, −1.81372965549993270738878948681, −1.68212548289882829496761789702, −1.68006319500303214066489101032, −1.52257208871247151055353974364, −1.37393026352352622891300728050, −1.33034746365028026617328465330, −1.28351097899423478984374087480, −1.18003727337079210744863866645, −1.07627109087777638648245068468, −1.04864946058350134490573142402, −0.828868009106063223174534884315, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.828868009106063223174534884315, 1.04864946058350134490573142402, 1.07627109087777638648245068468, 1.18003727337079210744863866645, 1.28351097899423478984374087480, 1.33034746365028026617328465330, 1.37393026352352622891300728050, 1.52257208871247151055353974364, 1.68006319500303214066489101032, 1.68212548289882829496761789702, 1.81372965549993270738878948681, 1.83405290224567076612171745758, 1.86624768419348607197380586582, 1.89864609277754449591693428977, 2.01594046550860401293166150827, 2.06292068996911688832947815866, 2.29945048511787238187050154150, 2.31972894576000714757123748669, 2.34035698214974575320674519410, 2.51439608492355699517108894232, 2.65597309781283640523391005694, 2.72714320875040103344217880430, 2.72994419209824229082769172358, 2.89619234879211889561300683218, 3.10242656701675296504885219058
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.