Dirichlet series
| L(s) = 1 | + 7·2-s + 17·4-s + 7·5-s + 12·7-s + 5·8-s + 49·10-s + 22·11-s + 84·14-s − 56·16-s + 6·17-s − 7·19-s + 119·20-s + 154·22-s + 29·23-s − 4·25-s + 204·28-s + 22·29-s − 16·31-s − 106·32-s + 42·34-s + 84·35-s − 4·37-s − 49·38-s + 35·40-s + 21·41-s + 11·43-s + 374·44-s + ⋯ |
| L(s) = 1 | + 4.94·2-s + 17/2·4-s + 3.13·5-s + 4.53·7-s + 1.76·8-s + 15.4·10-s + 6.63·11-s + 22.4·14-s − 14·16-s + 1.45·17-s − 1.60·19-s + 26.6·20-s + 32.8·22-s + 6.04·23-s − 4/5·25-s + 38.5·28-s + 4.08·29-s − 2.87·31-s − 18.7·32-s + 7.20·34-s + 14.1·35-s − 0.657·37-s − 7.94·38-s + 5.53·40-s + 3.27·41-s + 1.67·43-s + 56.3·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12} \cdot 127^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12} \cdot 127^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{24} \cdot 7^{12} \cdot 127^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.62441\times 10^{21}\) |
| Root analytic conductor: | \(7.99301\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{24} \cdot 7^{12} \cdot 127^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(39776.09810\) |
| \(L(\frac12)\) | \(\approx\) | \(39776.09810\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 7 | \( ( 1 - T )^{12} \) | |
| 127 | \( ( 1 - T )^{12} \) | |
| good | 2 | \( 1 - 7 T + p^{5} T^{2} - 55 p T^{3} + 317 T^{4} - 795 T^{5} + 1791 T^{6} - 1839 p T^{7} + 3487 p T^{8} - 12301 T^{9} + 1269 p^{4} T^{10} - 15733 p T^{11} + 45871 T^{12} - 15733 p^{2} T^{13} + 1269 p^{6} T^{14} - 12301 p^{3} T^{15} + 3487 p^{5} T^{16} - 1839 p^{6} T^{17} + 1791 p^{6} T^{18} - 795 p^{7} T^{19} + 317 p^{8} T^{20} - 55 p^{10} T^{21} + p^{15} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 - 7 T + 53 T^{2} - 246 T^{3} + 1136 T^{4} - 4123 T^{5} + 14704 T^{6} - 44997 T^{7} + 27009 p T^{8} - 361862 T^{9} + 190053 p T^{10} - 453794 p T^{11} + 5306511 T^{12} - 453794 p^{2} T^{13} + 190053 p^{3} T^{14} - 361862 p^{3} T^{15} + 27009 p^{5} T^{16} - 44997 p^{5} T^{17} + 14704 p^{6} T^{18} - 4123 p^{7} T^{19} + 1136 p^{8} T^{20} - 246 p^{9} T^{21} + 53 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 2 p T + 310 T^{2} - 3233 T^{3} + 27525 T^{4} - 198779 T^{5} + 1252854 T^{6} - 7006099 T^{7} + 35184038 T^{8} - 159974396 T^{9} + 662234726 T^{10} - 2505459725 T^{11} + 8680044717 T^{12} - 2505459725 p T^{13} + 662234726 p^{2} T^{14} - 159974396 p^{3} T^{15} + 35184038 p^{4} T^{16} - 7006099 p^{5} T^{17} + 1252854 p^{6} T^{18} - 198779 p^{7} T^{19} + 27525 p^{8} T^{20} - 3233 p^{9} T^{21} + 310 p^{10} T^{22} - 2 p^{12} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 6 p T^{2} + 6 T^{3} + 3142 T^{4} + 126 T^{5} + 85177 T^{6} - 5639 T^{7} + 1751144 T^{8} - 334994 T^{9} + 29187591 T^{10} - 597120 p T^{11} + 410472133 T^{12} - 597120 p^{2} T^{13} + 29187591 p^{2} T^{14} - 334994 p^{3} T^{15} + 1751144 p^{4} T^{16} - 5639 p^{5} T^{17} + 85177 p^{6} T^{18} + 126 p^{7} T^{19} + 3142 p^{8} T^{20} + 6 p^{9} T^{21} + 6 p^{11} T^{22} + p^{12} T^{24} \) | |
| 17 | \( 1 - 6 T + 150 T^{2} - 861 T^{3} + 11039 T^{4} - 58946 T^{5} + 524291 T^{6} - 2547955 T^{7} + 17782322 T^{8} - 77457818 T^{9} + 451150636 T^{10} - 1742252808 T^{11} + 8746331963 T^{12} - 1742252808 p T^{13} + 451150636 p^{2} T^{14} - 77457818 p^{3} T^{15} + 17782322 p^{4} T^{16} - 2547955 p^{5} T^{17} + 524291 p^{6} T^{18} - 58946 p^{7} T^{19} + 11039 p^{8} T^{20} - 861 p^{9} T^{21} + 150 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 7 T + 113 T^{2} + 557 T^{3} + 5849 T^{4} + 24530 T^{5} + 212023 T^{6} + 802042 T^{7} + 6029495 T^{8} + 21080957 T^{9} + 143612483 T^{10} + 471134705 T^{11} + 2944902893 T^{12} + 471134705 p T^{13} + 143612483 p^{2} T^{14} + 21080957 p^{3} T^{15} + 6029495 p^{4} T^{16} + 802042 p^{5} T^{17} + 212023 p^{6} T^{18} + 24530 p^{7} T^{19} + 5849 p^{8} T^{20} + 557 p^{9} T^{21} + 113 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 29 T + 542 T^{2} - 7573 T^{3} + 87342 T^{4} - 861461 T^{5} + 7482463 T^{6} - 58099514 T^{7} + 408414306 T^{8} - 2617981756 T^{9} + 15395107951 T^{10} - 83316425431 T^{11} + 415948195901 T^{12} - 83316425431 p T^{13} + 15395107951 p^{2} T^{14} - 2617981756 p^{3} T^{15} + 408414306 p^{4} T^{16} - 58099514 p^{5} T^{17} + 7482463 p^{6} T^{18} - 861461 p^{7} T^{19} + 87342 p^{8} T^{20} - 7573 p^{9} T^{21} + 542 p^{10} T^{22} - 29 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 22 T + 444 T^{2} - 5948 T^{3} + 73244 T^{4} - 735689 T^{5} + 6885195 T^{6} - 56276420 T^{7} + 432301143 T^{8} - 2987132731 T^{9} + 19491587322 T^{10} - 115935681738 T^{11} + 652645944087 T^{12} - 115935681738 p T^{13} + 19491587322 p^{2} T^{14} - 2987132731 p^{3} T^{15} + 432301143 p^{4} T^{16} - 56276420 p^{5} T^{17} + 6885195 p^{6} T^{18} - 735689 p^{7} T^{19} + 73244 p^{8} T^{20} - 5948 p^{9} T^{21} + 444 p^{10} T^{22} - 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 16 T + 287 T^{2} + 2874 T^{3} + 31158 T^{4} + 237487 T^{5} + 2009279 T^{6} + 12848798 T^{7} + 94191169 T^{8} + 534755984 T^{9} + 3587688040 T^{10} + 18769932379 T^{11} + 118295445101 T^{12} + 18769932379 p T^{13} + 3587688040 p^{2} T^{14} + 534755984 p^{3} T^{15} + 94191169 p^{4} T^{16} + 12848798 p^{5} T^{17} + 2009279 p^{6} T^{18} + 237487 p^{7} T^{19} + 31158 p^{8} T^{20} + 2874 p^{9} T^{21} + 287 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 4 T + 181 T^{2} + 510 T^{3} + 15686 T^{4} + 21336 T^{5} + 848235 T^{6} - 767510 T^{7} + 30486755 T^{8} - 151748820 T^{9} + 782416647 T^{10} - 9627135267 T^{11} + 21532144847 T^{12} - 9627135267 p T^{13} + 782416647 p^{2} T^{14} - 151748820 p^{3} T^{15} + 30486755 p^{4} T^{16} - 767510 p^{5} T^{17} + 848235 p^{6} T^{18} + 21336 p^{7} T^{19} + 15686 p^{8} T^{20} + 510 p^{9} T^{21} + 181 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 21 T + 397 T^{2} - 121 p T^{3} + 60353 T^{4} - 595870 T^{5} + 5804080 T^{6} - 48840010 T^{7} + 407453481 T^{8} - 3030450751 T^{9} + 22467228151 T^{10} - 150878853149 T^{11} + 1012729360251 T^{12} - 150878853149 p T^{13} + 22467228151 p^{2} T^{14} - 3030450751 p^{3} T^{15} + 407453481 p^{4} T^{16} - 48840010 p^{5} T^{17} + 5804080 p^{6} T^{18} - 595870 p^{7} T^{19} + 60353 p^{8} T^{20} - 121 p^{10} T^{21} + 397 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 11 T + 433 T^{2} - 3791 T^{3} + 84876 T^{4} - 619672 T^{5} + 10284562 T^{6} - 64446156 T^{7} + 874248113 T^{8} - 4782575521 T^{9} + 55476814996 T^{10} - 267170950199 T^{11} + 2707748351749 T^{12} - 267170950199 p T^{13} + 55476814996 p^{2} T^{14} - 4782575521 p^{3} T^{15} + 874248113 p^{4} T^{16} - 64446156 p^{5} T^{17} + 10284562 p^{6} T^{18} - 619672 p^{7} T^{19} + 84876 p^{8} T^{20} - 3791 p^{9} T^{21} + 433 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 31 T + 851 T^{2} - 15591 T^{3} + 257616 T^{4} - 3460144 T^{5} + 42762377 T^{6} - 458816246 T^{7} + 4589623268 T^{8} - 41052380290 T^{9} + 345041231405 T^{10} - 2628912170117 T^{11} + 18890968445127 T^{12} - 2628912170117 p T^{13} + 345041231405 p^{2} T^{14} - 41052380290 p^{3} T^{15} + 4589623268 p^{4} T^{16} - 458816246 p^{5} T^{17} + 42762377 p^{6} T^{18} - 3460144 p^{7} T^{19} + 257616 p^{8} T^{20} - 15591 p^{9} T^{21} + 851 p^{10} T^{22} - 31 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 38 T + 1114 T^{2} - 23200 T^{3} + 413253 T^{4} - 6166493 T^{5} + 82074452 T^{6} - 965485995 T^{7} + 10351010793 T^{8} - 100472784479 T^{9} + 899169946060 T^{10} - 7368541365910 T^{11} + 55963916662751 T^{12} - 7368541365910 p T^{13} + 899169946060 p^{2} T^{14} - 100472784479 p^{3} T^{15} + 10351010793 p^{4} T^{16} - 965485995 p^{5} T^{17} + 82074452 p^{6} T^{18} - 6166493 p^{7} T^{19} + 413253 p^{8} T^{20} - 23200 p^{9} T^{21} + 1114 p^{10} T^{22} - 38 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 15 T + 557 T^{2} - 7228 T^{3} + 148191 T^{4} - 1667699 T^{5} + 24849493 T^{6} - 244376286 T^{7} + 2931747227 T^{8} - 25366461556 T^{9} + 257317271117 T^{10} - 1963795581714 T^{11} + 17280433060643 T^{12} - 1963795581714 p T^{13} + 257317271117 p^{2} T^{14} - 25366461556 p^{3} T^{15} + 2931747227 p^{4} T^{16} - 244376286 p^{5} T^{17} + 24849493 p^{6} T^{18} - 1667699 p^{7} T^{19} + 148191 p^{8} T^{20} - 7228 p^{9} T^{21} + 557 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 3 T + 432 T^{2} + 1230 T^{3} + 90161 T^{4} + 273851 T^{5} + 12309892 T^{6} + 42041210 T^{7} + 1247608113 T^{8} + 4675036486 T^{9} + 100502461314 T^{10} + 383065843548 T^{11} + 6694800790131 T^{12} + 383065843548 p T^{13} + 100502461314 p^{2} T^{14} + 4675036486 p^{3} T^{15} + 1247608113 p^{4} T^{16} + 42041210 p^{5} T^{17} + 12309892 p^{6} T^{18} + 273851 p^{7} T^{19} + 90161 p^{8} T^{20} + 1230 p^{9} T^{21} + 432 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + T + 435 T^{2} + 1226 T^{3} + 93877 T^{4} + 376053 T^{5} + 13893378 T^{6} + 61837204 T^{7} + 1579932599 T^{8} + 6958693447 T^{9} + 143413068674 T^{10} + 595120533863 T^{11} + 10600062861109 T^{12} + 595120533863 p T^{13} + 143413068674 p^{2} T^{14} + 6958693447 p^{3} T^{15} + 1579932599 p^{4} T^{16} + 61837204 p^{5} T^{17} + 13893378 p^{6} T^{18} + 376053 p^{7} T^{19} + 93877 p^{8} T^{20} + 1226 p^{9} T^{21} + 435 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 57 T + 1863 T^{2} - 43083 T^{3} + 789480 T^{4} - 12133524 T^{5} + 163477864 T^{6} - 1985571923 T^{7} + 22205022013 T^{8} - 231246886523 T^{9} + 2259759437514 T^{10} - 20781791620062 T^{11} + 180293488087779 T^{12} - 20781791620062 p T^{13} + 2259759437514 p^{2} T^{14} - 231246886523 p^{3} T^{15} + 22205022013 p^{4} T^{16} - 1985571923 p^{5} T^{17} + 163477864 p^{6} T^{18} - 12133524 p^{7} T^{19} + 789480 p^{8} T^{20} - 43083 p^{9} T^{21} + 1863 p^{10} T^{22} - 57 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 7 T + 587 T^{2} + 3074 T^{3} + 157312 T^{4} + 583506 T^{5} + 26023030 T^{6} + 60949389 T^{7} + 3052742503 T^{8} + 3650320080 T^{9} + 280386426518 T^{10} + 138384443368 T^{11} + 21843447058245 T^{12} + 138384443368 p T^{13} + 280386426518 p^{2} T^{14} + 3650320080 p^{3} T^{15} + 3052742503 p^{4} T^{16} + 60949389 p^{5} T^{17} + 26023030 p^{6} T^{18} + 583506 p^{7} T^{19} + 157312 p^{8} T^{20} + 3074 p^{9} T^{21} + 587 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 18 T + 559 T^{2} + 8811 T^{3} + 152764 T^{4} + 2049448 T^{5} + 27037636 T^{6} + 311089904 T^{7} + 3508827540 T^{8} + 35662362397 T^{9} + 361263751482 T^{10} + 3338367560519 T^{11} + 30997455703321 T^{12} + 3338367560519 p T^{13} + 361263751482 p^{2} T^{14} + 35662362397 p^{3} T^{15} + 3508827540 p^{4} T^{16} + 311089904 p^{5} T^{17} + 27037636 p^{6} T^{18} + 2049448 p^{7} T^{19} + 152764 p^{8} T^{20} + 8811 p^{9} T^{21} + 559 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 21 T + 663 T^{2} - 10210 T^{3} + 205019 T^{4} - 2643048 T^{5} + 41783428 T^{6} - 470349966 T^{7} + 6284996835 T^{8} - 62840031715 T^{9} + 733381689758 T^{10} - 6562524873910 T^{11} + 68041602774789 T^{12} - 6562524873910 p T^{13} + 733381689758 p^{2} T^{14} - 62840031715 p^{3} T^{15} + 6284996835 p^{4} T^{16} - 470349966 p^{5} T^{17} + 41783428 p^{6} T^{18} - 2643048 p^{7} T^{19} + 205019 p^{8} T^{20} - 10210 p^{9} T^{21} + 663 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 6 T + 248 T^{2} + 1937 T^{3} + 46222 T^{4} + 282188 T^{5} + 5446195 T^{6} + 28943943 T^{7} + 518826385 T^{8} + 1848017586 T^{9} + 40371147531 T^{10} + 102610466949 T^{11} + 3352784006073 T^{12} + 102610466949 p T^{13} + 40371147531 p^{2} T^{14} + 1848017586 p^{3} T^{15} + 518826385 p^{4} T^{16} + 28943943 p^{5} T^{17} + 5446195 p^{6} T^{18} + 282188 p^{7} T^{19} + 46222 p^{8} T^{20} + 1937 p^{9} T^{21} + 248 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 4 T + 389 T^{2} - 889 T^{3} + 83868 T^{4} - 183451 T^{5} + 14101452 T^{6} - 37965828 T^{7} + 1915592806 T^{8} - 5833378176 T^{9} + 221723974008 T^{10} - 732128943698 T^{11} + 22870071327251 T^{12} - 732128943698 p T^{13} + 221723974008 p^{2} T^{14} - 5833378176 p^{3} T^{15} + 1915592806 p^{4} T^{16} - 37965828 p^{5} T^{17} + 14101452 p^{6} T^{18} - 183451 p^{7} T^{19} + 83868 p^{8} T^{20} - 889 p^{9} T^{21} + 389 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.30549484801610836968924695963, −2.25428613248775999429258415401, −2.16887549898752441709160052060, −2.12299460707049853788351716969, −2.07366394137467016602204149849, −2.03487830201194498807190464442, −1.71503167300157245333693157388, −1.66720911864615377492881028396, −1.65313976256958056361932662234, −1.62325398353671711753739870838, −1.56708833045715430323760860606, −1.55172794960242260602350354498, −1.45162707686168242195710382791, −1.26802055994153194805807363818, −1.18019459950740819203016857622, −1.01480445394391595422696921100, −0.958625929251646544754007120257, −0.885821341012196396600156838799, −0.858451393359907981109459486708, −0.802143987705254648564888050110, −0.73786102016329615616651339483, −0.71273814865375220637368326740, −0.58235554120775856657681116204, −0.52303471410951429693290836710, −0.28387965450738095360742542046, 0.28387965450738095360742542046, 0.52303471410951429693290836710, 0.58235554120775856657681116204, 0.71273814865375220637368326740, 0.73786102016329615616651339483, 0.802143987705254648564888050110, 0.858451393359907981109459486708, 0.885821341012196396600156838799, 0.958625929251646544754007120257, 1.01480445394391595422696921100, 1.18019459950740819203016857622, 1.26802055994153194805807363818, 1.45162707686168242195710382791, 1.55172794960242260602350354498, 1.56708833045715430323760860606, 1.62325398353671711753739870838, 1.65313976256958056361932662234, 1.66720911864615377492881028396, 1.71503167300157245333693157388, 2.03487830201194498807190464442, 2.07366394137467016602204149849, 2.12299460707049853788351716969, 2.16887549898752441709160052060, 2.25428613248775999429258415401, 2.30549484801610836968924695963
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.