Dirichlet series
| L(s) = 1 | − 12·2-s + 78·4-s − 5·5-s + 6·7-s − 364·8-s + 60·10-s − 6·11-s + 3·13-s − 72·14-s + 1.36e3·16-s − 6·17-s + 8·19-s − 390·20-s + 72·22-s − 11·23-s − 12·25-s − 36·26-s + 468·28-s − 29·29-s + 2·31-s − 4.36e3·32-s + 72·34-s − 30·35-s + 5·37-s − 96·38-s + 1.82e3·40-s − 22·41-s + ⋯ |
| L(s) = 1 | − 8.48·2-s + 39·4-s − 2.23·5-s + 2.26·7-s − 128.·8-s + 18.9·10-s − 1.80·11-s + 0.832·13-s − 19.2·14-s + 341.·16-s − 1.45·17-s + 1.83·19-s − 87.2·20-s + 15.3·22-s − 2.29·23-s − 2.39·25-s − 7.06·26-s + 88.4·28-s − 5.38·29-s + 0.359·31-s − 772.·32-s + 12.3·34-s − 5.07·35-s + 0.821·37-s − 15.5·38-s + 287.·40-s − 3.43·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36} \cdot 149^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36} \cdot 149^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{36} \cdot 149^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.94635\times 10^{21}\) |
| Root analytic conductor: | \(8.01546\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{36} \cdot 149^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
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 | 2 | \( ( 1 + T )^{12} \) |
| 3 | \( 1 \) | |
| 149 | \( ( 1 - T )^{12} \) | |
| good | 5 | \( 1 + p T + 37 T^{2} + 133 T^{3} + 604 T^{4} + 1787 T^{5} + 6392 T^{6} + 16572 T^{7} + 50924 T^{8} + 119058 T^{9} + 65492 p T^{10} + 701969 T^{11} + 1770847 T^{12} + 701969 p T^{13} + 65492 p^{3} T^{14} + 119058 p^{3} T^{15} + 50924 p^{4} T^{16} + 16572 p^{5} T^{17} + 6392 p^{6} T^{18} + 1787 p^{7} T^{19} + 604 p^{8} T^{20} + 133 p^{9} T^{21} + 37 p^{10} T^{22} + p^{12} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 - 6 T + 53 T^{2} - 242 T^{3} + 1336 T^{4} - 5118 T^{5} + 22241 T^{6} - 74476 T^{7} + 275221 T^{8} - 117375 p T^{9} + 2668448 T^{10} - 7159148 T^{11} + 20764979 T^{12} - 7159148 p T^{13} + 2668448 p^{2} T^{14} - 117375 p^{4} T^{15} + 275221 p^{4} T^{16} - 74476 p^{5} T^{17} + 22241 p^{6} T^{18} - 5118 p^{7} T^{19} + 1336 p^{8} T^{20} - 242 p^{9} T^{21} + 53 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 6 T + 83 T^{2} + 400 T^{3} + 3275 T^{4} + 13795 T^{5} + 85490 T^{6} + 324432 T^{7} + 1657369 T^{8} + 5718173 T^{9} + 25187716 T^{10} + 78968378 T^{11} + 307925583 T^{12} + 78968378 p T^{13} + 25187716 p^{2} T^{14} + 5718173 p^{3} T^{15} + 1657369 p^{4} T^{16} + 324432 p^{5} T^{17} + 85490 p^{6} T^{18} + 13795 p^{7} T^{19} + 3275 p^{8} T^{20} + 400 p^{9} T^{21} + 83 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 3 T + 93 T^{2} - 185 T^{3} + 4117 T^{4} - 5292 T^{5} + 120357 T^{6} - 96192 T^{7} + 2648158 T^{8} - 100457 p T^{9} + 46450531 T^{10} - 15709542 T^{11} + 665668655 T^{12} - 15709542 p T^{13} + 46450531 p^{2} T^{14} - 100457 p^{4} T^{15} + 2648158 p^{4} T^{16} - 96192 p^{5} T^{17} + 120357 p^{6} T^{18} - 5292 p^{7} T^{19} + 4117 p^{8} T^{20} - 185 p^{9} T^{21} + 93 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 6 T + 105 T^{2} + 597 T^{3} + 6053 T^{4} + 32404 T^{5} + 242061 T^{6} + 1194171 T^{7} + 7302766 T^{8} + 32901614 T^{9} + 173184434 T^{10} + 705647403 T^{11} + 3282052285 T^{12} + 705647403 p T^{13} + 173184434 p^{2} T^{14} + 32901614 p^{3} T^{15} + 7302766 p^{4} T^{16} + 1194171 p^{5} T^{17} + 242061 p^{6} T^{18} + 32404 p^{7} T^{19} + 6053 p^{8} T^{20} + 597 p^{9} T^{21} + 105 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 8 T + 129 T^{2} - 782 T^{3} + 7183 T^{4} - 38111 T^{5} + 260210 T^{6} - 1339878 T^{7} + 7370283 T^{8} - 37496113 T^{9} + 172035206 T^{10} - 849459602 T^{11} + 3458078643 T^{12} - 849459602 p T^{13} + 172035206 p^{2} T^{14} - 37496113 p^{3} T^{15} + 7370283 p^{4} T^{16} - 1339878 p^{5} T^{17} + 260210 p^{6} T^{18} - 38111 p^{7} T^{19} + 7183 p^{8} T^{20} - 782 p^{9} T^{21} + 129 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 11 T + 262 T^{2} + 2269 T^{3} + 30685 T^{4} + 221080 T^{5} + 2185501 T^{6} + 13469165 T^{7} + 106742913 T^{8} + 570337343 T^{9} + 164684779 p T^{10} + 17621235363 T^{11} + 100387404119 T^{12} + 17621235363 p T^{13} + 164684779 p^{3} T^{14} + 570337343 p^{3} T^{15} + 106742913 p^{4} T^{16} + 13469165 p^{5} T^{17} + 2185501 p^{6} T^{18} + 221080 p^{7} T^{19} + 30685 p^{8} T^{20} + 2269 p^{9} T^{21} + 262 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + p T + 21 p T^{2} + 9253 T^{3} + 117590 T^{4} + 1265253 T^{5} + 12043636 T^{6} + 102332654 T^{7} + 791357894 T^{8} + 5594431404 T^{9} + 36508906376 T^{10} + 220065448467 T^{11} + 1231373716743 T^{12} + 220065448467 p T^{13} + 36508906376 p^{2} T^{14} + 5594431404 p^{3} T^{15} + 791357894 p^{4} T^{16} + 102332654 p^{5} T^{17} + 12043636 p^{6} T^{18} + 1265253 p^{7} T^{19} + 117590 p^{8} T^{20} + 9253 p^{9} T^{21} + 21 p^{11} T^{22} + p^{12} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 2 T + 316 T^{2} - 629 T^{3} + 47111 T^{4} - 90807 T^{5} + 4384473 T^{6} - 7978244 T^{7} + 283776859 T^{8} - 475465865 T^{9} + 13457602963 T^{10} - 20221270731 T^{11} + 479490551393 T^{12} - 20221270731 p T^{13} + 13457602963 p^{2} T^{14} - 475465865 p^{3} T^{15} + 283776859 p^{4} T^{16} - 7978244 p^{5} T^{17} + 4384473 p^{6} T^{18} - 90807 p^{7} T^{19} + 47111 p^{8} T^{20} - 629 p^{9} T^{21} + 316 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 5 T + 119 T^{2} - 115 T^{3} + 7645 T^{4} - 1649 T^{5} + 440841 T^{6} + 321802 T^{7} + 19993330 T^{8} + 6656905 T^{9} + 950071077 T^{10} + 125428093 T^{11} + 34857558493 T^{12} + 125428093 p T^{13} + 950071077 p^{2} T^{14} + 6656905 p^{3} T^{15} + 19993330 p^{4} T^{16} + 321802 p^{5} T^{17} + 440841 p^{6} T^{18} - 1649 p^{7} T^{19} + 7645 p^{8} T^{20} - 115 p^{9} T^{21} + 119 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 22 T + 570 T^{2} + 8654 T^{3} + 134326 T^{4} + 1580751 T^{5} + 18404440 T^{6} + 177542004 T^{7} + 1677920778 T^{8} + 13669761819 T^{9} + 108769892738 T^{10} + 759716186158 T^{11} + 5178296674430 T^{12} + 759716186158 p T^{13} + 108769892738 p^{2} T^{14} + 13669761819 p^{3} T^{15} + 1677920778 p^{4} T^{16} + 177542004 p^{5} T^{17} + 18404440 p^{6} T^{18} + 1580751 p^{7} T^{19} + 134326 p^{8} T^{20} + 8654 p^{9} T^{21} + 570 p^{10} T^{22} + 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 9 T + 340 T^{2} - 3067 T^{3} + 59853 T^{4} - 506558 T^{5} + 6943178 T^{6} - 53812233 T^{7} + 581760563 T^{8} - 4070801777 T^{9} + 36810465985 T^{10} - 229950733232 T^{11} + 1796356034379 T^{12} - 229950733232 p T^{13} + 36810465985 p^{2} T^{14} - 4070801777 p^{3} T^{15} + 581760563 p^{4} T^{16} - 53812233 p^{5} T^{17} + 6943178 p^{6} T^{18} - 506558 p^{7} T^{19} + 59853 p^{8} T^{20} - 3067 p^{9} T^{21} + 340 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 15 T + 362 T^{2} + 4070 T^{3} + 57742 T^{4} + 536591 T^{5} + 5843321 T^{6} + 47584927 T^{7} + 441055404 T^{8} + 3260908938 T^{9} + 26980598283 T^{10} + 183650618078 T^{11} + 1382094670383 T^{12} + 183650618078 p T^{13} + 26980598283 p^{2} T^{14} + 3260908938 p^{3} T^{15} + 441055404 p^{4} T^{16} + 47584927 p^{5} T^{17} + 5843321 p^{6} T^{18} + 536591 p^{7} T^{19} + 57742 p^{8} T^{20} + 4070 p^{9} T^{21} + 362 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 12 T + 401 T^{2} + 4001 T^{3} + 1331 p T^{4} + 575383 T^{5} + 7018763 T^{6} + 45343642 T^{7} + 427988438 T^{8} + 2060058947 T^{9} + 17268440727 T^{10} + 61388213430 T^{11} + 693724505451 T^{12} + 61388213430 p T^{13} + 17268440727 p^{2} T^{14} + 2060058947 p^{3} T^{15} + 427988438 p^{4} T^{16} + 45343642 p^{5} T^{17} + 7018763 p^{6} T^{18} + 575383 p^{7} T^{19} + 1331 p^{9} T^{20} + 4001 p^{9} T^{21} + 401 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 34 T + 991 T^{2} + 19524 T^{3} + 5778 p T^{4} + 4900505 T^{5} + 64248442 T^{6} + 743642222 T^{7} + 7996728095 T^{8} + 78396120401 T^{9} + 720435496810 T^{10} + 6116626005439 T^{11} + 48799744500633 T^{12} + 6116626005439 p T^{13} + 720435496810 p^{2} T^{14} + 78396120401 p^{3} T^{15} + 7996728095 p^{4} T^{16} + 743642222 p^{5} T^{17} + 64248442 p^{6} T^{18} + 4900505 p^{7} T^{19} + 5778 p^{9} T^{20} + 19524 p^{9} T^{21} + 991 p^{10} T^{22} + 34 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 4 T + 406 T^{2} + 2298 T^{3} + 82658 T^{4} + 590587 T^{5} + 185243 p T^{6} + 92930194 T^{7} + 1165100774 T^{8} + 10151363202 T^{9} + 95808777573 T^{10} + 817946848303 T^{11} + 6441572717915 T^{12} + 817946848303 p T^{13} + 95808777573 p^{2} T^{14} + 10151363202 p^{3} T^{15} + 1165100774 p^{4} T^{16} + 92930194 p^{5} T^{17} + 185243 p^{7} T^{18} + 590587 p^{7} T^{19} + 82658 p^{8} T^{20} + 2298 p^{9} T^{21} + 406 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - T + 257 T^{2} + 45 T^{3} + 40286 T^{4} + 24251 T^{5} + 5084197 T^{6} + 4140364 T^{7} + 503637946 T^{8} + 544064397 T^{9} + 42191060791 T^{10} + 45687110740 T^{11} + 3079301020221 T^{12} + 45687110740 p T^{13} + 42191060791 p^{2} T^{14} + 544064397 p^{3} T^{15} + 503637946 p^{4} T^{16} + 4140364 p^{5} T^{17} + 5084197 p^{6} T^{18} + 24251 p^{7} T^{19} + 40286 p^{8} T^{20} + 45 p^{9} T^{21} + 257 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 21 T + 815 T^{2} + 13439 T^{3} + 298431 T^{4} + 4057110 T^{5} + 66280222 T^{6} + 764199396 T^{7} + 10056987962 T^{8} + 99917777007 T^{9} + 1105322506763 T^{10} + 9528643307964 T^{11} + 90523489032239 T^{12} + 9528643307964 p T^{13} + 1105322506763 p^{2} T^{14} + 99917777007 p^{3} T^{15} + 10056987962 p^{4} T^{16} + 764199396 p^{5} T^{17} + 66280222 p^{6} T^{18} + 4057110 p^{7} T^{19} + 298431 p^{8} T^{20} + 13439 p^{9} T^{21} + 815 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 2 T + 398 T^{2} + 1673 T^{3} + 77385 T^{4} + 460592 T^{5} + 10426140 T^{6} + 69484139 T^{7} + 1135186987 T^{8} + 7189648889 T^{9} + 104932563821 T^{10} + 595825738667 T^{11} + 8285671901785 T^{12} + 595825738667 p T^{13} + 104932563821 p^{2} T^{14} + 7189648889 p^{3} T^{15} + 1135186987 p^{4} T^{16} + 69484139 p^{5} T^{17} + 10426140 p^{6} T^{18} + 460592 p^{7} T^{19} + 77385 p^{8} T^{20} + 1673 p^{9} T^{21} + 398 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 9 T + 518 T^{2} - 3691 T^{3} + 131738 T^{4} - 781364 T^{5} + 22668483 T^{6} - 116720572 T^{7} + 2981526684 T^{8} - 13678163642 T^{9} + 315293662403 T^{10} - 1306179035946 T^{11} + 27391779560869 T^{12} - 1306179035946 p T^{13} + 315293662403 p^{2} T^{14} - 13678163642 p^{3} T^{15} + 2981526684 p^{4} T^{16} - 116720572 p^{5} T^{17} + 22668483 p^{6} T^{18} - 781364 p^{7} T^{19} + 131738 p^{8} T^{20} - 3691 p^{9} T^{21} + 518 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 10 T + 595 T^{2} + 4428 T^{3} + 153376 T^{4} + 876647 T^{5} + 24166252 T^{6} + 117887750 T^{7} + 34644621 p T^{8} + 14127122419 T^{9} + 295311784318 T^{10} + 1507783663471 T^{11} + 26514609855899 T^{12} + 1507783663471 p T^{13} + 295311784318 p^{2} T^{14} + 14127122419 p^{3} T^{15} + 34644621 p^{5} T^{16} + 117887750 p^{5} T^{17} + 24166252 p^{6} T^{18} + 876647 p^{7} T^{19} + 153376 p^{8} T^{20} + 4428 p^{9} T^{21} + 595 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 2 T + 662 T^{2} - 1003 T^{3} + 214775 T^{4} - 220301 T^{5} + 45647850 T^{6} - 27386804 T^{7} + 7148082988 T^{8} - 2124959770 T^{9} + 875420427455 T^{10} - 125970749265 T^{11} + 86394150305655 T^{12} - 125970749265 p T^{13} + 875420427455 p^{2} T^{14} - 2124959770 p^{3} T^{15} + 7148082988 p^{4} T^{16} - 27386804 p^{5} T^{17} + 45647850 p^{6} T^{18} - 220301 p^{7} T^{19} + 214775 p^{8} T^{20} - 1003 p^{9} T^{21} + 662 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 13 T + 909 T^{2} + 10619 T^{3} + 399660 T^{4} + 4192016 T^{5} + 111664562 T^{6} + 1049739005 T^{7} + 22001910224 T^{8} + 184572841939 T^{9} + 3213928058406 T^{10} + 23874378565506 T^{11} + 356479471151649 T^{12} + 23874378565506 p T^{13} + 3213928058406 p^{2} T^{14} + 184572841939 p^{3} T^{15} + 22001910224 p^{4} T^{16} + 1049739005 p^{5} T^{17} + 111664562 p^{6} T^{18} + 4192016 p^{7} T^{19} + 399660 p^{8} T^{20} + 10619 p^{9} T^{21} + 909 p^{10} T^{22} + 13 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.55988982401000849370830540684, −2.43591984141413704322074058579, −2.22163632812230125909633652173, −2.20224546773035508332562250269, −2.18744887627079361139361592394, −2.16250019123360919272784145443, −2.09592095627196113129488150130, −2.05075823063855764280234900746, −2.04957227814711625436591942018, −1.98927758027035264160756080897, −1.98127835726745639661146882492, −1.96860636199840591223081810938, −1.71117621923286916827951565757, −1.51478349460126275262217945661, −1.43108189431139248489230914217, −1.42375775350880749256728015808, −1.41106865631353533920862965922, −1.27631371533090052283724109610, −1.25897381282847814241757664138, −1.18619473954663238574352339496, −1.17367369039282922350417245919, −1.05988459654285081280349101946, −1.01843956080927589160624601124, −0.945495136341319071146647454673, −0.77325335231765937139230839026, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.77325335231765937139230839026, 0.945495136341319071146647454673, 1.01843956080927589160624601124, 1.05988459654285081280349101946, 1.17367369039282922350417245919, 1.18619473954663238574352339496, 1.25897381282847814241757664138, 1.27631371533090052283724109610, 1.41106865631353533920862965922, 1.42375775350880749256728015808, 1.43108189431139248489230914217, 1.51478349460126275262217945661, 1.71117621923286916827951565757, 1.96860636199840591223081810938, 1.98127835726745639661146882492, 1.98927758027035264160756080897, 2.04957227814711625436591942018, 2.05075823063855764280234900746, 2.09592095627196113129488150130, 2.16250019123360919272784145443, 2.18744887627079361139361592394, 2.20224546773035508332562250269, 2.22163632812230125909633652173, 2.43591984141413704322074058579, 2.55988982401000849370830540684
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.