Dirichlet series
| L(s) = 1 | − 2·2-s − 3·3-s + 4-s + 6·6-s + 7-s + 2·9-s − 2·11-s − 3·12-s + 13-s − 2·14-s − 12·17-s − 4·18-s − 6·19-s − 3·21-s + 4·22-s + 35·23-s + 2·25-s − 2·26-s + 12·27-s + 28-s − 14·29-s − 8·31-s + 6·33-s + 24·34-s + 2·36-s + 31·37-s + 12·38-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 1/2·4-s + 2.44·6-s + 0.377·7-s + 2/3·9-s − 0.603·11-s − 0.866·12-s + 0.277·13-s − 0.534·14-s − 2.91·17-s − 0.942·18-s − 1.37·19-s − 0.654·21-s + 0.852·22-s + 7.29·23-s + 2/5·25-s − 0.392·26-s + 2.30·27-s + 0.188·28-s − 2.59·29-s − 1.43·31-s + 1.04·33-s + 4.11·34-s + 1/3·36-s + 5.09·37-s + 1.94·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 29^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.73780\times 10^{-5}\) |
| Root analytic conductor: | \(0.680538\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 29^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.04891275001\) |
| \(L(\frac12)\) | \(\approx\) | \(0.04891275001\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 29 | \( 1 + 14 T + 71 T^{2} + 126 T^{3} - 769 T^{4} - 532 p T^{5} - 4255 p T^{6} - 532 p^{2} T^{7} - 769 p^{2} T^{8} + 126 p^{3} T^{9} + 71 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 3 | \( 1 + p T + 7 T^{2} + p T^{3} - 29 T^{4} - 92 T^{5} - 142 T^{6} - T^{7} + 526 T^{8} + 365 p T^{9} + 772 T^{10} - 568 p T^{11} - 5903 T^{12} - 568 p^{2} T^{13} + 772 p^{2} T^{14} + 365 p^{4} T^{15} + 526 p^{4} T^{16} - p^{5} T^{17} - 142 p^{6} T^{18} - 92 p^{7} T^{19} - 29 p^{8} T^{20} + p^{10} T^{21} + 7 p^{10} T^{22} + p^{12} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 - 2 T^{2} - 21 T^{4} - 84 T^{5} + 8 T^{6} + 616 T^{7} + 453 T^{8} + 616 T^{9} + 3458 T^{10} - 7448 T^{11} - 45709 T^{12} - 7448 p T^{13} + 3458 p^{2} T^{14} + 616 p^{3} T^{15} + 453 p^{4} T^{16} + 616 p^{5} T^{17} + 8 p^{6} T^{18} - 84 p^{7} T^{19} - 21 p^{8} T^{20} - 2 p^{10} T^{22} + p^{12} T^{24} \) | |
| 7 | \( 1 - T - 25 T^{2} + 16 T^{3} + 228 T^{4} - 12 p T^{5} - 587 T^{6} + 108 p T^{7} - 1762 T^{8} - 11013 T^{9} - 37248 T^{10} + 47184 T^{11} + 607909 T^{12} + 47184 p T^{13} - 37248 p^{2} T^{14} - 11013 p^{3} T^{15} - 1762 p^{4} T^{16} + 108 p^{6} T^{17} - 587 p^{6} T^{18} - 12 p^{8} T^{19} + 228 p^{8} T^{20} + 16 p^{9} T^{21} - 25 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 2 T + 18 T^{2} + 58 T^{3} + 43 p T^{4} + 1478 T^{5} + 622 p T^{6} + 22138 T^{7} + 110657 T^{8} + 324810 T^{9} + 1351578 T^{10} + 3710512 T^{11} + 16741707 T^{12} + 3710512 p T^{13} + 1351578 p^{2} T^{14} + 324810 p^{3} T^{15} + 110657 p^{4} T^{16} + 22138 p^{5} T^{17} + 622 p^{7} T^{18} + 1478 p^{7} T^{19} + 43 p^{9} T^{20} + 58 p^{9} T^{21} + 18 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - T - 18 T^{2} + 24 T^{3} - 146 T^{4} + 744 T^{5} + 3395 T^{6} - 13584 T^{7} + 25919 T^{8} - 213669 T^{9} - 37590 T^{10} + 3264394 T^{11} - 9363773 T^{12} + 3264394 p T^{13} - 37590 p^{2} T^{14} - 213669 p^{3} T^{15} + 25919 p^{4} T^{16} - 13584 p^{5} T^{17} + 3395 p^{6} T^{18} + 744 p^{7} T^{19} - 146 p^{8} T^{20} + 24 p^{9} T^{21} - 18 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( ( 1 + 6 T + 49 T^{2} + 286 T^{3} + 1569 T^{4} + 6950 T^{5} + 33591 T^{6} + 6950 p T^{7} + 1569 p^{2} T^{8} + 286 p^{3} T^{9} + 49 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( 1 + 6 T + p T^{2} - 189 T^{3} - 1190 T^{4} - 6174 T^{5} + 395 p T^{6} + 85995 T^{7} + 42376 p T^{8} + 1007102 T^{9} + 2495766 T^{10} - 54489512 T^{11} - 164472881 T^{12} - 54489512 p T^{13} + 2495766 p^{2} T^{14} + 1007102 p^{3} T^{15} + 42376 p^{5} T^{16} + 85995 p^{5} T^{17} + 395 p^{7} T^{18} - 6174 p^{7} T^{19} - 1190 p^{8} T^{20} - 189 p^{9} T^{21} + p^{11} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 35 T + 516 T^{2} - 3808 T^{3} + 8792 T^{4} + 88396 T^{5} - 844129 T^{6} + 1560356 T^{7} + 21496403 T^{8} - 185318595 T^{9} + 480127410 T^{10} + 2017404452 T^{11} - 20465833233 T^{12} + 2017404452 p T^{13} + 480127410 p^{2} T^{14} - 185318595 p^{3} T^{15} + 21496403 p^{4} T^{16} + 1560356 p^{5} T^{17} - 844129 p^{6} T^{18} + 88396 p^{7} T^{19} + 8792 p^{8} T^{20} - 3808 p^{9} T^{21} + 516 p^{10} T^{22} - 35 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 8 T - 42 T^{2} - 418 T^{3} + 1165 T^{4} + 18276 T^{5} + 11132 T^{6} - 847210 T^{7} - 117683 p T^{8} + 23595856 T^{9} + 172811272 T^{10} - 262259800 T^{11} - 5381168495 T^{12} - 262259800 p T^{13} + 172811272 p^{2} T^{14} + 23595856 p^{3} T^{15} - 117683 p^{5} T^{16} - 847210 p^{5} T^{17} + 11132 p^{6} T^{18} + 18276 p^{7} T^{19} + 1165 p^{8} T^{20} - 418 p^{9} T^{21} - 42 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 31 T + 429 T^{2} - 3421 T^{3} + 15002 T^{4} + 17000 T^{5} - 1022774 T^{6} + 10558493 T^{7} - 63523085 T^{8} + 190865363 T^{9} + 696668043 T^{10} - 14626722708 T^{11} + 115003206160 T^{12} - 14626722708 p T^{13} + 696668043 p^{2} T^{14} + 190865363 p^{3} T^{15} - 63523085 p^{4} T^{16} + 10558493 p^{5} T^{17} - 1022774 p^{6} T^{18} + 17000 p^{7} T^{19} + 15002 p^{8} T^{20} - 3421 p^{9} T^{21} + 429 p^{10} T^{22} - 31 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( ( 1 + 15 T + 265 T^{2} + 2346 T^{3} + 23564 T^{4} + 151066 T^{5} + 1169507 T^{6} + 151066 p T^{7} + 23564 p^{2} T^{8} + 2346 p^{3} T^{9} + 265 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 43 | \( 1 + 5 T - 95 T^{2} - 419 T^{3} + 6060 T^{4} + 24212 T^{5} - 199816 T^{6} - 971011 T^{7} - 2163977 T^{8} + 26051945 T^{9} + 644216507 T^{10} - 349053828 T^{11} - 38467894288 T^{12} - 349053828 p T^{13} + 644216507 p^{2} T^{14} + 26051945 p^{3} T^{15} - 2163977 p^{4} T^{16} - 971011 p^{5} T^{17} - 199816 p^{6} T^{18} + 24212 p^{7} T^{19} + 6060 p^{8} T^{20} - 419 p^{9} T^{21} - 95 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 33 T + 455 T^{2} + 3155 T^{3} + 7209 T^{4} - 64536 T^{5} - 610888 T^{6} + 24647 T^{7} + 41205554 T^{8} + 360337001 T^{9} + 657892628 T^{10} - 14509840136 T^{11} - 160598514331 T^{12} - 14509840136 p T^{13} + 657892628 p^{2} T^{14} + 360337001 p^{3} T^{15} + 41205554 p^{4} T^{16} + 24647 p^{5} T^{17} - 610888 p^{6} T^{18} - 64536 p^{7} T^{19} + 7209 p^{8} T^{20} + 3155 p^{9} T^{21} + 455 p^{10} T^{22} + 33 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 13 T + 20 T^{2} + 85 T^{3} + 3533 T^{4} - 20401 T^{5} + 30044 T^{6} - 1141488 T^{7} - 3747337 T^{8} + 137654842 T^{9} - 315733654 T^{10} + 1855530631 T^{11} - 43430583054 T^{12} + 1855530631 p T^{13} - 315733654 p^{2} T^{14} + 137654842 p^{3} T^{15} - 3747337 p^{4} T^{16} - 1141488 p^{5} T^{17} + 30044 p^{6} T^{18} - 20401 p^{7} T^{19} + 3533 p^{8} T^{20} + 85 p^{9} T^{21} + 20 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( ( 1 - 19 T + 430 T^{2} - 5270 T^{3} + 67351 T^{4} - 596271 T^{5} + 5360932 T^{6} - 596271 p T^{7} + 67351 p^{2} T^{8} - 5270 p^{3} T^{9} + 430 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 61 | \( 1 + 5 T + 28 T^{2} - 478 T^{3} + 2666 T^{4} - 27730 T^{5} + 39447 T^{6} - 3437854 T^{7} + 17676545 T^{8} - 164577291 T^{9} + 1112390350 T^{10} - 5446056302 T^{11} + 162829383287 T^{12} - 5446056302 p T^{13} + 1112390350 p^{2} T^{14} - 164577291 p^{3} T^{15} + 17676545 p^{4} T^{16} - 3437854 p^{5} T^{17} + 39447 p^{6} T^{18} - 27730 p^{7} T^{19} + 2666 p^{8} T^{20} - 478 p^{9} T^{21} + 28 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 7 T - 81 T^{2} - 1008 T^{3} + 2268 T^{4} + 76720 T^{5} + 42001 T^{6} - 11032 p T^{7} + 10851998 T^{8} - 178049641 T^{9} - 3188000284 T^{10} + 9157457484 T^{11} + 371121758785 T^{12} + 9157457484 p T^{13} - 3188000284 p^{2} T^{14} - 178049641 p^{3} T^{15} + 10851998 p^{4} T^{16} - 11032 p^{6} T^{17} + 42001 p^{6} T^{18} + 76720 p^{7} T^{19} + 2268 p^{8} T^{20} - 1008 p^{9} T^{21} - 81 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 3 T - 221 T^{2} + 741 T^{3} + 15302 T^{4} - 42420 T^{5} + 32746 T^{6} + 351537 T^{7} - 61049567 T^{8} - 199663935 T^{9} + 5612698675 T^{10} + 14511430920 T^{11} - 435605659216 T^{12} + 14511430920 p T^{13} + 5612698675 p^{2} T^{14} - 199663935 p^{3} T^{15} - 61049567 p^{4} T^{16} + 351537 p^{5} T^{17} + 32746 p^{6} T^{18} - 42420 p^{7} T^{19} + 15302 p^{8} T^{20} + 741 p^{9} T^{21} - 221 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 22 T + 315 T^{2} + 2340 T^{3} + 9845 T^{4} - 21155 T^{5} - 206673 T^{6} - 2739205 T^{7} - 38870619 T^{8} - 680686702 T^{9} - 2702419548 T^{10} + 24999159012 T^{11} + 527202315406 T^{12} + 24999159012 p T^{13} - 2702419548 p^{2} T^{14} - 680686702 p^{3} T^{15} - 38870619 p^{4} T^{16} - 2739205 p^{5} T^{17} - 206673 p^{6} T^{18} - 21155 p^{7} T^{19} + 9845 p^{8} T^{20} + 2340 p^{9} T^{21} + 315 p^{10} T^{22} + 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 60 T + 1850 T^{2} + 39188 T^{3} + 638583 T^{4} + 8355834 T^{5} + 87684348 T^{6} + 695495844 T^{7} + 3085917097 T^{8} - 17972154932 T^{9} - 617443507302 T^{10} - 8708064636960 T^{11} - 88267185380581 T^{12} - 8708064636960 p T^{13} - 617443507302 p^{2} T^{14} - 17972154932 p^{3} T^{15} + 3085917097 p^{4} T^{16} + 695495844 p^{5} T^{17} + 87684348 p^{6} T^{18} + 8355834 p^{7} T^{19} + 638583 p^{8} T^{20} + 39188 p^{9} T^{21} + 1850 p^{10} T^{22} + 60 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 39 T + 592 T^{2} + 2848 T^{3} - 37890 T^{4} - 685444 T^{5} - 2606723 T^{6} + 38911760 T^{7} + 536755419 T^{8} + 1660870215 T^{9} - 18387522258 T^{10} - 244448028524 T^{11} - 2020666744585 T^{12} - 244448028524 p T^{13} - 18387522258 p^{2} T^{14} + 1660870215 p^{3} T^{15} + 536755419 p^{4} T^{16} + 38911760 p^{5} T^{17} - 2606723 p^{6} T^{18} - 685444 p^{7} T^{19} - 37890 p^{8} T^{20} + 2848 p^{9} T^{21} + 592 p^{10} T^{22} + 39 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 39 T + 465 T^{2} + 1322 T^{3} - 78156 T^{4} + 442604 T^{5} + 6281677 T^{6} - 104397206 T^{7} + 139406354 T^{8} + 10269859891 T^{9} - 91182359374 T^{10} - 382531726410 T^{11} + 11016680533567 T^{12} - 382531726410 p T^{13} - 91182359374 p^{2} T^{14} + 10269859891 p^{3} T^{15} + 139406354 p^{4} T^{16} - 104397206 p^{5} T^{17} + 6281677 p^{6} T^{18} + 442604 p^{7} T^{19} - 78156 p^{8} T^{20} + 1322 p^{9} T^{21} + 465 p^{10} T^{22} - 39 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 19 T + 321 T^{2} + 4305 T^{3} + 60718 T^{4} + 565740 T^{5} + 5020268 T^{6} + 25232445 T^{7} + 11919959 T^{8} - 4488131307 T^{9} - 71197240023 T^{10} - 1000551796594 T^{11} - 9916513237784 T^{12} - 1000551796594 p T^{13} - 71197240023 p^{2} T^{14} - 4488131307 p^{3} T^{15} + 11919959 p^{4} T^{16} + 25232445 p^{5} T^{17} + 5020268 p^{6} T^{18} + 565740 p^{7} T^{19} + 60718 p^{8} T^{20} + 4305 p^{9} T^{21} + 321 p^{10} T^{22} + 19 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−5.64285772551687805729814546390, −5.62337168461694740137725333271, −5.57893596623414583010741519341, −5.54087034181358025019017811898, −5.34179659423460159677096180889, −5.26531205183255138421576235460, −5.20735773581590030516994544317, −4.81328645179728076489521002190, −4.79962161119483935944837319547, −4.74589366603281866618749223747, −4.39231051012775475224377202971, −4.36684241440737050875581389347, −4.15671795440660988422127086633, −4.14835157466600069837477845827, −3.96787236268700373413171601004, −3.59636949026535389544487755160, −3.17937074556836729539283677817, −3.11927520482895634263698173579, −2.97289032003055873712764771288, −2.88304899228522888301033541176, −2.59676443865007709016019987668, −2.42782369973587164779305623392, −2.04754554209974826976360601438, −1.46253132642499295750824365909, −1.29498353440283692095818641178, 1.29498353440283692095818641178, 1.46253132642499295750824365909, 2.04754554209974826976360601438, 2.42782369973587164779305623392, 2.59676443865007709016019987668, 2.88304899228522888301033541176, 2.97289032003055873712764771288, 3.11927520482895634263698173579, 3.17937074556836729539283677817, 3.59636949026535389544487755160, 3.96787236268700373413171601004, 4.14835157466600069837477845827, 4.15671795440660988422127086633, 4.36684241440737050875581389347, 4.39231051012775475224377202971, 4.74589366603281866618749223747, 4.79962161119483935944837319547, 4.81328645179728076489521002190, 5.20735773581590030516994544317, 5.26531205183255138421576235460, 5.34179659423460159677096180889, 5.54087034181358025019017811898, 5.57893596623414583010741519341, 5.62337168461694740137725333271, 5.64285772551687805729814546390
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.