Dirichlet series
| L(s) = 1 | + 3·2-s + 3-s + 3·4-s − 5·5-s + 3·6-s − 12·7-s + 8-s + 8·9-s − 15·10-s + 7·11-s + 3·12-s + 3·13-s − 36·14-s − 5·15-s + 4·17-s + 24·18-s + 4·19-s − 15·20-s − 12·21-s + 21·22-s − 23-s + 24-s + 10·25-s + 9·26-s + 15·27-s − 36·28-s + 22·29-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 0.577·3-s + 3/2·4-s − 2.23·5-s + 1.22·6-s − 4.53·7-s + 0.353·8-s + 8/3·9-s − 4.74·10-s + 2.11·11-s + 0.866·12-s + 0.832·13-s − 9.62·14-s − 1.29·15-s + 0.970·17-s + 5.65·18-s + 0.917·19-s − 3.35·20-s − 2.61·21-s + 4.47·22-s − 0.208·23-s + 0.204·24-s + 2·25-s + 1.76·26-s + 2.88·27-s − 6.80·28-s + 4.08·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 5^{24} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(227060.\) |
| Root analytic conductor: | \(1.67175\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 5^{24} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(20.57550539\) |
| \(L(\frac12)\) | \(\approx\) | \(20.57550539\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{3} \) |
| 5 | \( 1 + p T + 3 p T^{2} + 9 p T^{3} + 16 p T^{4} + p^{3} T^{5} + 13 p^{2} T^{6} + p^{4} T^{7} + 16 p^{3} T^{8} + 9 p^{4} T^{9} + 3 p^{5} T^{10} + p^{6} T^{11} + p^{6} T^{12} \) | |
| 7 | \( ( 1 + T )^{12} \) | |
| good | 3 | \( 1 - T - 7 T^{2} + p^{3} T^{4} + 2 p^{3} T^{5} - 85 T^{6} - 94 p T^{7} + 38 T^{8} + 248 p T^{9} + 115 p^{2} T^{10} - 917 T^{11} - 4667 T^{12} - 917 p T^{13} + 115 p^{4} T^{14} + 248 p^{4} T^{15} + 38 p^{4} T^{16} - 94 p^{6} T^{17} - 85 p^{6} T^{18} + 2 p^{10} T^{19} + p^{11} T^{20} - 7 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) |
| 11 | \( 1 - 7 T + 15 T^{2} - 48 T^{3} + 369 T^{4} - 1218 T^{5} + 3391 T^{6} - 1640 p T^{7} + 68194 T^{8} - 176652 T^{9} + 566355 T^{10} - 2220705 T^{11} + 7824965 T^{12} - 2220705 p T^{13} + 566355 p^{2} T^{14} - 176652 p^{3} T^{15} + 68194 p^{4} T^{16} - 1640 p^{6} T^{17} + 3391 p^{6} T^{18} - 1218 p^{7} T^{19} + 369 p^{8} T^{20} - 48 p^{9} T^{21} + 15 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 3 T - 20 T^{2} + 118 T^{3} + 161 T^{4} - 2778 T^{5} + 3 p^{2} T^{6} + 49992 T^{7} - 117950 T^{8} - 558088 T^{9} + 2845345 T^{10} + 2678925 T^{11} - 39869651 T^{12} + 2678925 p T^{13} + 2845345 p^{2} T^{14} - 558088 p^{3} T^{15} - 117950 p^{4} T^{16} + 49992 p^{5} T^{17} + 3 p^{8} T^{18} - 2778 p^{7} T^{19} + 161 p^{8} T^{20} + 118 p^{9} T^{21} - 20 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 4 T - 52 T^{2} + 212 T^{3} + 1204 T^{4} - 8304 T^{5} - 1499 T^{6} + 234288 T^{7} - 622968 T^{8} - 4319236 T^{9} + 20524788 T^{10} + 33120788 T^{11} - 413458971 T^{12} + 33120788 p T^{13} + 20524788 p^{2} T^{14} - 4319236 p^{3} T^{15} - 622968 p^{4} T^{16} + 234288 p^{5} T^{17} - 1499 p^{6} T^{18} - 8304 p^{7} T^{19} + 1204 p^{8} T^{20} + 212 p^{9} T^{21} - 52 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 4 T - 55 T^{2} + 322 T^{3} + 786 T^{4} - 10430 T^{5} + 19782 T^{6} + 121858 T^{7} - 830761 T^{8} + 935330 T^{9} + 9745914 T^{10} - 23118154 T^{11} - 60429309 T^{12} - 23118154 p T^{13} + 9745914 p^{2} T^{14} + 935330 p^{3} T^{15} - 830761 p^{4} T^{16} + 121858 p^{5} T^{17} + 19782 p^{6} T^{18} - 10430 p^{7} T^{19} + 786 p^{8} T^{20} + 322 p^{9} T^{21} - 55 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + T - 53 T^{2} - 306 T^{3} + 975 T^{4} + 11712 T^{5} + 26649 T^{6} - 153426 T^{7} - 961598 T^{8} - 2533552 T^{9} - 218007 T^{10} + 52959247 T^{11} + 441957817 T^{12} + 52959247 p T^{13} - 218007 p^{2} T^{14} - 2533552 p^{3} T^{15} - 961598 p^{4} T^{16} - 153426 p^{5} T^{17} + 26649 p^{6} T^{18} + 11712 p^{7} T^{19} + 975 p^{8} T^{20} - 306 p^{9} T^{21} - 53 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 22 T + 156 T^{2} + 68 T^{3} - 6774 T^{4} + 30632 T^{5} + 46573 T^{6} - 1017678 T^{7} + 5929708 T^{8} - 26575028 T^{9} + 50923194 T^{10} + 552546070 T^{11} - 5342883971 T^{12} + 552546070 p T^{13} + 50923194 p^{2} T^{14} - 26575028 p^{3} T^{15} + 5929708 p^{4} T^{16} - 1017678 p^{5} T^{17} + 46573 p^{6} T^{18} + 30632 p^{7} T^{19} - 6774 p^{8} T^{20} + 68 p^{9} T^{21} + 156 p^{10} T^{22} - 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - p T + 453 T^{2} - 4225 T^{3} + 29693 T^{4} - 192850 T^{5} + 1377427 T^{6} - 10076394 T^{7} + 65664126 T^{8} - 379719432 T^{9} + 2153666263 T^{10} - 12728315052 T^{11} + 73593545547 T^{12} - 12728315052 p T^{13} + 2153666263 p^{2} T^{14} - 379719432 p^{3} T^{15} + 65664126 p^{4} T^{16} - 10076394 p^{5} T^{17} + 1377427 p^{6} T^{18} - 192850 p^{7} T^{19} + 29693 p^{8} T^{20} - 4225 p^{9} T^{21} + 453 p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 9 T - 17 T^{2} + 47 T^{3} + 2009 T^{4} + 17896 T^{5} - 183604 T^{6} - 200352 T^{7} - 35484 p T^{8} + 29143544 T^{9} + 202793238 T^{10} - 1300954802 T^{11} - 1160573366 T^{12} - 1300954802 p T^{13} + 202793238 p^{2} T^{14} + 29143544 p^{3} T^{15} - 35484 p^{5} T^{16} - 200352 p^{5} T^{17} - 183604 p^{6} T^{18} + 17896 p^{7} T^{19} + 2009 p^{8} T^{20} + 47 p^{9} T^{21} - 17 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 19 T + 68 T^{2} - 655 T^{3} - 1387 T^{4} + 63285 T^{5} + 319397 T^{6} - 2378554 T^{7} - 20541144 T^{8} + 66494623 T^{9} + 976204363 T^{10} - 1644832272 T^{11} - 49442383823 T^{12} - 1644832272 p T^{13} + 976204363 p^{2} T^{14} + 66494623 p^{3} T^{15} - 20541144 p^{4} T^{16} - 2378554 p^{5} T^{17} + 319397 p^{6} T^{18} + 63285 p^{7} T^{19} - 1387 p^{8} T^{20} - 655 p^{9} T^{21} + 68 p^{10} T^{22} + 19 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( ( 1 - 25 T + 439 T^{2} - 5258 T^{3} + 52746 T^{4} - 426931 T^{5} + 3057464 T^{6} - 426931 p T^{7} + 52746 p^{2} T^{8} - 5258 p^{3} T^{9} + 439 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 + 24 T + 115 T^{2} - 2376 T^{3} - 35394 T^{4} - 95696 T^{5} + 1682298 T^{6} + 17182896 T^{7} + 34195735 T^{8} - 445798096 T^{9} - 3285259630 T^{10} - 981842400 T^{11} + 79567059979 T^{12} - 981842400 p T^{13} - 3285259630 p^{2} T^{14} - 445798096 p^{3} T^{15} + 34195735 p^{4} T^{16} + 17182896 p^{5} T^{17} + 1682298 p^{6} T^{18} - 95696 p^{7} T^{19} - 35394 p^{8} T^{20} - 2376 p^{9} T^{21} + 115 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 35 T + 588 T^{2} - 6446 T^{3} + 991 p T^{4} - 302216 T^{5} + 600290 T^{6} + 9127215 T^{7} - 111906848 T^{8} + 601751953 T^{9} - 6692433 T^{10} - 37037958071 T^{11} + 396587073686 T^{12} - 37037958071 p T^{13} - 6692433 p^{2} T^{14} + 601751953 p^{3} T^{15} - 111906848 p^{4} T^{16} + 9127215 p^{5} T^{17} + 600290 p^{6} T^{18} - 302216 p^{7} T^{19} + 991 p^{9} T^{20} - 6446 p^{9} T^{21} + 588 p^{10} T^{22} - 35 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - T - 45 T^{2} - 377 T^{3} + 5021 T^{4} - 8190 T^{5} - 34303 T^{6} - 1922028 T^{7} + 23142924 T^{8} - 85445370 T^{9} + 25142239 T^{10} - 5172383726 T^{11} + 106812734191 T^{12} - 5172383726 p T^{13} + 25142239 p^{2} T^{14} - 85445370 p^{3} T^{15} + 23142924 p^{4} T^{16} - 1922028 p^{5} T^{17} - 34303 p^{6} T^{18} - 8190 p^{7} T^{19} + 5021 p^{8} T^{20} - 377 p^{9} T^{21} - 45 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 8 T - 144 T^{2} + 1210 T^{3} + 15692 T^{4} - 91008 T^{5} - 1749157 T^{6} + 5810908 T^{7} + 166130048 T^{8} - 350338768 T^{9} - 12151497572 T^{10} + 9558414952 T^{11} + 772713424737 T^{12} + 9558414952 p T^{13} - 12151497572 p^{2} T^{14} - 350338768 p^{3} T^{15} + 166130048 p^{4} T^{16} + 5810908 p^{5} T^{17} - 1749157 p^{6} T^{18} - 91008 p^{7} T^{19} + 15692 p^{8} T^{20} + 1210 p^{9} T^{21} - 144 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 36 T + 448 T^{2} + 882 T^{3} - 30116 T^{4} - 288924 T^{5} - 1370359 T^{6} - 12802922 T^{7} - 99303768 T^{8} + 312986874 T^{9} + 9836955068 T^{10} + 57716972648 T^{11} + 232196539349 T^{12} + 57716972648 p T^{13} + 9836955068 p^{2} T^{14} + 312986874 p^{3} T^{15} - 99303768 p^{4} T^{16} - 12802922 p^{5} T^{17} - 1370359 p^{6} T^{18} - 288924 p^{7} T^{19} - 30116 p^{8} T^{20} + 882 p^{9} T^{21} + 448 p^{10} T^{22} + 36 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - T - 272 T^{2} + 1515 T^{3} + 31793 T^{4} - 390805 T^{5} - 1127933 T^{6} + 51435256 T^{7} - 189880844 T^{8} - 3895432897 T^{9} + 35948452553 T^{10} + 120778460288 T^{11} - 3267845694113 T^{12} + 120778460288 p T^{13} + 35948452553 p^{2} T^{14} - 3895432897 p^{3} T^{15} - 189880844 p^{4} T^{16} + 51435256 p^{5} T^{17} - 1127933 p^{6} T^{18} - 390805 p^{7} T^{19} + 31793 p^{8} T^{20} + 1515 p^{9} T^{21} - 272 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 31 T + 157 T^{2} - 4351 T^{3} - 56165 T^{4} + 21012 T^{5} + 4960409 T^{6} + 40392564 T^{7} - 66672508 T^{8} - 3587879652 T^{9} - 16845025467 T^{10} + 109824364092 T^{11} + 1752697474667 T^{12} + 109824364092 p T^{13} - 16845025467 p^{2} T^{14} - 3587879652 p^{3} T^{15} - 66672508 p^{4} T^{16} + 40392564 p^{5} T^{17} + 4960409 p^{6} T^{18} + 21012 p^{7} T^{19} - 56165 p^{8} T^{20} - 4351 p^{9} T^{21} + 157 p^{10} T^{22} + 31 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 2 T + 286 T^{2} - 1942 T^{3} + 45506 T^{4} - 458068 T^{5} + 6694253 T^{6} - 59915278 T^{7} + 860339148 T^{8} - 83098502 p T^{9} + 84094060634 T^{10} - 659242015430 T^{11} + 6879140123889 T^{12} - 659242015430 p T^{13} + 84094060634 p^{2} T^{14} - 83098502 p^{4} T^{15} + 860339148 p^{4} T^{16} - 59915278 p^{5} T^{17} + 6694253 p^{6} T^{18} - 458068 p^{7} T^{19} + 45506 p^{8} T^{20} - 1942 p^{9} T^{21} + 286 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 19 T + 3 T^{2} - 2230 T^{3} - 9853 T^{4} + 260324 T^{5} + 3150455 T^{6} - 5556832 T^{7} - 239317072 T^{8} + 68226904 T^{9} + 18813056155 T^{10} + 47128776483 T^{11} - 758831244287 T^{12} + 47128776483 p T^{13} + 18813056155 p^{2} T^{14} + 68226904 p^{3} T^{15} - 239317072 p^{4} T^{16} - 5556832 p^{5} T^{17} + 3150455 p^{6} T^{18} + 260324 p^{7} T^{19} - 9853 p^{8} T^{20} - 2230 p^{9} T^{21} + 3 p^{10} T^{22} + 19 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 40 T + 783 T^{2} - 11380 T^{3} + 136646 T^{4} - 1233320 T^{5} + 8628700 T^{6} - 61036520 T^{7} + 518583425 T^{8} - 6988908095 T^{9} + 118589802798 T^{10} - 1538822973165 T^{11} + 15615590135134 T^{12} - 1538822973165 p T^{13} + 118589802798 p^{2} T^{14} - 6988908095 p^{3} T^{15} + 518583425 p^{4} T^{16} - 61036520 p^{5} T^{17} + 8628700 p^{6} T^{18} - 1233320 p^{7} T^{19} + 136646 p^{8} T^{20} - 11380 p^{9} T^{21} + 783 p^{10} T^{22} - 40 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 28 T + 403 T^{2} - 4628 T^{3} + 55650 T^{4} - 570136 T^{5} + 5049586 T^{6} - 55109928 T^{7} + 751828627 T^{8} - 8579658424 T^{9} + 887690146 p T^{10} - 959504128864 T^{11} + 10485381483807 T^{12} - 959504128864 p T^{13} + 887690146 p^{3} T^{14} - 8579658424 p^{3} T^{15} + 751828627 p^{4} T^{16} - 55109928 p^{5} T^{17} + 5049586 p^{6} T^{18} - 570136 p^{7} T^{19} + 55650 p^{8} T^{20} - 4628 p^{9} T^{21} + 403 p^{10} T^{22} - 28 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−4.00932608176713005258173849792, −3.67671339112078566598186620860, −3.63622073433343928747130297556, −3.61312398084146614452856130743, −3.59519968222820123704331681822, −3.41666646070924694273931121466, −3.35446769426285471129075146413, −3.16511770183992870226450089474, −3.05966497519910207359228993507, −2.96772729830404625783721518342, −2.83635416049766491213294006262, −2.73440251880966496197598693014, −2.67954215323168162518530684817, −2.65223116769960031825119717443, −2.54926066185725510067979645716, −2.26084145896832469221021942119, −2.02904665657229155899156515999, −1.90962453624360825875513681438, −1.48115898942119609256960287851, −1.26322995903064245395740198975, −1.21876531884075683793706926074, −0.986537433317681608510853430986, −0.75364913466258876825673635147, −0.73172569485840210894964311647, −0.72196000015573435812770309429, 0.72196000015573435812770309429, 0.73172569485840210894964311647, 0.75364913466258876825673635147, 0.986537433317681608510853430986, 1.21876531884075683793706926074, 1.26322995903064245395740198975, 1.48115898942119609256960287851, 1.90962453624360825875513681438, 2.02904665657229155899156515999, 2.26084145896832469221021942119, 2.54926066185725510067979645716, 2.65223116769960031825119717443, 2.67954215323168162518530684817, 2.73440251880966496197598693014, 2.83635416049766491213294006262, 2.96772729830404625783721518342, 3.05966497519910207359228993507, 3.16511770183992870226450089474, 3.35446769426285471129075146413, 3.41666646070924694273931121466, 3.59519968222820123704331681822, 3.61312398084146614452856130743, 3.63622073433343928747130297556, 3.67671339112078566598186620860, 4.00932608176713005258173849792
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.