Dirichlet series
| L(s) = 1 | + 4·4-s − 12·11-s + 4·16-s + 20·19-s + 4·25-s − 12·29-s − 16·31-s − 32·41-s − 48·44-s + 44·49-s + 44·59-s − 52·61-s − 6·64-s − 20·71-s + 80·76-s + 4·79-s + 68·89-s + 16·100-s − 92·101-s − 8·109-s − 48·116-s + 8·121-s − 64·124-s − 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 2·4-s − 3.61·11-s + 16-s + 4.58·19-s + 4/5·25-s − 2.22·29-s − 2.87·31-s − 4.99·41-s − 7.23·44-s + 44/7·49-s + 5.72·59-s − 6.65·61-s − 3/4·64-s − 2.37·71-s + 9.17·76-s + 0.450·79-s + 7.20·89-s + 8/5·100-s − 9.15·101-s − 0.766·109-s − 4.45·116-s + 8/11·121-s − 5.74·124-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{24} \cdot 5^{12} \cdot 29^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.63927\times 10^{12}\) |
| Root analytic conductor: | \(3.22807\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{24} \cdot 5^{12} \cdot 29^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.341305395\) |
| \(L(\frac12)\) | \(\approx\) | \(1.341305395\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 4 T^{2} + 4 T^{3} + 19 T^{4} - 36 T^{5} - 16 p T^{6} - 36 p T^{7} + 19 p^{2} T^{8} + 4 p^{3} T^{9} - 4 p^{4} T^{10} + p^{6} T^{12} \) | |
| 29 | \( ( 1 + T )^{12} \) | |
| good | 2 | \( 1 - p^{2} T^{2} + 3 p^{2} T^{4} - 13 p T^{6} + p^{6} T^{8} - 21 p^{3} T^{10} + 365 T^{12} - 21 p^{5} T^{14} + p^{10} T^{16} - 13 p^{7} T^{18} + 3 p^{10} T^{20} - p^{12} T^{22} + p^{12} T^{24} \) |
| 7 | \( 1 - 44 T^{2} + 136 p T^{4} - 13856 T^{6} + 156004 T^{8} - 1443108 T^{10} + 11092730 T^{12} - 1443108 p^{2} T^{14} + 156004 p^{4} T^{16} - 13856 p^{6} T^{18} + 136 p^{9} T^{20} - 44 p^{10} T^{22} + p^{12} T^{24} \) | |
| 11 | \( ( 1 + 6 T + 50 T^{2} + 200 T^{3} + 992 T^{4} + 2994 T^{5} + 12416 T^{6} + 2994 p T^{7} + 992 p^{2} T^{8} + 200 p^{3} T^{9} + 50 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 13 | \( 1 - 56 T^{2} + 1396 T^{4} - 22796 T^{6} + 289444 T^{8} - 2802576 T^{10} + 27392294 T^{12} - 2802576 p^{2} T^{14} + 289444 p^{4} T^{16} - 22796 p^{6} T^{18} + 1396 p^{8} T^{20} - 56 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( 1 - 60 T^{2} + 1844 T^{4} - 50404 T^{6} + 1267244 T^{8} - 25697188 T^{10} + 450597318 T^{12} - 25697188 p^{2} T^{14} + 1267244 p^{4} T^{16} - 50404 p^{6} T^{18} + 1844 p^{8} T^{20} - 60 p^{10} T^{22} + p^{12} T^{24} \) | |
| 19 | \( ( 1 - 10 T + 84 T^{2} - 462 T^{3} + 2723 T^{4} - 13348 T^{5} + 66376 T^{6} - 13348 p T^{7} + 2723 p^{2} T^{8} - 462 p^{3} T^{9} + 84 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 23 | \( 1 - 180 T^{2} + 14562 T^{4} - 697796 T^{6} + 22380559 T^{8} - 544017928 T^{10} + 12197286428 T^{12} - 544017928 p^{2} T^{14} + 22380559 p^{4} T^{16} - 697796 p^{6} T^{18} + 14562 p^{8} T^{20} - 180 p^{10} T^{22} + p^{12} T^{24} \) | |
| 31 | \( ( 1 + 8 T + 98 T^{2} + 356 T^{3} + 3639 T^{4} + 11108 T^{5} + 130068 T^{6} + 11108 p T^{7} + 3639 p^{2} T^{8} + 356 p^{3} T^{9} + 98 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( 1 - 80 T^{2} + 7606 T^{4} - 392720 T^{6} + 21943231 T^{8} - 855501024 T^{10} + 36696567668 T^{12} - 855501024 p^{2} T^{14} + 21943231 p^{4} T^{16} - 392720 p^{6} T^{18} + 7606 p^{8} T^{20} - 80 p^{10} T^{22} + p^{12} T^{24} \) | |
| 41 | \( ( 1 + 16 T + 222 T^{2} + 2320 T^{3} + 21567 T^{4} + 166432 T^{5} + 1134052 T^{6} + 166432 p T^{7} + 21567 p^{2} T^{8} + 2320 p^{3} T^{9} + 222 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 43 | \( 1 - 156 T^{2} + 13582 T^{4} - 861916 T^{6} + 47246079 T^{8} - 2406241768 T^{10} + 110221496004 T^{12} - 2406241768 p^{2} T^{14} + 47246079 p^{4} T^{16} - 861916 p^{6} T^{18} + 13582 p^{8} T^{20} - 156 p^{10} T^{22} + p^{12} T^{24} \) | |
| 47 | \( 1 - 420 T^{2} + 85364 T^{4} - 11109004 T^{6} + 1030089164 T^{8} - 71622208108 T^{10} + 3826287085638 T^{12} - 71622208108 p^{2} T^{14} + 1030089164 p^{4} T^{16} - 11109004 p^{6} T^{18} + 85364 p^{8} T^{20} - 420 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( 1 - 320 T^{2} + 53538 T^{4} - 6084688 T^{6} + 523808815 T^{8} - 36292173872 T^{10} + 2094938778300 T^{12} - 36292173872 p^{2} T^{14} + 523808815 p^{4} T^{16} - 6084688 p^{6} T^{18} + 53538 p^{8} T^{20} - 320 p^{10} T^{22} + p^{12} T^{24} \) | |
| 59 | \( ( 1 - 22 T + 372 T^{2} - 4298 T^{3} + 45907 T^{4} - 410572 T^{5} + 3442400 T^{6} - 410572 p T^{7} + 45907 p^{2} T^{8} - 4298 p^{3} T^{9} + 372 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 61 | \( ( 1 + 26 T + 570 T^{2} + 8494 T^{3} + 106251 T^{4} + 1070992 T^{5} + 9174692 T^{6} + 1070992 p T^{7} + 106251 p^{2} T^{8} + 8494 p^{3} T^{9} + 570 p^{4} T^{10} + 26 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( 1 - 236 T^{2} + 29224 T^{4} - 2399408 T^{6} + 140031028 T^{8} - 5673900036 T^{10} + 249184874186 T^{12} - 5673900036 p^{2} T^{14} + 140031028 p^{4} T^{16} - 2399408 p^{6} T^{18} + 29224 p^{8} T^{20} - 236 p^{10} T^{22} + p^{12} T^{24} \) | |
| 71 | \( ( 1 + 10 T + 174 T^{2} + 1530 T^{3} + 22907 T^{4} + 157072 T^{5} + 1675308 T^{6} + 157072 p T^{7} + 22907 p^{2} T^{8} + 1530 p^{3} T^{9} + 174 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 73 | \( 1 - 616 T^{2} + 184870 T^{4} - 35787240 T^{6} + 4984957839 T^{8} - 527115511152 T^{10} + 43424297181268 T^{12} - 527115511152 p^{2} T^{14} + 4984957839 p^{4} T^{16} - 35787240 p^{6} T^{18} + 184870 p^{8} T^{20} - 616 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( ( 1 - 2 T + 240 T^{2} - 886 T^{3} + 28123 T^{4} - 144908 T^{5} + 2416704 T^{6} - 144908 p T^{7} + 28123 p^{2} T^{8} - 886 p^{3} T^{9} + 240 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 - 552 T^{2} + 149066 T^{4} - 26151976 T^{6} + 3380877503 T^{8} - 350762692336 T^{10} + 31097886726348 T^{12} - 350762692336 p^{2} T^{14} + 3380877503 p^{4} T^{16} - 26151976 p^{6} T^{18} + 149066 p^{8} T^{20} - 552 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( ( 1 - 34 T + 884 T^{2} - 15808 T^{3} + 239316 T^{4} - 2875498 T^{5} + 30004398 T^{6} - 2875498 p T^{7} + 239316 p^{2} T^{8} - 15808 p^{3} T^{9} + 884 p^{4} T^{10} - 34 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 - 592 T^{2} + 150674 T^{4} - 20358752 T^{6} + 1263823711 T^{8} + 34692758896 T^{10} - 11527615876708 T^{12} + 34692758896 p^{2} T^{14} + 1263823711 p^{4} T^{16} - 20358752 p^{6} T^{18} + 150674 p^{8} T^{20} - 592 p^{10} T^{22} + p^{12} T^{24} \) | |
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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
−2.99845054150616283521683998093, −2.94554163812445357247440289525, −2.82944837680644355067929172673, −2.81346818761928824551722964537, −2.62504634502053370994945099204, −2.46098236379939347301758520425, −2.42216914732772039065148939808, −2.39179740552220864057214526050, −2.25798200554752673514102601346, −2.23199338019455519856135134316, −2.17120930831220949092096380969, −2.16636464431100601227427607249, −1.88452240575834175204841029215, −1.77611952244692809886065153407, −1.55004825545321641840850171508, −1.42584367192593605363377023856, −1.35850127342481252984351475844, −1.31302907255214297481008184144, −1.25547384090140798996355541467, −1.16773014761747608494162207437, −1.00137814177975605781126531849, −0.50508313319113510134729190445, −0.47359066027052097360828361037, −0.30119890612260922712815559865, −0.099821713690000123534637619815, 0.099821713690000123534637619815, 0.30119890612260922712815559865, 0.47359066027052097360828361037, 0.50508313319113510134729190445, 1.00137814177975605781126531849, 1.16773014761747608494162207437, 1.25547384090140798996355541467, 1.31302907255214297481008184144, 1.35850127342481252984351475844, 1.42584367192593605363377023856, 1.55004825545321641840850171508, 1.77611952244692809886065153407, 1.88452240575834175204841029215, 2.16636464431100601227427607249, 2.17120930831220949092096380969, 2.23199338019455519856135134316, 2.25798200554752673514102601346, 2.39179740552220864057214526050, 2.42216914732772039065148939808, 2.46098236379939347301758520425, 2.62504634502053370994945099204, 2.81346818761928824551722964537, 2.82944837680644355067929172673, 2.94554163812445357247440289525, 2.99845054150616283521683998093
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.