Dirichlet series
| L(s) = 1 | + 3·4-s − 2·5-s − 6·9-s + 16-s − 4·19-s − 6·20-s + 3·25-s + 24·29-s + 36·31-s − 18·36-s + 12·41-s + 12·45-s + 30·49-s + 36·59-s − 8·61-s − 6·64-s − 12·76-s + 4·79-s − 2·80-s + 21·81-s + 32·89-s + 8·95-s + 9·100-s + 64·101-s + 72·116-s + 108·124-s − 10·125-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 0.894·5-s − 2·9-s + 1/4·16-s − 0.917·19-s − 1.34·20-s + 3/5·25-s + 4.45·29-s + 6.46·31-s − 3·36-s + 1.87·41-s + 1.78·45-s + 30/7·49-s + 4.68·59-s − 1.02·61-s − 3/4·64-s − 1.37·76-s + 0.450·79-s − 0.223·80-s + 7/3·81-s + 3.39·89-s + 0.820·95-s + 9/10·100-s + 6.36·101-s + 6.68·116-s + 9.69·124-s − 0.894·125-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{12} \cdot 11^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{12} \cdot 11^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{12} \cdot 5^{12} \cdot 11^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.58706\times 10^{13}\) |
| Root analytic conductor: | \(3.80694\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{12} \cdot 5^{12} \cdot 11^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(29.03967229\) |
| \(L(\frac12)\) | \(\approx\) | \(29.03967229\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T^{2} )^{6} \) |
| 5 | \( 1 + 2 T + T^{2} + 6 T^{3} + 33 T^{4} + 32 T^{5} - 78 T^{6} + 32 p T^{7} + 33 p^{2} T^{8} + 6 p^{3} T^{9} + p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) | |
| 11 | \( 1 \) | |
| good | 2 | \( 1 - 3 T^{2} + p^{3} T^{4} - 15 T^{6} + 25 T^{8} - 15 p^{2} T^{10} + 23 p^{2} T^{12} - 15 p^{4} T^{14} + 25 p^{4} T^{16} - 15 p^{6} T^{18} + p^{11} T^{20} - 3 p^{10} T^{22} + p^{12} T^{24} \) |
| 7 | \( 1 - 30 T^{2} + 583 T^{4} - 8010 T^{6} + 88723 T^{8} - 801720 T^{10} + 6133066 T^{12} - 801720 p^{2} T^{14} + 88723 p^{4} T^{16} - 8010 p^{6} T^{18} + 583 p^{8} T^{20} - 30 p^{10} T^{22} + p^{12} T^{24} \) | |
| 13 | \( 1 - 58 T^{2} + 1663 T^{4} - 2614 p T^{6} + 584083 T^{8} - 8929144 T^{10} + 122478106 T^{12} - 8929144 p^{2} T^{14} + 584083 p^{4} T^{16} - 2614 p^{7} T^{18} + 1663 p^{8} T^{20} - 58 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( 1 - 64 T^{2} + 2372 T^{4} - 66800 T^{6} + 1531336 T^{8} - 30272480 T^{10} + 537773962 T^{12} - 30272480 p^{2} T^{14} + 1531336 p^{4} T^{16} - 66800 p^{6} T^{18} + 2372 p^{8} T^{20} - 64 p^{10} T^{22} + p^{12} T^{24} \) | |
| 19 | \( ( 1 + 2 T + 67 T^{2} + 126 T^{3} + 2211 T^{4} + 3700 T^{5} + 48850 T^{6} + 3700 p T^{7} + 2211 p^{2} T^{8} + 126 p^{3} T^{9} + 67 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 23 | \( 1 - 90 T^{2} + 5055 T^{4} - 216666 T^{6} + 7361979 T^{8} - 211341996 T^{10} + 5239535114 T^{12} - 211341996 p^{2} T^{14} + 7361979 p^{4} T^{16} - 216666 p^{6} T^{18} + 5055 p^{8} T^{20} - 90 p^{10} T^{22} + p^{12} T^{24} \) | |
| 29 | \( ( 1 - 12 T + 133 T^{2} - 18 p T^{3} + 1337 T^{4} + 18242 T^{5} - 105070 T^{6} + 18242 p T^{7} + 1337 p^{2} T^{8} - 18 p^{4} T^{9} + 133 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 31 | \( ( 1 - 18 T + 234 T^{2} - 2340 T^{3} + 19526 T^{4} - 135426 T^{5} + 817790 T^{6} - 135426 p T^{7} + 19526 p^{2} T^{8} - 2340 p^{3} T^{9} + 234 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( 1 - 120 T^{2} + 12160 T^{4} - 820408 T^{6} + 48325280 T^{8} - 2235703544 T^{10} + 91881331406 T^{12} - 2235703544 p^{2} T^{14} + 48325280 p^{4} T^{16} - 820408 p^{6} T^{18} + 12160 p^{8} T^{20} - 120 p^{10} T^{22} + p^{12} T^{24} \) | |
| 41 | \( ( 1 - 6 T + 127 T^{2} - 630 T^{3} + 9743 T^{4} - 38660 T^{5} + 457250 T^{6} - 38660 p T^{7} + 9743 p^{2} T^{8} - 630 p^{3} T^{9} + 127 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 43 | \( 1 - 232 T^{2} + 718 p T^{4} - 2883272 T^{6} + 4814045 p T^{8} - 11943432464 T^{10} + 564867962988 T^{12} - 11943432464 p^{2} T^{14} + 4814045 p^{5} T^{16} - 2883272 p^{6} T^{18} + 718 p^{9} T^{20} - 232 p^{10} T^{22} + p^{12} T^{24} \) | |
| 47 | \( 1 - 10 p T^{2} + 104507 T^{4} - 14552402 T^{6} + 1414245543 T^{8} - 101085594760 T^{10} + 5449761421146 T^{12} - 101085594760 p^{2} T^{14} + 1414245543 p^{4} T^{16} - 14552402 p^{6} T^{18} + 104507 p^{8} T^{20} - 10 p^{11} T^{22} + p^{12} T^{24} \) | |
| 53 | \( 1 - 436 T^{2} + 92464 T^{4} - 12635380 T^{6} + 1245734344 T^{8} - 93927058228 T^{10} + 5581218860710 T^{12} - 93927058228 p^{2} T^{14} + 1245734344 p^{4} T^{16} - 12635380 p^{6} T^{18} + 92464 p^{8} T^{20} - 436 p^{10} T^{22} + p^{12} T^{24} \) | |
| 59 | \( ( 1 - 18 T + 348 T^{2} - 4098 T^{3} + 44207 T^{4} - 399048 T^{5} + 3196184 T^{6} - 399048 p T^{7} + 44207 p^{2} T^{8} - 4098 p^{3} T^{9} + 348 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 61 | \( ( 1 + 4 T + 120 T^{2} + 1164 T^{3} + 13296 T^{4} + 92004 T^{5} + 1078750 T^{6} + 92004 p T^{7} + 13296 p^{2} T^{8} + 1164 p^{3} T^{9} + 120 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( 1 - 542 T^{2} + 144839 T^{4} - 25288042 T^{6} + 3205888451 T^{8} - 310065373704 T^{10} + 23422990953258 T^{12} - 310065373704 p^{2} T^{14} + 3205888451 p^{4} T^{16} - 25288042 p^{6} T^{18} + 144839 p^{8} T^{20} - 542 p^{10} T^{22} + p^{12} T^{24} \) | |
| 71 | \( ( 1 + 296 T^{2} - 240 T^{3} + 42919 T^{4} - 35824 T^{5} + 3837792 T^{6} - 35824 p T^{7} + 42919 p^{2} T^{8} - 240 p^{3} T^{9} + 296 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 73 | \( 1 - 250 T^{2} + 35215 T^{4} - 4058186 T^{6} + 407838299 T^{8} - 35416244636 T^{10} + 2717178469674 T^{12} - 35416244636 p^{2} T^{14} + 407838299 p^{4} T^{16} - 4058186 p^{6} T^{18} + 35215 p^{8} T^{20} - 250 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( ( 1 - 2 T + 174 T^{2} + 148 T^{3} + 16146 T^{4} + 69774 T^{5} + 1168654 T^{6} + 69774 p T^{7} + 16146 p^{2} T^{8} + 148 p^{3} T^{9} + 174 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 - 392 T^{2} + 90734 T^{4} - 15217352 T^{6} + 2008228335 T^{8} - 217017094384 T^{10} + 19650680023908 T^{12} - 217017094384 p^{2} T^{14} + 2008228335 p^{4} T^{16} - 15217352 p^{6} T^{18} + 90734 p^{8} T^{20} - 392 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( ( 1 - 16 T + 273 T^{2} - 2194 T^{3} + 18589 T^{4} - 386 p T^{5} + 313138 T^{6} - 386 p^{2} T^{7} + 18589 p^{2} T^{8} - 2194 p^{3} T^{9} + 273 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 - 472 T^{2} + 95472 T^{4} - 11875064 T^{6} + 1293761392 T^{8} - 154615438168 T^{10} + 16862419502750 T^{12} - 154615438168 p^{2} T^{14} + 1293761392 p^{4} T^{16} - 11875064 p^{6} T^{18} + 95472 p^{8} T^{20} - 472 p^{10} T^{22} + p^{12} T^{24} \) | |
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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
−2.80988174081740452811588036846, −2.78494449304593057071298686417, −2.74499120503993300317804079611, −2.64145348021814313126417858167, −2.45943143301129218108558798307, −2.43029354987895018439824391659, −2.27028940292004398600784340909, −2.25998748086015675501048206095, −2.21613812168179755560459571821, −2.15094448357204167713223125485, −2.08983078715098855703841046951, −2.00228663934252416442980908631, −1.94379823776531954730014515453, −1.62451535187303954401403002393, −1.31259597059321063871959919540, −1.30570499593881183016786972227, −1.28413478539086750149541925742, −1.10706008708979422574109632496, −0.847621137827232349408458084751, −0.78362258930882130011551753291, −0.76788218119759543165555564684, −0.67892676202920843920144243663, −0.63616842835739201742640803861, −0.59937940481318619709372700081, −0.17886591080993960409634424019, 0.17886591080993960409634424019, 0.59937940481318619709372700081, 0.63616842835739201742640803861, 0.67892676202920843920144243663, 0.76788218119759543165555564684, 0.78362258930882130011551753291, 0.847621137827232349408458084751, 1.10706008708979422574109632496, 1.28413478539086750149541925742, 1.30570499593881183016786972227, 1.31259597059321063871959919540, 1.62451535187303954401403002393, 1.94379823776531954730014515453, 2.00228663934252416442980908631, 2.08983078715098855703841046951, 2.15094448357204167713223125485, 2.21613812168179755560459571821, 2.25998748086015675501048206095, 2.27028940292004398600784340909, 2.43029354987895018439824391659, 2.45943143301129218108558798307, 2.64145348021814313126417858167, 2.74499120503993300317804079611, 2.78494449304593057071298686417, 2.80988174081740452811588036846
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.