Dirichlet series
| L(s) = 1 | − 2·4-s − 6·5-s − 2·7-s + 12·11-s + 2·16-s + 6·19-s + 12·20-s + 12·23-s + 15·25-s + 4·28-s + 6·31-s + 12·32-s + 12·35-s − 10·37-s + 24·41-s − 4·43-s − 24·44-s + 5·49-s + 12·53-s − 72·55-s − 24·59-s + 6·67-s − 42·73-s − 12·76-s − 24·77-s + 18·79-s − 12·80-s + ⋯ |
| L(s) = 1 | − 4-s − 2.68·5-s − 0.755·7-s + 3.61·11-s + 1/2·16-s + 1.37·19-s + 2.68·20-s + 2.50·23-s + 3·25-s + 0.755·28-s + 1.07·31-s + 2.12·32-s + 2.02·35-s − 1.64·37-s + 3.74·41-s − 0.609·43-s − 3.61·44-s + 5/7·49-s + 1.64·53-s − 9.70·55-s − 3.12·59-s + 0.733·67-s − 4.91·73-s − 1.37·76-s − 2.73·77-s + 2.02·79-s − 1.34·80-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{24} \cdot 5^{12} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(64128.6\) |
| Root analytic conductor: | \(1.58596\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{24} \cdot 5^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(3.606401249\) |
| \(L(\frac12)\) | \(\approx\) | \(3.606401249\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 + T + T^{2} )^{6} \) | |
| 7 | \( 1 + 2 T - T^{2} + 6 T^{3} + 130 T^{4} + 130 T^{5} - 131 T^{6} + 130 p T^{7} + 130 p^{2} T^{8} + 6 p^{3} T^{9} - p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 2 | \( 1 + p T^{2} + p T^{4} - 3 p^{2} T^{5} - 3 p^{3} T^{7} - p^{3} T^{8} - 3 p^{3} T^{9} + 13 p^{2} T^{10} + 3 p^{3} T^{11} + 37 p^{2} T^{12} + 3 p^{4} T^{13} + 13 p^{4} T^{14} - 3 p^{6} T^{15} - p^{7} T^{16} - 3 p^{8} T^{17} - 3 p^{9} T^{19} + p^{9} T^{20} + p^{11} T^{22} + p^{12} T^{24} \) |
| 11 | \( 1 - 12 T + 104 T^{2} - 672 T^{3} + 331 p T^{4} - 16332 T^{5} + 62340 T^{6} - 195852 T^{7} + 477010 T^{8} - 649368 T^{9} - 1262450 T^{10} + 12891636 T^{11} - 53694803 T^{12} + 12891636 p T^{13} - 1262450 p^{2} T^{14} - 649368 p^{3} T^{15} + 477010 p^{4} T^{16} - 195852 p^{5} T^{17} + 62340 p^{6} T^{18} - 16332 p^{7} T^{19} + 331 p^{9} T^{20} - 672 p^{9} T^{21} + 104 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 54 T^{2} + 1725 T^{4} - 42910 T^{6} + 855078 T^{8} - 14180646 T^{10} + 200139573 T^{12} - 14180646 p^{2} T^{14} + 855078 p^{4} T^{16} - 42910 p^{6} T^{18} + 1725 p^{8} T^{20} - 54 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( 1 - 42 T^{2} + 120 T^{3} + 831 T^{4} - 5256 T^{5} - 3958 T^{6} + 127332 T^{7} - 227496 T^{8} - 2018952 T^{9} + 9082140 T^{10} + 14965308 T^{11} - 192427863 T^{12} + 14965308 p T^{13} + 9082140 p^{2} T^{14} - 2018952 p^{3} T^{15} - 227496 p^{4} T^{16} + 127332 p^{5} T^{17} - 3958 p^{6} T^{18} - 5256 p^{7} T^{19} + 831 p^{8} T^{20} + 120 p^{9} T^{21} - 42 p^{10} T^{22} + p^{12} T^{24} \) | |
| 19 | \( 1 - 6 T + 75 T^{2} - 378 T^{3} + 2703 T^{4} - 15096 T^{5} + 236 p^{2} T^{6} - 494928 T^{7} + 125331 p T^{8} - 12436578 T^{9} + 2998275 p T^{10} - 267312150 T^{11} + 1188448446 T^{12} - 267312150 p T^{13} + 2998275 p^{3} T^{14} - 12436578 p^{3} T^{15} + 125331 p^{5} T^{16} - 494928 p^{5} T^{17} + 236 p^{8} T^{18} - 15096 p^{7} T^{19} + 2703 p^{8} T^{20} - 378 p^{9} T^{21} + 75 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 12 T + 110 T^{2} - 744 T^{3} + 3947 T^{4} - 16104 T^{5} + 39450 T^{6} + 32328 T^{7} - 53596 p T^{8} + 9040500 T^{9} - 50848868 T^{10} + 256937364 T^{11} - 1217123567 T^{12} + 256937364 p T^{13} - 50848868 p^{2} T^{14} + 9040500 p^{3} T^{15} - 53596 p^{5} T^{16} + 32328 p^{5} T^{17} + 39450 p^{6} T^{18} - 16104 p^{7} T^{19} + 3947 p^{8} T^{20} - 744 p^{9} T^{21} + 110 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 184 T^{2} + 16934 T^{4} - 1029132 T^{6} + 46624399 T^{8} - 1703051756 T^{10} + 52962115192 T^{12} - 1703051756 p^{2} T^{14} + 46624399 p^{4} T^{16} - 1029132 p^{6} T^{18} + 16934 p^{8} T^{20} - 184 p^{10} T^{22} + p^{12} T^{24} \) | |
| 31 | \( 1 - 6 T + 75 T^{2} - 378 T^{3} + 2451 T^{4} - 14268 T^{5} + 15464 T^{6} - 69696 T^{7} - 2002827 T^{8} + 11519550 T^{9} - 72958179 T^{10} + 685842102 T^{11} - 2463303858 T^{12} + 685842102 p T^{13} - 72958179 p^{2} T^{14} + 11519550 p^{3} T^{15} - 2002827 p^{4} T^{16} - 69696 p^{5} T^{17} + 15464 p^{6} T^{18} - 14268 p^{7} T^{19} + 2451 p^{8} T^{20} - 378 p^{9} T^{21} + 75 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 10 T - T^{2} + 206 T^{3} + 1893 T^{4} - 19356 T^{5} - 25070 T^{6} - 11384 T^{7} - 4236169 T^{8} + 22256290 T^{9} - 6662617 T^{10} - 355484390 T^{11} + 9259385786 T^{12} - 355484390 p T^{13} - 6662617 p^{2} T^{14} + 22256290 p^{3} T^{15} - 4236169 p^{4} T^{16} - 11384 p^{5} T^{17} - 25070 p^{6} T^{18} - 19356 p^{7} T^{19} + 1893 p^{8} T^{20} + 206 p^{9} T^{21} - p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( ( 1 - 12 T + 192 T^{2} - 1704 T^{3} + 18087 T^{4} - 122340 T^{5} + 939022 T^{6} - 122340 p T^{7} + 18087 p^{2} T^{8} - 1704 p^{3} T^{9} + 192 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 43 | \( ( 1 + 2 T + 65 T^{2} + 318 T^{3} + 5080 T^{4} + 14554 T^{5} + 241357 T^{6} + 14554 p T^{7} + 5080 p^{2} T^{8} + 318 p^{3} T^{9} + 65 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 - 162 T^{2} + 192 T^{3} + 11733 T^{4} - 23328 T^{5} - 672838 T^{6} + 760896 T^{7} + 42600858 T^{8} + 8087904 T^{9} - 2486828586 T^{10} - 573202080 T^{11} + 123583933965 T^{12} - 573202080 p T^{13} - 2486828586 p^{2} T^{14} + 8087904 p^{3} T^{15} + 42600858 p^{4} T^{16} + 760896 p^{5} T^{17} - 672838 p^{6} T^{18} - 23328 p^{7} T^{19} + 11733 p^{8} T^{20} + 192 p^{9} T^{21} - 162 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( 1 - 12 T + 290 T^{2} - 2904 T^{3} + 41501 T^{4} - 314064 T^{5} + 3427926 T^{6} - 18869580 T^{7} + 172940890 T^{8} - 534860628 T^{9} + 5470600330 T^{10} + 1694225472 T^{11} + 171503876389 T^{12} + 1694225472 p T^{13} + 5470600330 p^{2} T^{14} - 534860628 p^{3} T^{15} + 172940890 p^{4} T^{16} - 18869580 p^{5} T^{17} + 3427926 p^{6} T^{18} - 314064 p^{7} T^{19} + 41501 p^{8} T^{20} - 2904 p^{9} T^{21} + 290 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 24 T + 84 T^{2} - 1128 T^{3} + 9885 T^{4} + 197484 T^{5} - 900376 T^{6} - 207936 p T^{7} + 92002038 T^{8} + 411544404 T^{9} - 9205294494 T^{10} - 19136016684 T^{11} + 471190351413 T^{12} - 19136016684 p T^{13} - 9205294494 p^{2} T^{14} + 411544404 p^{3} T^{15} + 92002038 p^{4} T^{16} - 207936 p^{6} T^{17} - 900376 p^{6} T^{18} + 197484 p^{7} T^{19} + 9885 p^{8} T^{20} - 1128 p^{9} T^{21} + 84 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 138 T^{2} + 5889 T^{4} - 41004 T^{5} - 102418 T^{6} - 5059224 T^{7} - 8898402 T^{8} - 158765292 T^{9} + 1660743870 T^{10} + 9579987756 T^{11} + 189206615181 T^{12} + 9579987756 p T^{13} + 1660743870 p^{2} T^{14} - 158765292 p^{3} T^{15} - 8898402 p^{4} T^{16} - 5059224 p^{5} T^{17} - 102418 p^{6} T^{18} - 41004 p^{7} T^{19} + 5889 p^{8} T^{20} + 138 p^{10} T^{22} + p^{12} T^{24} \) | |
| 67 | \( 1 - 6 T - 219 T^{2} + 1306 T^{3} + 22971 T^{4} - 120540 T^{5} - 1928636 T^{6} + 7225512 T^{7} + 163638939 T^{8} - 309627182 T^{9} - 13816766313 T^{10} + 6348931902 T^{11} + 1027244050774 T^{12} + 6348931902 p T^{13} - 13816766313 p^{2} T^{14} - 309627182 p^{3} T^{15} + 163638939 p^{4} T^{16} + 7225512 p^{5} T^{17} - 1928636 p^{6} T^{18} - 120540 p^{7} T^{19} + 22971 p^{8} T^{20} + 1306 p^{9} T^{21} - 219 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 460 T^{2} + 108410 T^{4} - 17445312 T^{6} + 2127087511 T^{8} - 205744741412 T^{10} + 16152616390624 T^{12} - 205744741412 p^{2} T^{14} + 2127087511 p^{4} T^{16} - 17445312 p^{6} T^{18} + 108410 p^{8} T^{20} - 460 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 + 42 T + 1119 T^{2} + 22302 T^{3} + 373665 T^{4} + 5463480 T^{5} + 72046694 T^{6} + 868793688 T^{7} + 9715685451 T^{8} + 101446836642 T^{9} + 996453337299 T^{10} + 9237792089286 T^{11} + 81107402998458 T^{12} + 9237792089286 p T^{13} + 996453337299 p^{2} T^{14} + 101446836642 p^{3} T^{15} + 9715685451 p^{4} T^{16} + 868793688 p^{5} T^{17} + 72046694 p^{6} T^{18} + 5463480 p^{7} T^{19} + 373665 p^{8} T^{20} + 22302 p^{9} T^{21} + 1119 p^{10} T^{22} + 42 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 18 T - 177 T^{2} + 3346 T^{3} + 40923 T^{4} - 454344 T^{5} - 6959816 T^{6} + 45880152 T^{7} + 889910193 T^{8} - 3054243566 T^{9} - 93786004551 T^{10} + 82100387598 T^{11} + 8378036930038 T^{12} + 82100387598 p T^{13} - 93786004551 p^{2} T^{14} - 3054243566 p^{3} T^{15} + 889910193 p^{4} T^{16} + 45880152 p^{5} T^{17} - 6959816 p^{6} T^{18} - 454344 p^{7} T^{19} + 40923 p^{8} T^{20} + 3346 p^{9} T^{21} - 177 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( ( 1 + 12 T + 300 T^{2} + 3204 T^{3} + 50769 T^{4} + 448284 T^{5} + 5149954 T^{6} + 448284 p T^{7} + 50769 p^{2} T^{8} + 3204 p^{3} T^{9} + 300 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( 1 - 12 T - 282 T^{2} + 648 T^{3} + 81261 T^{4} + 90852 T^{5} - 10133890 T^{6} - 68061132 T^{7} + 906987750 T^{8} + 7493819688 T^{9} - 22404081252 T^{10} - 403242866028 T^{11} + 351385075713 T^{12} - 403242866028 p T^{13} - 22404081252 p^{2} T^{14} + 7493819688 p^{3} T^{15} + 906987750 p^{4} T^{16} - 68061132 p^{5} T^{17} - 10133890 p^{6} T^{18} + 90852 p^{7} T^{19} + 81261 p^{8} T^{20} + 648 p^{9} T^{21} - 282 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 432 T^{2} + 85686 T^{4} - 12118408 T^{6} + 1564041327 T^{8} - 184182428136 T^{10} + 19001698536324 T^{12} - 184182428136 p^{2} T^{14} + 1564041327 p^{4} T^{16} - 12118408 p^{6} T^{18} + 85686 p^{8} T^{20} - 432 p^{10} T^{22} + p^{12} T^{24} \) | |
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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
−4.01320982844363675021388177986, −3.96197904869894137333314099441, −3.83458590407039347543858751173, −3.80030275356118282434597690069, −3.62836201455728176382213265522, −3.27573541768090673233899984818, −3.20880212359953827730113454599, −3.13900545690388046244830085234, −3.11583926667486190702458578315, −3.11192984906428994650566435402, −3.00692023534358361605134489536, −2.94449276711554811129332939093, −2.68438838250904170695637518259, −2.51377934380092110552034345965, −2.36027148500643467610439258854, −2.21152750919681444538961738393, −1.73485247185641737065712846508, −1.71922891199266515324985713402, −1.59186211811408795049387845759, −1.58415642738614257214605220331, −1.27767157052647124085875307977, −0.810393157021652804444693309800, −0.72476080859476545216441006191, −0.68530791909967995513137095681, −0.65794057923429440276648395211, 0.65794057923429440276648395211, 0.68530791909967995513137095681, 0.72476080859476545216441006191, 0.810393157021652804444693309800, 1.27767157052647124085875307977, 1.58415642738614257214605220331, 1.59186211811408795049387845759, 1.71922891199266515324985713402, 1.73485247185641737065712846508, 2.21152750919681444538961738393, 2.36027148500643467610439258854, 2.51377934380092110552034345965, 2.68438838250904170695637518259, 2.94449276711554811129332939093, 3.00692023534358361605134489536, 3.11192984906428994650566435402, 3.11583926667486190702458578315, 3.13900545690388046244830085234, 3.20880212359953827730113454599, 3.27573541768090673233899984818, 3.62836201455728176382213265522, 3.80030275356118282434597690069, 3.83458590407039347543858751173, 3.96197904869894137333314099441, 4.01320982844363675021388177986
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.