Dirichlet series
| L(s) = 1 | + 12·2-s + 12·3-s + 78·4-s + 8·5-s + 144·6-s + 5·7-s + 364·8-s + 78·9-s + 96·10-s + 11·11-s + 936·12-s + 6·13-s + 60·14-s + 96·15-s + 1.36e3·16-s − 12·17-s + 936·18-s + 3·19-s + 624·20-s + 60·21-s + 132·22-s + 22·23-s + 4.36e3·24-s + 13·25-s + 72·26-s + 364·27-s + 390·28-s + ⋯ |
| L(s) = 1 | + 8.48·2-s + 6.92·3-s + 39·4-s + 3.57·5-s + 58.7·6-s + 1.88·7-s + 128.·8-s + 26·9-s + 30.3·10-s + 3.31·11-s + 270.·12-s + 1.66·13-s + 16.0·14-s + 24.7·15-s + 341.·16-s − 2.91·17-s + 220.·18-s + 0.688·19-s + 139.·20-s + 13.0·21-s + 28.1·22-s + 4.58·23-s + 891.·24-s + 13/5·25-s + 14.1·26-s + 70.0·27-s + 73.7·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 17^{12} \cdot 59^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 17^{12} \cdot 59^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{12} \cdot 17^{12} \cdot 59^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.51618\times 10^{20}\) |
| Root analytic conductor: | \(6.93209\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{12} \cdot 17^{12} \cdot 59^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(6.307424061\times10^{7}\) |
| \(L(\frac12)\) | \(\approx\) | \(6.307424061\times10^{7}\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{12} \) |
| 3 | \( ( 1 - T )^{12} \) | |
| 17 | \( ( 1 + T )^{12} \) | |
| 59 | \( ( 1 - T )^{12} \) | |
| good | 5 | \( 1 - 8 T + 51 T^{2} - 241 T^{3} + 204 p T^{4} - 748 p T^{5} + 509 p^{2} T^{6} - 39442 T^{7} + 115511 T^{8} - 62796 p T^{9} + 811532 T^{10} - 1964637 T^{11} + 4534576 T^{12} - 1964637 p T^{13} + 811532 p^{2} T^{14} - 62796 p^{4} T^{15} + 115511 p^{4} T^{16} - 39442 p^{5} T^{17} + 509 p^{8} T^{18} - 748 p^{8} T^{19} + 204 p^{9} T^{20} - 241 p^{9} T^{21} + 51 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 - 5 T + p^{2} T^{2} - 27 p T^{3} + 1098 T^{4} - 493 p T^{5} + 15214 T^{6} - 39726 T^{7} + 21205 p T^{8} - 331017 T^{9} + 1147558 T^{10} - 2327810 T^{11} + 8085246 T^{12} - 2327810 p T^{13} + 1147558 p^{2} T^{14} - 331017 p^{3} T^{15} + 21205 p^{5} T^{16} - 39726 p^{5} T^{17} + 15214 p^{6} T^{18} - 493 p^{8} T^{19} + 1098 p^{8} T^{20} - 27 p^{10} T^{21} + p^{12} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - p T + 136 T^{2} - 1029 T^{3} + 7610 T^{4} - 44406 T^{5} + 247245 T^{6} - 1183845 T^{7} + 5408719 T^{8} - 22119640 T^{9} + 86817715 T^{10} - 311347963 T^{11} + 1076672844 T^{12} - 311347963 p T^{13} + 86817715 p^{2} T^{14} - 22119640 p^{3} T^{15} + 5408719 p^{4} T^{16} - 1183845 p^{5} T^{17} + 247245 p^{6} T^{18} - 44406 p^{7} T^{19} + 7610 p^{8} T^{20} - 1029 p^{9} T^{21} + 136 p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 6 T + 87 T^{2} - 500 T^{3} + 4049 T^{4} - 21192 T^{5} + 9755 p T^{6} - 601590 T^{7} + 2945467 T^{8} - 12658772 T^{9} + 53317026 T^{10} - 207298060 T^{11} + 771807910 T^{12} - 207298060 p T^{13} + 53317026 p^{2} T^{14} - 12658772 p^{3} T^{15} + 2945467 p^{4} T^{16} - 601590 p^{5} T^{17} + 9755 p^{7} T^{18} - 21192 p^{7} T^{19} + 4049 p^{8} T^{20} - 500 p^{9} T^{21} + 87 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 3 T + 99 T^{2} - 6 p T^{3} + 4257 T^{4} + 2409 T^{5} + 119748 T^{6} + 13075 p T^{7} + 2894779 T^{8} + 7766318 T^{9} + 68362009 T^{10} + 8392559 p T^{11} + 75661058 p T^{12} + 8392559 p^{2} T^{13} + 68362009 p^{2} T^{14} + 7766318 p^{3} T^{15} + 2894779 p^{4} T^{16} + 13075 p^{6} T^{17} + 119748 p^{6} T^{18} + 2409 p^{7} T^{19} + 4257 p^{8} T^{20} - 6 p^{10} T^{21} + 99 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 22 T + 366 T^{2} - 4422 T^{3} + 45491 T^{4} - 398585 T^{5} + 3123504 T^{6} - 21985462 T^{7} + 142319212 T^{8} - 849851556 T^{9} + 206542679 p T^{10} - 24858529091 T^{11} + 122908912502 T^{12} - 24858529091 p T^{13} + 206542679 p^{3} T^{14} - 849851556 p^{3} T^{15} + 142319212 p^{4} T^{16} - 21985462 p^{5} T^{17} + 3123504 p^{6} T^{18} - 398585 p^{7} T^{19} + 45491 p^{8} T^{20} - 4422 p^{9} T^{21} + 366 p^{10} T^{22} - 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 26 T + 561 T^{2} - 8371 T^{3} + 109292 T^{4} - 1182768 T^{5} + 11552999 T^{6} - 99231528 T^{7} + 783090761 T^{8} - 5577822816 T^{9} + 36861839364 T^{10} - 222507792339 T^{11} + 1251583504156 T^{12} - 222507792339 p T^{13} + 36861839364 p^{2} T^{14} - 5577822816 p^{3} T^{15} + 783090761 p^{4} T^{16} - 99231528 p^{5} T^{17} + 11552999 p^{6} T^{18} - 1182768 p^{7} T^{19} + 109292 p^{8} T^{20} - 8371 p^{9} T^{21} + 561 p^{10} T^{22} - 26 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - T + 154 T^{2} - 5 T^{3} + 12533 T^{4} + 2875 T^{5} + 730314 T^{6} + 116231 T^{7} + 33454419 T^{8} - 392016 T^{9} + 1281066780 T^{10} - 190706776 T^{11} + 42476071278 T^{12} - 190706776 p T^{13} + 1281066780 p^{2} T^{14} - 392016 p^{3} T^{15} + 33454419 p^{4} T^{16} + 116231 p^{5} T^{17} + 730314 p^{6} T^{18} + 2875 p^{7} T^{19} + 12533 p^{8} T^{20} - 5 p^{9} T^{21} + 154 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 10 T + 253 T^{2} - 1959 T^{3} + 31588 T^{4} - 214354 T^{5} + 2711359 T^{6} - 16447028 T^{7} + 174724441 T^{8} - 956315644 T^{9} + 8886357896 T^{10} - 44047315133 T^{11} + 364945806300 T^{12} - 44047315133 p T^{13} + 8886357896 p^{2} T^{14} - 956315644 p^{3} T^{15} + 174724441 p^{4} T^{16} - 16447028 p^{5} T^{17} + 2711359 p^{6} T^{18} - 214354 p^{7} T^{19} + 31588 p^{8} T^{20} - 1959 p^{9} T^{21} + 253 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 16 T + 8 p T^{2} - 3890 T^{3} + 50483 T^{4} - 488847 T^{5} + 4980234 T^{6} - 41569976 T^{7} + 359619120 T^{8} - 2659016966 T^{9} + 20274652749 T^{10} - 134713999933 T^{11} + 921246452002 T^{12} - 134713999933 p T^{13} + 20274652749 p^{2} T^{14} - 2659016966 p^{3} T^{15} + 359619120 p^{4} T^{16} - 41569976 p^{5} T^{17} + 4980234 p^{6} T^{18} - 488847 p^{7} T^{19} + 50483 p^{8} T^{20} - 3890 p^{9} T^{21} + 8 p^{11} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 23 T + 494 T^{2} - 6945 T^{3} + 93452 T^{4} - 1025738 T^{5} + 10856849 T^{6} - 100414985 T^{7} + 894019619 T^{8} - 7179083176 T^{9} + 55580772905 T^{10} - 394278746763 T^{11} + 2696050439680 T^{12} - 394278746763 p T^{13} + 55580772905 p^{2} T^{14} - 7179083176 p^{3} T^{15} + 894019619 p^{4} T^{16} - 100414985 p^{5} T^{17} + 10856849 p^{6} T^{18} - 1025738 p^{7} T^{19} + 93452 p^{8} T^{20} - 6945 p^{9} T^{21} + 494 p^{10} T^{22} - 23 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 6 T + 278 T^{2} - 639 T^{3} + 33521 T^{4} + 27941 T^{5} + 2761510 T^{6} + 8534956 T^{7} + 199123987 T^{8} + 742030442 T^{9} + 12685955780 T^{10} + 43652207820 T^{11} + 664921509750 T^{12} + 43652207820 p T^{13} + 12685955780 p^{2} T^{14} + 742030442 p^{3} T^{15} + 199123987 p^{4} T^{16} + 8534956 p^{5} T^{17} + 2761510 p^{6} T^{18} + 27941 p^{7} T^{19} + 33521 p^{8} T^{20} - 639 p^{9} T^{21} + 278 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 10 T + 329 T^{2} - 40 p T^{3} + 46936 T^{4} - 206497 T^{5} + 4523843 T^{6} - 14954354 T^{7} + 357531419 T^{8} - 939806598 T^{9} + 23562464052 T^{10} - 50164644249 T^{11} + 1326160748168 T^{12} - 50164644249 p T^{13} + 23562464052 p^{2} T^{14} - 939806598 p^{3} T^{15} + 357531419 p^{4} T^{16} - 14954354 p^{5} T^{17} + 4523843 p^{6} T^{18} - 206497 p^{7} T^{19} + 46936 p^{8} T^{20} - 40 p^{10} T^{21} + 329 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 15 T + 444 T^{2} - 6249 T^{3} + 108672 T^{4} - 1331380 T^{5} + 17730175 T^{6} - 188692169 T^{7} + 2096224607 T^{8} - 19561240074 T^{9} + 187215880333 T^{10} - 1543732697357 T^{11} + 12938468416416 T^{12} - 1543732697357 p T^{13} + 187215880333 p^{2} T^{14} - 19561240074 p^{3} T^{15} + 2096224607 p^{4} T^{16} - 188692169 p^{5} T^{17} + 17730175 p^{6} T^{18} - 1331380 p^{7} T^{19} + 108672 p^{8} T^{20} - 6249 p^{9} T^{21} + 444 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 4 T + 376 T^{2} - 791 T^{3} + 72851 T^{4} - 52483 T^{5} + 9803691 T^{6} + 3388737 T^{7} + 1022308687 T^{8} + 1096681215 T^{9} + 87425467265 T^{10} + 123277363470 T^{11} + 6325605398946 T^{12} + 123277363470 p T^{13} + 87425467265 p^{2} T^{14} + 1096681215 p^{3} T^{15} + 1022308687 p^{4} T^{16} + 3388737 p^{5} T^{17} + 9803691 p^{6} T^{18} - 52483 p^{7} T^{19} + 72851 p^{8} T^{20} - 791 p^{9} T^{21} + 376 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 10 T + 446 T^{2} - 4173 T^{3} + 94859 T^{4} - 759207 T^{5} + 12478030 T^{6} - 1122094 p T^{7} + 1128037419 T^{8} - 5439162632 T^{9} + 79041506420 T^{10} - 296710912754 T^{11} + 5315201586386 T^{12} - 296710912754 p T^{13} + 79041506420 p^{2} T^{14} - 5439162632 p^{3} T^{15} + 1128037419 p^{4} T^{16} - 1122094 p^{6} T^{17} + 12478030 p^{6} T^{18} - 759207 p^{7} T^{19} + 94859 p^{8} T^{20} - 4173 p^{9} T^{21} + 446 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 24 T + 738 T^{2} - 13420 T^{3} + 252035 T^{4} - 3720767 T^{5} + 53632468 T^{6} - 667254760 T^{7} + 7982391378 T^{8} - 85498289908 T^{9} + 878056045481 T^{10} - 8191908374589 T^{11} + 73293039390982 T^{12} - 8191908374589 p T^{13} + 878056045481 p^{2} T^{14} - 85498289908 p^{3} T^{15} + 7982391378 p^{4} T^{16} - 667254760 p^{5} T^{17} + 53632468 p^{6} T^{18} - 3720767 p^{7} T^{19} + 252035 p^{8} T^{20} - 13420 p^{9} T^{21} + 738 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 23 T + 838 T^{2} - 14682 T^{3} + 308243 T^{4} - 4383261 T^{5} + 68301864 T^{6} - 821272379 T^{7} + 10455345593 T^{8} - 109310267326 T^{9} + 1193837241830 T^{10} - 11038263616301 T^{11} + 106132418360590 T^{12} - 11038263616301 p T^{13} + 1193837241830 p^{2} T^{14} - 109310267326 p^{3} T^{15} + 10455345593 p^{4} T^{16} - 821272379 p^{5} T^{17} + 68301864 p^{6} T^{18} - 4383261 p^{7} T^{19} + 308243 p^{8} T^{20} - 14682 p^{9} T^{21} + 838 p^{10} T^{22} - 23 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 394 T^{2} + 838 T^{3} + 85658 T^{4} + 242147 T^{5} + 13279713 T^{6} + 41583946 T^{7} + 1591614515 T^{8} + 5134778456 T^{9} + 159560820573 T^{10} + 506534429703 T^{11} + 13989731333412 T^{12} + 506534429703 p T^{13} + 159560820573 p^{2} T^{14} + 5134778456 p^{3} T^{15} + 1591614515 p^{4} T^{16} + 41583946 p^{5} T^{17} + 13279713 p^{6} T^{18} + 242147 p^{7} T^{19} + 85658 p^{8} T^{20} + 838 p^{9} T^{21} + 394 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( 1 - 13 T + 640 T^{2} - 5864 T^{3} + 186798 T^{4} - 1284097 T^{5} + 35315134 T^{6} - 184266914 T^{7} + 4949887363 T^{8} - 19681831200 T^{9} + 557257097299 T^{10} - 1799959084980 T^{11} + 53294527225442 T^{12} - 1799959084980 p T^{13} + 557257097299 p^{2} T^{14} - 19681831200 p^{3} T^{15} + 4949887363 p^{4} T^{16} - 184266914 p^{5} T^{17} + 35315134 p^{6} T^{18} - 1284097 p^{7} T^{19} + 186798 p^{8} T^{20} - 5864 p^{9} T^{21} + 640 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 13 T + 624 T^{2} - 6199 T^{3} + 182188 T^{4} - 1381153 T^{5} + 33152915 T^{6} - 182225652 T^{7} + 4296773011 T^{8} - 15422891933 T^{9} + 445946914693 T^{10} - 1004584770090 T^{11} + 43003903870192 T^{12} - 1004584770090 p T^{13} + 445946914693 p^{2} T^{14} - 15422891933 p^{3} T^{15} + 4296773011 p^{4} T^{16} - 182225652 p^{5} T^{17} + 33152915 p^{6} T^{18} - 1381153 p^{7} T^{19} + 182188 p^{8} T^{20} - 6199 p^{9} T^{21} + 624 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.54233373152404360131319930829, −2.28235022087887955860722183651, −2.27410387503473007503350102347, −2.21297837280853149736232207829, −2.12523795204134110774183244217, −2.10966684196152310775379631463, −2.07680280953947950214350468362, −2.06300150365788598499098330423, −1.99952701444079296957066327294, −1.96659329965428230990828543494, −1.94931234938579054252242977225, −1.94492166898634058371129410217, −1.79301271320798307341508595704, −1.45013660088239179936172210418, −1.26415055700069465141148624907, −1.25338977569637865823051522351, −1.23125530890269007388235354459, −1.19942123326799244765080431792, −1.16520247220627701255813156371, −1.14711214674877648245304090249, −1.13818569483158793239177409395, −0.844702994041360756846088709039, −0.76966994480678243795231189134, −0.76071360635245539395335503538, −0.63864528853654740976915332922, 0.63864528853654740976915332922, 0.76071360635245539395335503538, 0.76966994480678243795231189134, 0.844702994041360756846088709039, 1.13818569483158793239177409395, 1.14711214674877648245304090249, 1.16520247220627701255813156371, 1.19942123326799244765080431792, 1.23125530890269007388235354459, 1.25338977569637865823051522351, 1.26415055700069465141148624907, 1.45013660088239179936172210418, 1.79301271320798307341508595704, 1.94492166898634058371129410217, 1.94931234938579054252242977225, 1.96659329965428230990828543494, 1.99952701444079296957066327294, 2.06300150365788598499098330423, 2.07680280953947950214350468362, 2.10966684196152310775379631463, 2.12523795204134110774183244217, 2.21297837280853149736232207829, 2.27410387503473007503350102347, 2.28235022087887955860722183651, 2.54233373152404360131319930829
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.