Dirichlet series
| L(s) = 1 | − 4·3-s − 4·5-s − 7-s − 2·9-s − 3·11-s − 3·13-s + 16·15-s − 8·17-s + 2·19-s + 4·21-s + 2·23-s − 16·25-s + 31·27-s − 29·31-s + 12·33-s + 4·35-s − 38·37-s + 12·39-s − 11·41-s − 9·43-s + 8·45-s − 34·47-s − 25·49-s + 32·51-s − 15·53-s + 12·55-s − 8·57-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1.78·5-s − 0.377·7-s − 2/3·9-s − 0.904·11-s − 0.832·13-s + 4.13·15-s − 1.94·17-s + 0.458·19-s + 0.872·21-s + 0.417·23-s − 3.19·25-s + 5.96·27-s − 5.20·31-s + 2.08·33-s + 0.676·35-s − 6.24·37-s + 1.92·39-s − 1.71·41-s − 1.37·43-s + 1.19·45-s − 4.95·47-s − 3.57·49-s + 4.48·51-s − 2.06·53-s + 1.61·55-s − 1.05·57-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 29^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 29^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{36} \cdot 29^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.78038\times 10^{20}\) |
| Root analytic conductor: | \(7.32962\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 2^{36} \cdot 29^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 \) | |
| good | 3 | \( 1 + 4 T + 2 p^{2} T^{2} + 49 T^{3} + 137 T^{4} + 316 T^{5} + 763 T^{6} + 181 p^{2} T^{7} + 3580 T^{8} + 2318 p T^{9} + 4565 p T^{10} + 24440 T^{11} + 44092 T^{12} + 24440 p T^{13} + 4565 p^{3} T^{14} + 2318 p^{4} T^{15} + 3580 p^{4} T^{16} + 181 p^{7} T^{17} + 763 p^{6} T^{18} + 316 p^{7} T^{19} + 137 p^{8} T^{20} + 49 p^{9} T^{21} + 2 p^{12} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 + 4 T + 32 T^{2} + 88 T^{3} + 439 T^{4} + 942 T^{5} + 3982 T^{6} + 1492 p T^{7} + 29557 T^{8} + 10176 p T^{9} + 37484 p T^{10} + 296804 T^{11} + 1011893 T^{12} + 296804 p T^{13} + 37484 p^{3} T^{14} + 10176 p^{4} T^{15} + 29557 p^{4} T^{16} + 1492 p^{6} T^{17} + 3982 p^{6} T^{18} + 942 p^{7} T^{19} + 439 p^{8} T^{20} + 88 p^{9} T^{21} + 32 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 + T + 26 T^{2} + 27 T^{3} + p^{3} T^{4} + 46 p T^{5} + 447 p T^{6} + p^{4} T^{7} + 526 p^{2} T^{8} + 313 p^{2} T^{9} + 627 p^{3} T^{10} + 316 p^{3} T^{11} + 4796 p^{3} T^{12} + 316 p^{4} T^{13} + 627 p^{5} T^{14} + 313 p^{5} T^{15} + 526 p^{6} T^{16} + p^{9} T^{17} + 447 p^{7} T^{18} + 46 p^{8} T^{19} + p^{11} T^{20} + 27 p^{9} T^{21} + 26 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 3 T + 65 T^{2} + 147 T^{3} + 180 p T^{4} + 2856 T^{5} + 3420 p T^{6} + 21119 T^{7} + 515659 T^{8} - 187597 T^{9} + 531285 p T^{10} - 6406536 T^{11} + 63343832 T^{12} - 6406536 p T^{13} + 531285 p^{3} T^{14} - 187597 p^{3} T^{15} + 515659 p^{4} T^{16} + 21119 p^{5} T^{17} + 3420 p^{7} T^{18} + 2856 p^{7} T^{19} + 180 p^{9} T^{20} + 147 p^{9} T^{21} + 65 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 3 T + 84 T^{2} + 197 T^{3} + 3528 T^{4} + 6183 T^{5} + 96757 T^{6} + 116653 T^{7} + 1956849 T^{8} + 1464819 T^{9} + 31752296 T^{10} + 14901426 T^{11} + 440546781 T^{12} + 14901426 p T^{13} + 31752296 p^{2} T^{14} + 1464819 p^{3} T^{15} + 1956849 p^{4} T^{16} + 116653 p^{5} T^{17} + 96757 p^{6} T^{18} + 6183 p^{7} T^{19} + 3528 p^{8} T^{20} + 197 p^{9} T^{21} + 84 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 8 T + 159 T^{2} + 954 T^{3} + 10954 T^{4} + 52736 T^{5} + 459708 T^{6} + 1868400 T^{7} + 13802677 T^{8} + 49250636 T^{9} + 323108661 T^{10} + 1033449538 T^{11} + 6098298784 T^{12} + 1033449538 p T^{13} + 323108661 p^{2} T^{14} + 49250636 p^{3} T^{15} + 13802677 p^{4} T^{16} + 1868400 p^{5} T^{17} + 459708 p^{6} T^{18} + 52736 p^{7} T^{19} + 10954 p^{8} T^{20} + 954 p^{9} T^{21} + 159 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 2 T + 136 T^{2} - 317 T^{3} + 9127 T^{4} - 22974 T^{5} + 407373 T^{6} - 1039169 T^{7} + 13598524 T^{8} - 33406702 T^{9} + 357593151 T^{10} - 814210068 T^{11} + 7559275408 T^{12} - 814210068 p T^{13} + 357593151 p^{2} T^{14} - 33406702 p^{3} T^{15} + 13598524 p^{4} T^{16} - 1039169 p^{5} T^{17} + 407373 p^{6} T^{18} - 22974 p^{7} T^{19} + 9127 p^{8} T^{20} - 317 p^{9} T^{21} + 136 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 2 T + 180 T^{2} - 337 T^{3} + 15997 T^{4} - 28968 T^{5} + 930445 T^{6} - 1636973 T^{7} + 1716144 p T^{8} - 66419842 T^{9} + 1288169887 T^{10} - 2012988946 T^{11} + 33204381092 T^{12} - 2012988946 p T^{13} + 1288169887 p^{2} T^{14} - 66419842 p^{3} T^{15} + 1716144 p^{5} T^{16} - 1636973 p^{5} T^{17} + 930445 p^{6} T^{18} - 28968 p^{7} T^{19} + 15997 p^{8} T^{20} - 337 p^{9} T^{21} + 180 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 29 T + 547 T^{2} + 7567 T^{3} + 86522 T^{4} + 26928 p T^{5} + 7070598 T^{6} + 53105719 T^{7} + 362795437 T^{8} + 2278341693 T^{9} + 13519358295 T^{10} + 77005555792 T^{11} + 431927430512 T^{12} + 77005555792 p T^{13} + 13519358295 p^{2} T^{14} + 2278341693 p^{3} T^{15} + 362795437 p^{4} T^{16} + 53105719 p^{5} T^{17} + 7070598 p^{6} T^{18} + 26928 p^{8} T^{19} + 86522 p^{8} T^{20} + 7567 p^{9} T^{21} + 547 p^{10} T^{22} + 29 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 38 T + 926 T^{2} + 16684 T^{3} + 247075 T^{4} + 3115134 T^{5} + 34534130 T^{6} + 341517128 T^{7} + 3055408071 T^{8} + 24895130694 T^{9} + 186004552420 T^{10} + 1277746407940 T^{11} + 8094319674245 T^{12} + 1277746407940 p T^{13} + 186004552420 p^{2} T^{14} + 24895130694 p^{3} T^{15} + 3055408071 p^{4} T^{16} + 341517128 p^{5} T^{17} + 34534130 p^{6} T^{18} + 3115134 p^{7} T^{19} + 247075 p^{8} T^{20} + 16684 p^{9} T^{21} + 926 p^{10} T^{22} + 38 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 11 T + 355 T^{2} + 3312 T^{3} + 60045 T^{4} + 486951 T^{5} + 6484726 T^{6} + 46518864 T^{7} + 503200817 T^{8} + 3221622057 T^{9} + 29720368743 T^{10} + 169979045757 T^{11} + 1372590632898 T^{12} + 169979045757 p T^{13} + 29720368743 p^{2} T^{14} + 3221622057 p^{3} T^{15} + 503200817 p^{4} T^{16} + 46518864 p^{5} T^{17} + 6484726 p^{6} T^{18} + 486951 p^{7} T^{19} + 60045 p^{8} T^{20} + 3312 p^{9} T^{21} + 355 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 9 T + 259 T^{2} + 2295 T^{3} + 35330 T^{4} + 309552 T^{5} + 3363586 T^{6} + 27938603 T^{7} + 246336921 T^{8} + 1878684697 T^{9} + 14472155727 T^{10} + 99589324156 T^{11} + 689652471872 T^{12} + 99589324156 p T^{13} + 14472155727 p^{2} T^{14} + 1878684697 p^{3} T^{15} + 246336921 p^{4} T^{16} + 27938603 p^{5} T^{17} + 3363586 p^{6} T^{18} + 309552 p^{7} T^{19} + 35330 p^{8} T^{20} + 2295 p^{9} T^{21} + 259 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 34 T + 904 T^{2} + 16933 T^{3} + 274915 T^{4} + 3731902 T^{5} + 45705087 T^{6} + 494764899 T^{7} + 4926671632 T^{8} + 44423485582 T^{9} + 372092073765 T^{10} + 2853317776674 T^{11} + 20411197599488 T^{12} + 2853317776674 p T^{13} + 372092073765 p^{2} T^{14} + 44423485582 p^{3} T^{15} + 4926671632 p^{4} T^{16} + 494764899 p^{5} T^{17} + 45705087 p^{6} T^{18} + 3731902 p^{7} T^{19} + 274915 p^{8} T^{20} + 16933 p^{9} T^{21} + 904 p^{10} T^{22} + 34 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 15 T + 358 T^{2} + 4351 T^{3} + 62898 T^{4} + 653495 T^{5} + 7401029 T^{6} + 67478723 T^{7} + 653345457 T^{8} + 5341839491 T^{9} + 45980891822 T^{10} + 342371237486 T^{11} + 2669724439403 T^{12} + 342371237486 p T^{13} + 45980891822 p^{2} T^{14} + 5341839491 p^{3} T^{15} + 653345457 p^{4} T^{16} + 67478723 p^{5} T^{17} + 7401029 p^{6} T^{18} + 653495 p^{7} T^{19} + 62898 p^{8} T^{20} + 4351 p^{9} T^{21} + 358 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 57 T + 1840 T^{2} - 42776 T^{3} + 795422 T^{4} - 12466184 T^{5} + 170362544 T^{6} - 2075130991 T^{7} + 22885986895 T^{8} - 230900345818 T^{9} + 2146304741280 T^{10} - 18456805476680 T^{11} + 147179152577028 T^{12} - 18456805476680 p T^{13} + 2146304741280 p^{2} T^{14} - 230900345818 p^{3} T^{15} + 22885986895 p^{4} T^{16} - 2075130991 p^{5} T^{17} + 170362544 p^{6} T^{18} - 12466184 p^{7} T^{19} + 795422 p^{8} T^{20} - 42776 p^{9} T^{21} + 1840 p^{10} T^{22} - 57 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 37 T + 982 T^{2} + 18165 T^{3} + 282614 T^{4} + 3650435 T^{5} + 42633197 T^{6} + 447141745 T^{7} + 4456752749 T^{8} + 41572331405 T^{9} + 374436228198 T^{10} + 3164502283230 T^{11} + 25594652350183 T^{12} + 3164502283230 p T^{13} + 374436228198 p^{2} T^{14} + 41572331405 p^{3} T^{15} + 4456752749 p^{4} T^{16} + 447141745 p^{5} T^{17} + 42633197 p^{6} T^{18} + 3650435 p^{7} T^{19} + 282614 p^{8} T^{20} + 18165 p^{9} T^{21} + 982 p^{10} T^{22} + 37 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 33 T + 1052 T^{2} - 22431 T^{3} + 433275 T^{4} - 6947156 T^{5} + 101023175 T^{6} - 1304917001 T^{7} + 15414192020 T^{8} - 166359064357 T^{9} + 1653209627381 T^{10} - 15197454548230 T^{11} + 129107332141616 T^{12} - 15197454548230 p T^{13} + 1653209627381 p^{2} T^{14} - 166359064357 p^{3} T^{15} + 15414192020 p^{4} T^{16} - 1304917001 p^{5} T^{17} + 101023175 p^{6} T^{18} - 6947156 p^{7} T^{19} + 433275 p^{8} T^{20} - 22431 p^{9} T^{21} + 1052 p^{10} T^{22} - 33 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 21 T + 627 T^{2} + 8419 T^{3} + 150262 T^{4} + 1483792 T^{5} + 20670658 T^{6} + 159415087 T^{7} + 1990660841 T^{8} + 12586629513 T^{9} + 156673174899 T^{10} + 876700626424 T^{11} + 11342852436528 T^{12} + 876700626424 p T^{13} + 156673174899 p^{2} T^{14} + 12586629513 p^{3} T^{15} + 1990660841 p^{4} T^{16} + 159415087 p^{5} T^{17} + 20670658 p^{6} T^{18} + 1483792 p^{7} T^{19} + 150262 p^{8} T^{20} + 8419 p^{9} T^{21} + 627 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 13 T + 576 T^{2} + 7231 T^{3} + 171034 T^{4} + 1950581 T^{5} + 33455463 T^{6} + 341386059 T^{7} + 4721896193 T^{8} + 43028710845 T^{9} + 504274880064 T^{10} + 4087760062562 T^{11} + 41698609404305 T^{12} + 4087760062562 p T^{13} + 504274880064 p^{2} T^{14} + 43028710845 p^{3} T^{15} + 4721896193 p^{4} T^{16} + 341386059 p^{5} T^{17} + 33455463 p^{6} T^{18} + 1950581 p^{7} T^{19} + 171034 p^{8} T^{20} + 7231 p^{9} T^{21} + 576 p^{10} T^{22} + 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 32 T + 979 T^{2} + 19110 T^{3} + 348378 T^{4} + 5079072 T^{5} + 70298656 T^{6} + 850083936 T^{7} + 9924743393 T^{8} + 105720902932 T^{9} + 1091974777789 T^{10} + 10440902255886 T^{11} + 96370929231016 T^{12} + 10440902255886 p T^{13} + 1091974777789 p^{2} T^{14} + 105720902932 p^{3} T^{15} + 9924743393 p^{4} T^{16} + 850083936 p^{5} T^{17} + 70298656 p^{6} T^{18} + 5079072 p^{7} T^{19} + 348378 p^{8} T^{20} + 19110 p^{9} T^{21} + 979 p^{10} T^{22} + 32 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 48 T + 1554 T^{2} - 36863 T^{3} + 723173 T^{4} - 12058644 T^{5} + 177856439 T^{6} - 2353645307 T^{7} + 28531001648 T^{8} - 319638445562 T^{9} + 3353357614935 T^{10} - 33108730959672 T^{11} + 309891275940500 T^{12} - 33108730959672 p T^{13} + 3353357614935 p^{2} T^{14} - 319638445562 p^{3} T^{15} + 28531001648 p^{4} T^{16} - 2353645307 p^{5} T^{17} + 177856439 p^{6} T^{18} - 12058644 p^{7} T^{19} + 723173 p^{8} T^{20} - 36863 p^{9} T^{21} + 1554 p^{10} T^{22} - 48 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 20 T + 729 T^{2} + 13237 T^{3} + 277262 T^{4} + 4321247 T^{5} + 68608021 T^{6} + 921445853 T^{7} + 12097202594 T^{8} + 141677129602 T^{9} + 1596424147142 T^{10} + 16423711061358 T^{11} + 161599480537675 T^{12} + 16423711061358 p T^{13} + 1596424147142 p^{2} T^{14} + 141677129602 p^{3} T^{15} + 12097202594 p^{4} T^{16} + 921445853 p^{5} T^{17} + 68608021 p^{6} T^{18} + 4321247 p^{7} T^{19} + 277262 p^{8} T^{20} + 13237 p^{9} T^{21} + 729 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 7 T + 609 T^{2} + 1775 T^{3} + 158992 T^{4} - 105088 T^{5} + 26579864 T^{6} - 965683 p T^{7} + 3554340491 T^{8} - 18740187787 T^{9} + 408335116235 T^{10} - 2473633895328 T^{11} + 41490552185032 T^{12} - 2473633895328 p T^{13} + 408335116235 p^{2} T^{14} - 18740187787 p^{3} T^{15} + 3554340491 p^{4} T^{16} - 965683 p^{6} T^{17} + 26579864 p^{6} T^{18} - 105088 p^{7} T^{19} + 158992 p^{8} T^{20} + 1775 p^{9} T^{21} + 609 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.75120734963227578097727383696, −2.68366230938495228130943945404, −2.67520800664267799665931226151, −2.54632729522130284991383989405, −2.42651899367129654071121404048, −2.26591564288039428286288467608, −2.26555425088884962067170441784, −2.20372080447830081229164637467, −2.18838390567474996795934916234, −2.16149388477465111432826323380, −2.16057439767609111571418103136, −1.92044652169018754435997954438, −1.91098928316612124112654341759, −1.78542510457644736500499297471, −1.64534405833810060974367071570, −1.52676235099646070593468362459, −1.50572794262075710601128474495, −1.46680130656724277688989769003, −1.36902151384363430358741899070, −1.25973150933715641796560706708, −1.23320086872040611562686102322, −1.02797201721409320120210983825, −1.01721949721087926494841467918, −0.913695907106670338989976202569, −0.66050168958016725704426356687, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.66050168958016725704426356687, 0.913695907106670338989976202569, 1.01721949721087926494841467918, 1.02797201721409320120210983825, 1.23320086872040611562686102322, 1.25973150933715641796560706708, 1.36902151384363430358741899070, 1.46680130656724277688989769003, 1.50572794262075710601128474495, 1.52676235099646070593468362459, 1.64534405833810060974367071570, 1.78542510457644736500499297471, 1.91098928316612124112654341759, 1.92044652169018754435997954438, 2.16057439767609111571418103136, 2.16149388477465111432826323380, 2.18838390567474996795934916234, 2.20372080447830081229164637467, 2.26555425088884962067170441784, 2.26591564288039428286288467608, 2.42651899367129654071121404048, 2.54632729522130284991383989405, 2.67520800664267799665931226151, 2.68366230938495228130943945404, 2.75120734963227578097727383696
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.