Dirichlet series
| L(s) = 1 | + 12·2-s − 2·3-s + 78·4-s − 12·5-s − 24·6-s − 9·7-s + 364·8-s − 16·9-s − 144·10-s + 11-s − 156·12-s − 6·13-s − 108·14-s + 24·15-s + 1.36e3·16-s − 11·17-s − 192·18-s − 13·19-s − 936·20-s + 18·21-s + 12·22-s − 21·23-s − 728·24-s + 78·25-s − 72·26-s + 36·27-s − 702·28-s + ⋯ |
| L(s) = 1 | + 8.48·2-s − 1.15·3-s + 39·4-s − 5.36·5-s − 9.79·6-s − 3.40·7-s + 128.·8-s − 5.33·9-s − 45.5·10-s + 0.301·11-s − 45.0·12-s − 1.66·13-s − 28.8·14-s + 6.19·15-s + 341.·16-s − 2.66·17-s − 45.2·18-s − 2.98·19-s − 209.·20-s + 3.92·21-s + 2.55·22-s − 4.37·23-s − 148.·24-s + 78/5·25-s − 14.1·26-s + 6.92·27-s − 132.·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 401^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 401^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 5^{12} \cdot 401^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.16160\times 10^{18}\) |
| Root analytic conductor: | \(5.65862\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 2^{12} \cdot 5^{12} \cdot 401^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
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 | \( ( 1 - T )^{12} \) |
| 5 | \( ( 1 + T )^{12} \) | |
| 401 | \( ( 1 - T )^{12} \) | |
| good | 3 | \( 1 + 2 T + 20 T^{2} + 4 p^{2} T^{3} + 23 p^{2} T^{4} + 38 p^{2} T^{5} + 1454 T^{6} + 2201 T^{7} + 2549 p T^{8} + 3527 p T^{9} + 10540 p T^{10} + 13268 p T^{11} + 105310 T^{12} + 13268 p^{2} T^{13} + 10540 p^{3} T^{14} + 3527 p^{4} T^{15} + 2549 p^{5} T^{16} + 2201 p^{5} T^{17} + 1454 p^{6} T^{18} + 38 p^{9} T^{19} + 23 p^{10} T^{20} + 4 p^{11} T^{21} + 20 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 + 9 T + 12 p T^{2} + 507 T^{3} + 415 p T^{4} + 13423 T^{5} + 58623 T^{6} + 222202 T^{7} + 801881 T^{8} + 2601035 T^{9} + 8104477 T^{10} + 23118678 T^{11} + 63732314 T^{12} + 23118678 p T^{13} + 8104477 p^{2} T^{14} + 2601035 p^{3} T^{15} + 801881 p^{4} T^{16} + 222202 p^{5} T^{17} + 58623 p^{6} T^{18} + 13423 p^{7} T^{19} + 415 p^{9} T^{20} + 507 p^{9} T^{21} + 12 p^{11} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - T + 74 T^{2} - 101 T^{3} + 2792 T^{4} - 4233 T^{5} + 70930 T^{6} - 9963 p T^{7} + 1345312 T^{8} - 2025517 T^{9} + 166338 p^{2} T^{10} - 28468225 T^{11} + 244570862 T^{12} - 28468225 p T^{13} + 166338 p^{4} T^{14} - 2025517 p^{3} T^{15} + 1345312 p^{4} T^{16} - 9963 p^{6} T^{17} + 70930 p^{6} T^{18} - 4233 p^{7} T^{19} + 2792 p^{8} T^{20} - 101 p^{9} T^{21} + 74 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 6 T + 132 T^{2} + 705 T^{3} + 8226 T^{4} + 39361 T^{5} + 320360 T^{6} + 1372984 T^{7} + 8683928 T^{8} + 2551763 p T^{9} + 172580900 T^{10} + 582177315 T^{11} + 2578069132 T^{12} + 582177315 p T^{13} + 172580900 p^{2} T^{14} + 2551763 p^{4} T^{15} + 8683928 p^{4} T^{16} + 1372984 p^{5} T^{17} + 320360 p^{6} T^{18} + 39361 p^{7} T^{19} + 8226 p^{8} T^{20} + 705 p^{9} T^{21} + 132 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 11 T + 162 T^{2} + 1315 T^{3} + 11926 T^{4} + 78747 T^{5} + 549770 T^{6} + 3089744 T^{7} + 17978263 T^{8} + 88128905 T^{9} + 443487026 T^{10} + 1920230718 T^{11} + 8509135824 T^{12} + 1920230718 p T^{13} + 443487026 p^{2} T^{14} + 88128905 p^{3} T^{15} + 17978263 p^{4} T^{16} + 3089744 p^{5} T^{17} + 549770 p^{6} T^{18} + 78747 p^{7} T^{19} + 11926 p^{8} T^{20} + 1315 p^{9} T^{21} + 162 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 13 T + 196 T^{2} + 1734 T^{3} + 16020 T^{4} + 111826 T^{5} + 798253 T^{6} + 4700466 T^{7} + 28178827 T^{8} + 144966631 T^{9} + 758572667 T^{10} + 3468912550 T^{11} + 16130954328 T^{12} + 3468912550 p T^{13} + 758572667 p^{2} T^{14} + 144966631 p^{3} T^{15} + 28178827 p^{4} T^{16} + 4700466 p^{5} T^{17} + 798253 p^{6} T^{18} + 111826 p^{7} T^{19} + 16020 p^{8} T^{20} + 1734 p^{9} T^{21} + 196 p^{10} T^{22} + 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 21 T + 349 T^{2} + 4069 T^{3} + 41236 T^{4} + 351240 T^{5} + 2716341 T^{6} + 18804819 T^{7} + 121076813 T^{8} + 718665218 T^{9} + 4018862412 T^{10} + 21002818223 T^{11} + 103985618488 T^{12} + 21002818223 p T^{13} + 4018862412 p^{2} T^{14} + 718665218 p^{3} T^{15} + 121076813 p^{4} T^{16} + 18804819 p^{5} T^{17} + 2716341 p^{6} T^{18} + 351240 p^{7} T^{19} + 41236 p^{8} T^{20} + 4069 p^{9} T^{21} + 349 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 10 T + 9 p T^{2} + 1966 T^{3} + 30097 T^{4} + 183996 T^{5} + 2161060 T^{6} + 11255937 T^{7} + 112103665 T^{8} + 513250943 T^{9} + 4517680475 T^{10} + 18482818840 T^{11} + 145826528258 T^{12} + 18482818840 p T^{13} + 4517680475 p^{2} T^{14} + 513250943 p^{3} T^{15} + 112103665 p^{4} T^{16} + 11255937 p^{5} T^{17} + 2161060 p^{6} T^{18} + 183996 p^{7} T^{19} + 30097 p^{8} T^{20} + 1966 p^{9} T^{21} + 9 p^{11} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 11 T + 227 T^{2} + 1615 T^{3} + 20900 T^{4} + 112977 T^{5} + 1271977 T^{6} + 5946911 T^{7} + 62721163 T^{8} + 260661987 T^{9} + 2517960900 T^{10} + 9282982813 T^{11} + 83977696464 T^{12} + 9282982813 p T^{13} + 2517960900 p^{2} T^{14} + 260661987 p^{3} T^{15} + 62721163 p^{4} T^{16} + 5946911 p^{5} T^{17} + 1271977 p^{6} T^{18} + 112977 p^{7} T^{19} + 20900 p^{8} T^{20} + 1615 p^{9} T^{21} + 227 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 29 T + 634 T^{2} + 10187 T^{3} + 139602 T^{4} + 1636549 T^{5} + 17180810 T^{6} + 161860688 T^{7} + 1397336497 T^{8} + 11054079853 T^{9} + 81045685614 T^{10} + 549653977234 T^{11} + 3471535946572 T^{12} + 549653977234 p T^{13} + 81045685614 p^{2} T^{14} + 11054079853 p^{3} T^{15} + 1397336497 p^{4} T^{16} + 161860688 p^{5} T^{17} + 17180810 p^{6} T^{18} + 1636549 p^{7} T^{19} + 139602 p^{8} T^{20} + 10187 p^{9} T^{21} + 634 p^{10} T^{22} + 29 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + T + 227 T^{2} + 611 T^{3} + 26416 T^{4} + 110933 T^{5} + 2147305 T^{6} + 11550595 T^{7} + 136264421 T^{8} + 830907243 T^{9} + 7141720354 T^{10} + 44376965029 T^{11} + 317294625780 T^{12} + 44376965029 p T^{13} + 7141720354 p^{2} T^{14} + 830907243 p^{3} T^{15} + 136264421 p^{4} T^{16} + 11550595 p^{5} T^{17} + 2147305 p^{6} T^{18} + 110933 p^{7} T^{19} + 26416 p^{8} T^{20} + 611 p^{9} T^{21} + 227 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 23 T + 710 T^{2} + 11408 T^{3} + 202944 T^{4} + 2518327 T^{5} + 32679638 T^{6} + 329144290 T^{7} + 3389916092 T^{8} + 28462469768 T^{9} + 242020145644 T^{10} + 1716470265728 T^{11} + 12282801803942 T^{12} + 1716470265728 p T^{13} + 242020145644 p^{2} T^{14} + 28462469768 p^{3} T^{15} + 3389916092 p^{4} T^{16} + 329144290 p^{5} T^{17} + 32679638 p^{6} T^{18} + 2518327 p^{7} T^{19} + 202944 p^{8} T^{20} + 11408 p^{9} T^{21} + 710 p^{10} T^{22} + 23 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 17 T + 392 T^{2} + 4877 T^{3} + 68596 T^{4} + 711925 T^{5} + 7758454 T^{6} + 70749814 T^{7} + 647419321 T^{8} + 5286513139 T^{9} + 42174957618 T^{10} + 310368551078 T^{11} + 2203921132484 T^{12} + 310368551078 p T^{13} + 42174957618 p^{2} T^{14} + 5286513139 p^{3} T^{15} + 647419321 p^{4} T^{16} + 70749814 p^{5} T^{17} + 7758454 p^{6} T^{18} + 711925 p^{7} T^{19} + 68596 p^{8} T^{20} + 4877 p^{9} T^{21} + 392 p^{10} T^{22} + 17 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 47 T + 1464 T^{2} + 32355 T^{3} + 578376 T^{4} + 8518229 T^{5} + 108098757 T^{6} + 1193284165 T^{7} + 11776836649 T^{8} + 104961651929 T^{9} + 867297393463 T^{10} + 6724540398031 T^{11} + 50010776341666 T^{12} + 6724540398031 p T^{13} + 867297393463 p^{2} T^{14} + 104961651929 p^{3} T^{15} + 11776836649 p^{4} T^{16} + 1193284165 p^{5} T^{17} + 108098757 p^{6} T^{18} + 8518229 p^{7} T^{19} + 578376 p^{8} T^{20} + 32355 p^{9} T^{21} + 1464 p^{10} T^{22} + 47 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 14 T + 524 T^{2} - 6332 T^{3} + 130961 T^{4} - 1388671 T^{5} + 20825935 T^{6} - 196142214 T^{7} + 2364188051 T^{8} - 19891125494 T^{9} + 202847297785 T^{10} - 1522847324069 T^{11} + 13522000135294 T^{12} - 1522847324069 p T^{13} + 202847297785 p^{2} T^{14} - 19891125494 p^{3} T^{15} + 2364188051 p^{4} T^{16} - 196142214 p^{5} T^{17} + 20825935 p^{6} T^{18} - 1388671 p^{7} T^{19} + 130961 p^{8} T^{20} - 6332 p^{9} T^{21} + 524 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 22 T + 747 T^{2} + 12648 T^{3} + 249186 T^{4} + 3419700 T^{5} + 49881736 T^{6} + 572715499 T^{7} + 6736313418 T^{8} + 65910665846 T^{9} + 649967327993 T^{10} + 5468019382925 T^{11} + 46067190229586 T^{12} + 5468019382925 p T^{13} + 649967327993 p^{2} T^{14} + 65910665846 p^{3} T^{15} + 6736313418 p^{4} T^{16} + 572715499 p^{5} T^{17} + 49881736 p^{6} T^{18} + 3419700 p^{7} T^{19} + 249186 p^{8} T^{20} + 12648 p^{9} T^{21} + 747 p^{10} T^{22} + 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 28 T + 689 T^{2} + 10891 T^{3} + 161345 T^{4} + 1947730 T^{5} + 23126622 T^{6} + 246374126 T^{7} + 2578192547 T^{8} + 24871330110 T^{9} + 231020161059 T^{10} + 2012513191251 T^{11} + 16898953980082 T^{12} + 2012513191251 p T^{13} + 231020161059 p^{2} T^{14} + 24871330110 p^{3} T^{15} + 2578192547 p^{4} T^{16} + 246374126 p^{5} T^{17} + 23126622 p^{6} T^{18} + 1947730 p^{7} T^{19} + 161345 p^{8} T^{20} + 10891 p^{9} T^{21} + 689 p^{10} T^{22} + 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 18 T + 377 T^{2} + 4677 T^{3} + 69343 T^{4} + 736011 T^{5} + 8984892 T^{6} + 86228572 T^{7} + 948563309 T^{8} + 8397539729 T^{9} + 83744230371 T^{10} + 690029166881 T^{11} + 6400390234166 T^{12} + 690029166881 p T^{13} + 83744230371 p^{2} T^{14} + 8397539729 p^{3} T^{15} + 948563309 p^{4} T^{16} + 86228572 p^{5} T^{17} + 8984892 p^{6} T^{18} + 736011 p^{7} T^{19} + 69343 p^{8} T^{20} + 4677 p^{9} T^{21} + 377 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 2 T + 607 T^{2} + 546 T^{3} + 180491 T^{4} + 38574 T^{5} + 34940912 T^{6} - 6463501 T^{7} + 4890792591 T^{8} - 1878185724 T^{9} + 519673391213 T^{10} - 228111031597 T^{11} + 42893697332658 T^{12} - 228111031597 p T^{13} + 519673391213 p^{2} T^{14} - 1878185724 p^{3} T^{15} + 4890792591 p^{4} T^{16} - 6463501 p^{5} T^{17} + 34940912 p^{6} T^{18} + 38574 p^{7} T^{19} + 180491 p^{8} T^{20} + 546 p^{9} T^{21} + 607 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 39 T + 1280 T^{2} + 28792 T^{3} + 576222 T^{4} + 9524009 T^{5} + 144520164 T^{6} + 1929852318 T^{7} + 24089980562 T^{8} + 272304248612 T^{9} + 2904521634968 T^{10} + 28411976889090 T^{11} + 263234863218358 T^{12} + 28411976889090 p T^{13} + 2904521634968 p^{2} T^{14} + 272304248612 p^{3} T^{15} + 24089980562 p^{4} T^{16} + 1929852318 p^{5} T^{17} + 144520164 p^{6} T^{18} + 9524009 p^{7} T^{19} + 576222 p^{8} T^{20} + 28792 p^{9} T^{21} + 1280 p^{10} T^{22} + 39 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 5 T + 320 T^{2} + 2558 T^{3} + 65050 T^{4} + 598605 T^{5} + 10645776 T^{6} + 95974594 T^{7} + 1402228546 T^{8} + 143562174 p T^{9} + 152449757362 T^{10} + 1201994024908 T^{11} + 13850165462714 T^{12} + 1201994024908 p T^{13} + 152449757362 p^{2} T^{14} + 143562174 p^{4} T^{15} + 1402228546 p^{4} T^{16} + 95974594 p^{5} T^{17} + 10645776 p^{6} T^{18} + 598605 p^{7} T^{19} + 65050 p^{8} T^{20} + 2558 p^{9} T^{21} + 320 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 8 T + 514 T^{2} + 4157 T^{3} + 136934 T^{4} + 1110429 T^{5} + 25021642 T^{6} + 201108030 T^{7} + 3508301548 T^{8} + 27557573181 T^{9} + 400998482658 T^{10} + 3009007792643 T^{11} + 38641229406062 T^{12} + 3009007792643 p T^{13} + 400998482658 p^{2} T^{14} + 27557573181 p^{3} T^{15} + 3508301548 p^{4} T^{16} + 201108030 p^{5} T^{17} + 25021642 p^{6} T^{18} + 1110429 p^{7} T^{19} + 136934 p^{8} T^{20} + 4157 p^{9} T^{21} + 514 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 32 T + 1110 T^{2} + 23592 T^{3} + 5139 p T^{4} + 8248923 T^{5} + 133984853 T^{6} + 1854692468 T^{7} + 25171997923 T^{8} + 301987005962 T^{9} + 3554048841815 T^{10} + 37560019017727 T^{11} + 389515614831966 T^{12} + 37560019017727 p T^{13} + 3554048841815 p^{2} T^{14} + 301987005962 p^{3} T^{15} + 25171997923 p^{4} T^{16} + 1854692468 p^{5} T^{17} + 133984853 p^{6} T^{18} + 8248923 p^{7} T^{19} + 5139 p^{9} T^{20} + 23592 p^{9} T^{21} + 1110 p^{10} T^{22} + 32 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.16365345447012397856640903380, −2.85851737851110432336270136014, −2.85591668879680225001431037712, −2.79406383920401352230444910294, −2.79242793107429401642970400228, −2.73512803582452535536804666957, −2.70518396063675799109818462275, −2.70307577274699567633661052600, −2.66671076205377683897196060629, −2.61679890559348993466804564039, −2.61474193370765391059020731105, −2.54681619154558301097024768079, −2.32209115942569046715072934994, −1.95868328364000158724264878832, −1.91738838178545038911104912942, −1.87759096507658055105667345235, −1.86878461584591449843592520213, −1.70341702849131624979417957290, −1.65480223416249292098120722626, −1.61258298493836190456644654992, −1.50338848745475014967223559464, −1.34174244843812120751505487476, −1.27708684416439217718869980421, −1.27080034839403260274910062598, −1.20580773751597064168460785300, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.20580773751597064168460785300, 1.27080034839403260274910062598, 1.27708684416439217718869980421, 1.34174244843812120751505487476, 1.50338848745475014967223559464, 1.61258298493836190456644654992, 1.65480223416249292098120722626, 1.70341702849131624979417957290, 1.86878461584591449843592520213, 1.87759096507658055105667345235, 1.91738838178545038911104912942, 1.95868328364000158724264878832, 2.32209115942569046715072934994, 2.54681619154558301097024768079, 2.61474193370765391059020731105, 2.61679890559348993466804564039, 2.66671076205377683897196060629, 2.70307577274699567633661052600, 2.70518396063675799109818462275, 2.73512803582452535536804666957, 2.79242793107429401642970400228, 2.79406383920401352230444910294, 2.85591668879680225001431037712, 2.85851737851110432336270136014, 3.16365345447012397856640903380
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.