Dirichlet series
| L(s) = 1 | − 2·2-s − 6·3-s + 9·4-s + 12·6-s − 2·7-s − 22·8-s − 2·9-s − 14·11-s − 54·12-s + 4·14-s + 57·16-s + 48·17-s + 4·18-s − 30·19-s + 12·21-s + 28·22-s − 14·23-s + 132·24-s + 15·25-s + 84·27-s − 18·28-s + 64·29-s + 132·31-s − 140·32-s + 84·33-s − 96·34-s − 18·36-s + ⋯ |
| L(s) = 1 | − 2-s − 2·3-s + 9/4·4-s + 2·6-s − 2/7·7-s − 2.75·8-s − 2/9·9-s − 1.27·11-s − 9/2·12-s + 2/7·14-s + 3.56·16-s + 2.82·17-s + 2/9·18-s − 1.57·19-s + 4/7·21-s + 1.27·22-s − 0.608·23-s + 11/2·24-s + 3/5·25-s + 28/9·27-s − 0.642·28-s + 2.20·29-s + 4.25·31-s − 4.37·32-s + 2.54·33-s − 2.82·34-s − 1/2·36-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(5^{12} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.566027\) |
| Root analytic conductor: | \(0.976565\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 5^{12} \cdot 7^{12} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.3479203783\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3479203783\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \) |
| 7 | \( 1 + 2 T + 2 p T^{2} + 348 T^{3} + 1588 T^{4} + 2062 p T^{5} + 2782 p^{2} T^{6} + 2062 p^{3} T^{7} + 1588 p^{4} T^{8} + 348 p^{6} T^{9} + 2 p^{9} T^{10} + 2 p^{10} T^{11} + p^{12} T^{12} \) | |
| good | 2 | \( 1 + p T - 5 T^{2} - 3 p T^{3} + 5 p^{2} T^{4} + 5 p T^{5} + 25 T^{6} + 29 p T^{7} - 109 p^{2} T^{8} - 235 p T^{9} + 1107 T^{10} - 51 p T^{11} - 5463 T^{12} - 51 p^{3} T^{13} + 1107 p^{4} T^{14} - 235 p^{7} T^{15} - 109 p^{10} T^{16} + 29 p^{11} T^{17} + 25 p^{12} T^{18} + 5 p^{15} T^{19} + 5 p^{18} T^{20} - 3 p^{19} T^{21} - 5 p^{20} T^{22} + p^{23} T^{23} + p^{24} T^{24} \) |
| 3 | \( 1 + 2 p T + 38 T^{2} + 52 p T^{3} + 220 p T^{4} + 230 p^{2} T^{5} + 5744 T^{6} + 382 p^{3} T^{7} - 68 T^{8} - 12692 p^{2} T^{9} - 8746 p^{4} T^{10} - 11218 p^{5} T^{11} - 114022 p^{4} T^{12} - 11218 p^{7} T^{13} - 8746 p^{8} T^{14} - 12692 p^{8} T^{15} - 68 p^{8} T^{16} + 382 p^{13} T^{17} + 5744 p^{12} T^{18} + 230 p^{16} T^{19} + 220 p^{17} T^{20} + 52 p^{19} T^{21} + 38 p^{20} T^{22} + 2 p^{23} T^{23} + p^{24} T^{24} \) | |
| 11 | \( 1 + 14 T - 179 T^{2} - 3378 T^{3} - 10510 T^{4} - 101510 T^{5} - 1940909 T^{6} + 49596994 T^{7} + 1461472640 T^{8} + 7377470974 T^{9} - 78163118391 T^{10} - 1070296414578 T^{11} - 9257941759776 T^{12} - 1070296414578 p^{2} T^{13} - 78163118391 p^{4} T^{14} + 7377470974 p^{6} T^{15} + 1461472640 p^{8} T^{16} + 49596994 p^{10} T^{17} - 1940909 p^{12} T^{18} - 101510 p^{14} T^{19} - 10510 p^{16} T^{20} - 3378 p^{18} T^{21} - 179 p^{20} T^{22} + 14 p^{22} T^{23} + p^{24} T^{24} \) | |
| 13 | \( 1 - 114 p T^{2} + 1042167 T^{4} - 464849386 T^{6} + 148108329555 T^{8} - 35857522447788 T^{10} + 6808838187253242 T^{12} - 35857522447788 p^{4} T^{14} + 148108329555 p^{8} T^{16} - 464849386 p^{12} T^{18} + 1042167 p^{16} T^{20} - 114 p^{21} T^{22} + p^{24} T^{24} \) | |
| 17 | \( 1 - 48 T + 2382 T^{2} - 77472 T^{3} + 2498625 T^{4} - 67135632 T^{5} + 1727446034 T^{6} - 40136784912 T^{7} + 880931070942 T^{8} - 18174432672288 T^{9} + 351810098240190 T^{10} - 6513275972033712 T^{11} + 112870694757995013 T^{12} - 6513275972033712 p^{2} T^{13} + 351810098240190 p^{4} T^{14} - 18174432672288 p^{6} T^{15} + 880931070942 p^{8} T^{16} - 40136784912 p^{10} T^{17} + 1727446034 p^{12} T^{18} - 67135632 p^{14} T^{19} + 2498625 p^{16} T^{20} - 77472 p^{18} T^{21} + 2382 p^{20} T^{22} - 48 p^{22} T^{23} + p^{24} T^{24} \) | |
| 19 | \( 1 + 30 T + 1401 T^{2} + 33030 T^{3} + 884826 T^{4} + 20001714 T^{5} + 344601839 T^{6} + 6453829386 T^{7} + 66537113172 T^{8} + 653188957326 T^{9} - 1006933743831 T^{10} - 384405046252650 T^{11} - 5242806750443784 T^{12} - 384405046252650 p^{2} T^{13} - 1006933743831 p^{4} T^{14} + 653188957326 p^{6} T^{15} + 66537113172 p^{8} T^{16} + 6453829386 p^{10} T^{17} + 344601839 p^{12} T^{18} + 20001714 p^{14} T^{19} + 884826 p^{16} T^{20} + 33030 p^{18} T^{21} + 1401 p^{20} T^{22} + 30 p^{22} T^{23} + p^{24} T^{24} \) | |
| 23 | \( 1 + 14 T - 1631 T^{2} - 3594 T^{3} + 1665797 T^{4} - 8196524 T^{5} - 890248094 T^{6} + 13524083944 T^{7} + 223091831501 T^{8} - 7643356730246 T^{9} + 95428737941649 T^{10} + 2006796517495686 T^{11} - 92105247923668542 T^{12} + 2006796517495686 p^{2} T^{13} + 95428737941649 p^{4} T^{14} - 7643356730246 p^{6} T^{15} + 223091831501 p^{8} T^{16} + 13524083944 p^{10} T^{17} - 890248094 p^{12} T^{18} - 8196524 p^{14} T^{19} + 1665797 p^{16} T^{20} - 3594 p^{18} T^{21} - 1631 p^{20} T^{22} + 14 p^{22} T^{23} + p^{24} T^{24} \) | |
| 29 | \( ( 1 - 32 T + 3552 T^{2} - 122256 T^{3} + 5916460 T^{4} - 196641728 T^{5} + 6117094942 T^{6} - 196641728 p^{2} T^{7} + 5916460 p^{4} T^{8} - 122256 p^{6} T^{9} + 3552 p^{8} T^{10} - 32 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 31 | \( 1 - 132 T + 12894 T^{2} - 935352 T^{3} + 58065489 T^{4} - 3050976288 T^{5} + 143999803298 T^{6} - 6064589395908 T^{7} + 236052211330302 T^{8} - 8471733336563436 T^{9} + 289433514790600350 T^{10} - 9384030507705444000 T^{11} + \)\(29\!\cdots\!29\)\( T^{12} - 9384030507705444000 p^{2} T^{13} + 289433514790600350 p^{4} T^{14} - 8471733336563436 p^{6} T^{15} + 236052211330302 p^{8} T^{16} - 6064589395908 p^{10} T^{17} + 143999803298 p^{12} T^{18} - 3050976288 p^{14} T^{19} + 58065489 p^{16} T^{20} - 935352 p^{18} T^{21} + 12894 p^{20} T^{22} - 132 p^{22} T^{23} + p^{24} T^{24} \) | |
| 37 | \( 1 - 44 T - 4291 T^{2} + 241124 T^{3} + 9668742 T^{4} - 689019636 T^{5} - 324999377 p T^{6} + 1280625867436 T^{7} + 4009330536896 T^{8} - 1538422889903564 T^{9} + 15464652696139049 T^{10} + 849812472525660964 T^{11} - 34333638656751212332 T^{12} + 849812472525660964 p^{2} T^{13} + 15464652696139049 p^{4} T^{14} - 1538422889903564 p^{6} T^{15} + 4009330536896 p^{8} T^{16} + 1280625867436 p^{10} T^{17} - 324999377 p^{13} T^{18} - 689019636 p^{14} T^{19} + 9668742 p^{16} T^{20} + 241124 p^{18} T^{21} - 4291 p^{20} T^{22} - 44 p^{22} T^{23} + p^{24} T^{24} \) | |
| 41 | \( 1 - 6978 T^{2} + 29395173 T^{4} - 89601849670 T^{6} + 218675773246986 T^{8} - 448976563577597898 T^{10} + \)\(80\!\cdots\!77\)\( T^{12} - 448976563577597898 p^{4} T^{14} + 218675773246986 p^{8} T^{16} - 89601849670 p^{12} T^{18} + 29395173 p^{16} T^{20} - 6978 p^{20} T^{22} + p^{24} T^{24} \) | |
| 43 | \( ( 1 + 2 T + 7538 T^{2} + 1596 T^{3} + 27008008 T^{4} - 11306606 T^{5} + 60967237342 T^{6} - 11306606 p^{2} T^{7} + 27008008 p^{4} T^{8} + 1596 p^{6} T^{9} + 7538 p^{8} T^{10} + 2 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 47 | \( 1 - 204 T + 29397 T^{2} - 3167100 T^{3} + 290418402 T^{4} - 23370921444 T^{5} + 1705985151887 T^{6} - 114387377070276 T^{7} + 7121727778488432 T^{8} - 413844362647971516 T^{9} + 22564123030969504557 T^{10} - \)\(11\!\cdots\!96\)\( T^{11} + \)\(56\!\cdots\!04\)\( T^{12} - \)\(11\!\cdots\!96\)\( p^{2} T^{13} + 22564123030969504557 p^{4} T^{14} - 413844362647971516 p^{6} T^{15} + 7121727778488432 p^{8} T^{16} - 114387377070276 p^{10} T^{17} + 1705985151887 p^{12} T^{18} - 23370921444 p^{14} T^{19} + 290418402 p^{16} T^{20} - 3167100 p^{18} T^{21} + 29397 p^{20} T^{22} - 204 p^{22} T^{23} + p^{24} T^{24} \) | |
| 53 | \( 1 - 196 T + 13213 T^{2} - 531828 T^{3} + 46624490 T^{4} - 3460227932 T^{5} + 114727083151 T^{6} - 3187972234364 T^{7} + 109323786787880 T^{8} + 2196815143901836 T^{9} - 301463170054474635 T^{10} + 37355670780781789692 T^{11} - \)\(32\!\cdots\!76\)\( T^{12} + 37355670780781789692 p^{2} T^{13} - 301463170054474635 p^{4} T^{14} + 2196815143901836 p^{6} T^{15} + 109323786787880 p^{8} T^{16} - 3187972234364 p^{10} T^{17} + 114727083151 p^{12} T^{18} - 3460227932 p^{14} T^{19} + 46624490 p^{16} T^{20} - 531828 p^{18} T^{21} + 13213 p^{20} T^{22} - 196 p^{22} T^{23} + p^{24} T^{24} \) | |
| 59 | \( 1 - 72 T + 17682 T^{2} - 1148688 T^{3} + 164787345 T^{4} - 10566780840 T^{5} + 1093761678926 T^{6} - 70159293997992 T^{7} + 5748831154849182 T^{8} - 368559907967058624 T^{9} + 25324575120331389906 T^{10} - 26585041793186477592 p T^{11} + 27297799098536813853 p^{2} T^{12} - 26585041793186477592 p^{3} T^{13} + 25324575120331389906 p^{4} T^{14} - 368559907967058624 p^{6} T^{15} + 5748831154849182 p^{8} T^{16} - 70159293997992 p^{10} T^{17} + 1093761678926 p^{12} T^{18} - 10566780840 p^{14} T^{19} + 164787345 p^{16} T^{20} - 1148688 p^{18} T^{21} + 17682 p^{20} T^{22} - 72 p^{22} T^{23} + p^{24} T^{24} \) | |
| 61 | \( 1 - 72 T + 19176 T^{2} - 1256256 T^{3} + 196053300 T^{4} - 13507864560 T^{5} + 1475851108916 T^{6} - 102662314431264 T^{7} + 8714109428620692 T^{8} - 590990079276027360 T^{9} + 42479838876985802160 T^{10} - \)\(27\!\cdots\!84\)\( T^{11} + \)\(17\!\cdots\!82\)\( T^{12} - \)\(27\!\cdots\!84\)\( p^{2} T^{13} + 42479838876985802160 p^{4} T^{14} - 590990079276027360 p^{6} T^{15} + 8714109428620692 p^{8} T^{16} - 102662314431264 p^{10} T^{17} + 1475851108916 p^{12} T^{18} - 13507864560 p^{14} T^{19} + 196053300 p^{16} T^{20} - 1256256 p^{18} T^{21} + 19176 p^{20} T^{22} - 72 p^{22} T^{23} + p^{24} T^{24} \) | |
| 67 | \( 1 + 138 T - 9270 T^{2} - 1437548 T^{3} + 115886436 T^{4} + 10557292578 T^{5} - 1139954744456 T^{6} - 58817128757610 T^{7} + 8471104062197196 T^{8} + 229786064617296868 T^{9} - 50182738990417249110 T^{10} - \)\(37\!\cdots\!50\)\( T^{11} + \)\(25\!\cdots\!50\)\( T^{12} - \)\(37\!\cdots\!50\)\( p^{2} T^{13} - 50182738990417249110 p^{4} T^{14} + 229786064617296868 p^{6} T^{15} + 8471104062197196 p^{8} T^{16} - 58817128757610 p^{10} T^{17} - 1139954744456 p^{12} T^{18} + 10557292578 p^{14} T^{19} + 115886436 p^{16} T^{20} - 1437548 p^{18} T^{21} - 9270 p^{20} T^{22} + 138 p^{22} T^{23} + p^{24} T^{24} \) | |
| 71 | \( ( 1 + 4 T + 14826 T^{2} - 442692 T^{3} + 122154931 T^{4} - 3956292368 T^{5} + 781022121076 T^{6} - 3956292368 p^{2} T^{7} + 122154931 p^{4} T^{8} - 442692 p^{6} T^{9} + 14826 p^{8} T^{10} + 4 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 73 | \( 1 + 528 T + 147318 T^{2} + 28717920 T^{3} + 4362863697 T^{4} + 544924287120 T^{5} + 57462036213962 T^{6} + 5186233581298704 T^{7} + 403863565075955790 T^{8} + 27374688378441994752 T^{9} + \)\(16\!\cdots\!86\)\( T^{10} + \)\(95\!\cdots\!24\)\( T^{11} + \)\(62\!\cdots\!73\)\( T^{12} + \)\(95\!\cdots\!24\)\( p^{2} T^{13} + \)\(16\!\cdots\!86\)\( p^{4} T^{14} + 27374688378441994752 p^{6} T^{15} + 403863565075955790 p^{8} T^{16} + 5186233581298704 p^{10} T^{17} + 57462036213962 p^{12} T^{18} + 544924287120 p^{14} T^{19} + 4362863697 p^{16} T^{20} + 28717920 p^{18} T^{21} + 147318 p^{20} T^{22} + 528 p^{22} T^{23} + p^{24} T^{24} \) | |
| 79 | \( 1 + 12 T - 26574 T^{2} - 366848 T^{3} + 390887217 T^{4} + 5621569728 T^{5} - 3680496208466 T^{6} - 54811745357412 T^{7} + 24735046819850526 T^{8} + 328584508359816652 T^{9} - \)\(13\!\cdots\!54\)\( T^{10} - \)\(88\!\cdots\!28\)\( T^{11} + \)\(71\!\cdots\!73\)\( T^{12} - \)\(88\!\cdots\!28\)\( p^{2} T^{13} - \)\(13\!\cdots\!54\)\( p^{4} T^{14} + 328584508359816652 p^{6} T^{15} + 24735046819850526 p^{8} T^{16} - 54811745357412 p^{10} T^{17} - 3680496208466 p^{12} T^{18} + 5621569728 p^{14} T^{19} + 390887217 p^{16} T^{20} - 366848 p^{18} T^{21} - 26574 p^{20} T^{22} + 12 p^{22} T^{23} + p^{24} T^{24} \) | |
| 83 | \( 1 - 43200 T^{2} + 987057204 T^{4} - 15370870045228 T^{6} + 180607989502531968 T^{8} - \)\(16\!\cdots\!36\)\( T^{10} + \)\(12\!\cdots\!66\)\( T^{12} - \)\(16\!\cdots\!36\)\( p^{4} T^{14} + 180607989502531968 p^{8} T^{16} - 15370870045228 p^{12} T^{18} + 987057204 p^{16} T^{20} - 43200 p^{20} T^{22} + p^{24} T^{24} \) | |
| 89 | \( 1 - 204 T + 49104 T^{2} - 7187328 T^{3} + 1045283496 T^{4} - 127643879244 T^{5} + 15255527568428 T^{6} - 1684635845896692 T^{7} + 183174646224057864 T^{8} - 18563328978636389184 T^{9} + \)\(18\!\cdots\!76\)\( T^{10} - \)\(17\!\cdots\!52\)\( T^{11} + \)\(16\!\cdots\!74\)\( T^{12} - \)\(17\!\cdots\!52\)\( p^{2} T^{13} + \)\(18\!\cdots\!76\)\( p^{4} T^{14} - 18563328978636389184 p^{6} T^{15} + 183174646224057864 p^{8} T^{16} - 1684635845896692 p^{10} T^{17} + 15255527568428 p^{12} T^{18} - 127643879244 p^{14} T^{19} + 1045283496 p^{16} T^{20} - 7187328 p^{18} T^{21} + 49104 p^{20} T^{22} - 204 p^{22} T^{23} + p^{24} T^{24} \) | |
| 97 | \( 1 - 64284 T^{2} + 2051576418 T^{4} - 44019752488108 T^{6} + 711686325387159663 T^{8} - \)\(91\!\cdots\!92\)\( T^{10} + \)\(94\!\cdots\!68\)\( T^{12} - \)\(91\!\cdots\!92\)\( p^{4} T^{14} + 711686325387159663 p^{8} T^{16} - 44019752488108 p^{12} T^{18} + 2051576418 p^{16} T^{20} - 64284 p^{20} T^{22} + p^{24} T^{24} \) | |
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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
−6.28568831326878156375543387186, −6.20692470184162015125425576547, −5.85773713791331389985062891464, −5.81648850069236098402121624755, −5.79532006443254472280295179654, −5.72381244159281797793558866364, −5.71763165968427736534948983525, −5.27035878004779938526391022854, −5.19776355699938965592974587622, −5.19518137808725939525715956849, −4.64762650962743563804669559187, −4.54741080603320411772198223109, −4.48874981715854331686926267498, −4.24826913804754478942692652722, −4.13109165277238082418957881702, −3.66085429610605330003981008529, −3.48534140172571030409630816687, −3.12022686675335999537069427628, −2.93283753868484626913008327884, −2.74594152394609113460031098320, −2.65955873603944789062487385177, −2.45300743904323791433935688585, −2.28033796020376678385424390076, −1.14791504190157114832869762290, −0.867534250729885271776125483445, 0.867534250729885271776125483445, 1.14791504190157114832869762290, 2.28033796020376678385424390076, 2.45300743904323791433935688585, 2.65955873603944789062487385177, 2.74594152394609113460031098320, 2.93283753868484626913008327884, 3.12022686675335999537069427628, 3.48534140172571030409630816687, 3.66085429610605330003981008529, 4.13109165277238082418957881702, 4.24826913804754478942692652722, 4.48874981715854331686926267498, 4.54741080603320411772198223109, 4.64762650962743563804669559187, 5.19518137808725939525715956849, 5.19776355699938965592974587622, 5.27035878004779938526391022854, 5.71763165968427736534948983525, 5.72381244159281797793558866364, 5.79532006443254472280295179654, 5.81648850069236098402121624755, 5.85773713791331389985062891464, 6.20692470184162015125425576547, 6.28568831326878156375543387186
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.