Dirichlet series
| L(s) = 1 | − 3.72e8·3-s + 4.20e12·4-s − 9.43e16·7-s − 1.17e19·9-s − 1.56e21·12-s + 7.04e20·13-s + 9.04e24·16-s + 3.76e25·19-s + 3.51e25·21-s + 5.89e28·25-s − 1.91e28·27-s − 3.96e29·28-s − 9.14e29·31-s − 4.93e31·36-s + 5.67e30·37-s − 2.62e29·39-s + 1.01e33·43-s − 3.36e33·48-s − 2.95e34·49-s + 2.96e33·52-s − 1.39e34·57-s − 4.94e35·61-s + 1.10e36·63-s + 1.33e37·64-s + 5.40e36·67-s + 3.00e37·73-s − 2.19e37·75-s + ⋯ |
| L(s) = 1 | − 0.106·3-s + 3.82·4-s − 1.18·7-s − 0.965·9-s − 0.408·12-s + 0.0370·13-s + 7.48·16-s + 1.00·19-s + 0.126·21-s + 6.48·25-s − 0.452·27-s − 4.52·28-s − 1.36·31-s − 3.69·36-s + 0.245·37-s − 0.00395·39-s + 2.16·43-s − 0.798·48-s − 4.63·49-s + 0.141·52-s − 0.106·57-s − 0.971·61-s + 1.14·63-s + 10.0·64-s + 1.62·67-s + 1.62·73-s − 0.692·75-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr =\mathstrut & \, \Lambda(41-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s+20)^{12} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(531441\) = \(3^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.23653\times 10^{17}\) |
| Root analytic conductor: | \(5.51386\) |
| Motivic weight: | \(40\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 531441,\ (\ :[20]^{12}),\ 1)\) |
Particular Values
| \(L(\frac{41}{2})\) | \(\approx\) | \(41.94578963\) |
| \(L(\frac12)\) | \(\approx\) | \(41.94578963\) |
| \(L(21)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 + 1531204 p^{5} T + 67017639876566 p^{11} T^{2} + 72202019129683173376 p^{18} T^{3} - \)\(63\!\cdots\!64\)\( p^{31} T^{4} + \)\(25\!\cdots\!96\)\( p^{46} T^{5} - \)\(48\!\cdots\!60\)\( p^{64} T^{6} + \)\(25\!\cdots\!96\)\( p^{86} T^{7} - \)\(63\!\cdots\!64\)\( p^{111} T^{8} + 72202019129683173376 p^{138} T^{9} + 67017639876566 p^{171} T^{10} + 1531204 p^{205} T^{11} + p^{240} T^{12} \) |
| good | 2 | \( 1 - 525797867265 p^{3} T^{2} + \)\(84\!\cdots\!71\)\( p^{10} T^{4} - \)\(13\!\cdots\!55\)\( p^{23} T^{6} + \)\(19\!\cdots\!25\)\( p^{39} T^{8} - \)\(24\!\cdots\!85\)\( p^{58} T^{10} + \)\(18\!\cdots\!95\)\( p^{81} T^{12} - \)\(24\!\cdots\!85\)\( p^{138} T^{14} + \)\(19\!\cdots\!25\)\( p^{199} T^{16} - \)\(13\!\cdots\!55\)\( p^{263} T^{18} + \)\(84\!\cdots\!71\)\( p^{330} T^{20} - 525797867265 p^{403} T^{22} + p^{480} T^{24} \) |
| 5 | \( 1 - \)\(23\!\cdots\!44\)\( p^{2} T^{2} + \)\(53\!\cdots\!06\)\( p^{5} T^{4} - \)\(15\!\cdots\!84\)\( p^{9} T^{6} + \)\(11\!\cdots\!31\)\( p^{18} T^{8} - \)\(12\!\cdots\!04\)\( p^{28} T^{10} + \)\(25\!\cdots\!76\)\( p^{39} T^{12} - \)\(12\!\cdots\!04\)\( p^{108} T^{14} + \)\(11\!\cdots\!31\)\( p^{178} T^{16} - \)\(15\!\cdots\!84\)\( p^{249} T^{18} + \)\(53\!\cdots\!06\)\( p^{325} T^{20} - \)\(23\!\cdots\!44\)\( p^{402} T^{22} + p^{480} T^{24} \) | |
| 7 | \( ( 1 + 6738696366054756 p T + \)\(52\!\cdots\!34\)\( p^{3} T^{2} + \)\(32\!\cdots\!12\)\( p^{4} T^{3} + \)\(14\!\cdots\!11\)\( p^{6} T^{4} + \)\(19\!\cdots\!24\)\( p^{10} T^{5} + \)\(26\!\cdots\!92\)\( p^{15} T^{6} + \)\(19\!\cdots\!24\)\( p^{50} T^{7} + \)\(14\!\cdots\!11\)\( p^{86} T^{8} + \)\(32\!\cdots\!12\)\( p^{124} T^{9} + \)\(52\!\cdots\!34\)\( p^{163} T^{10} + 6738696366054756 p^{201} T^{11} + p^{240} T^{12} )^{2} \) | |
| 11 | \( 1 - \)\(27\!\cdots\!32\)\( T^{2} + \)\(31\!\cdots\!46\)\( p^{2} T^{4} - \)\(24\!\cdots\!20\)\( p^{4} T^{6} + \)\(12\!\cdots\!95\)\( p^{8} T^{8} - \)\(44\!\cdots\!32\)\( p^{13} T^{10} + \)\(12\!\cdots\!04\)\( p^{19} T^{12} - \)\(44\!\cdots\!32\)\( p^{93} T^{14} + \)\(12\!\cdots\!95\)\( p^{168} T^{16} - \)\(24\!\cdots\!20\)\( p^{244} T^{18} + \)\(31\!\cdots\!46\)\( p^{322} T^{20} - \)\(27\!\cdots\!32\)\( p^{400} T^{22} + p^{480} T^{24} \) | |
| 13 | \( ( 1 - 27091109456313650076 p T + \)\(93\!\cdots\!78\)\( p^{2} T^{2} + \)\(94\!\cdots\!64\)\( p^{5} T^{3} + \)\(31\!\cdots\!03\)\( p^{5} T^{4} + \)\(49\!\cdots\!88\)\( p^{7} T^{5} + \)\(66\!\cdots\!76\)\( p^{8} T^{6} + \)\(49\!\cdots\!88\)\( p^{47} T^{7} + \)\(31\!\cdots\!03\)\( p^{85} T^{8} + \)\(94\!\cdots\!64\)\( p^{125} T^{9} + \)\(93\!\cdots\!78\)\( p^{162} T^{10} - 27091109456313650076 p^{201} T^{11} + p^{240} T^{12} )^{2} \) | |
| 17 | \( 1 - \)\(81\!\cdots\!80\)\( T^{2} + \)\(11\!\cdots\!06\)\( p^{2} T^{4} - \)\(11\!\cdots\!60\)\( p^{4} T^{6} + \)\(51\!\cdots\!35\)\( p^{7} T^{8} - \)\(12\!\cdots\!20\)\( p^{11} T^{10} + \)\(25\!\cdots\!40\)\( p^{15} T^{12} - \)\(12\!\cdots\!20\)\( p^{91} T^{14} + \)\(51\!\cdots\!35\)\( p^{167} T^{16} - \)\(11\!\cdots\!60\)\( p^{244} T^{18} + \)\(11\!\cdots\!06\)\( p^{322} T^{20} - \)\(81\!\cdots\!80\)\( p^{400} T^{22} + p^{480} T^{24} \) | |
| 19 | \( ( 1 - \)\(98\!\cdots\!40\)\( p T + \)\(10\!\cdots\!34\)\( p^{2} T^{2} + \)\(24\!\cdots\!20\)\( p^{3} T^{3} + \)\(32\!\cdots\!05\)\( p^{5} T^{4} + \)\(29\!\cdots\!20\)\( p^{7} T^{5} + \)\(44\!\cdots\!40\)\( p^{9} T^{6} + \)\(29\!\cdots\!20\)\( p^{47} T^{7} + \)\(32\!\cdots\!05\)\( p^{85} T^{8} + \)\(24\!\cdots\!20\)\( p^{123} T^{9} + \)\(10\!\cdots\!34\)\( p^{162} T^{10} - \)\(98\!\cdots\!40\)\( p^{201} T^{11} + p^{240} T^{12} )^{2} \) | |
| 23 | \( 1 - \)\(19\!\cdots\!20\)\( T^{2} + \)\(17\!\cdots\!54\)\( T^{4} - \)\(47\!\cdots\!80\)\( p T^{6} + \)\(40\!\cdots\!25\)\( p^{3} T^{8} - \)\(28\!\cdots\!80\)\( p^{5} T^{10} + \)\(16\!\cdots\!20\)\( p^{7} T^{12} - \)\(28\!\cdots\!80\)\( p^{85} T^{14} + \)\(40\!\cdots\!25\)\( p^{163} T^{16} - \)\(47\!\cdots\!80\)\( p^{241} T^{18} + \)\(17\!\cdots\!54\)\( p^{320} T^{20} - \)\(19\!\cdots\!20\)\( p^{400} T^{22} + p^{480} T^{24} \) | |
| 29 | \( 1 - \)\(11\!\cdots\!92\)\( T^{2} + \)\(23\!\cdots\!54\)\( p T^{4} - \)\(11\!\cdots\!80\)\( p^{3} T^{6} + \)\(47\!\cdots\!55\)\( p^{5} T^{8} - \)\(18\!\cdots\!88\)\( p^{7} T^{10} + \)\(67\!\cdots\!36\)\( p^{9} T^{12} - \)\(18\!\cdots\!88\)\( p^{87} T^{14} + \)\(47\!\cdots\!55\)\( p^{165} T^{16} - \)\(11\!\cdots\!80\)\( p^{243} T^{18} + \)\(23\!\cdots\!54\)\( p^{321} T^{20} - \)\(11\!\cdots\!92\)\( p^{400} T^{22} + p^{480} T^{24} \) | |
| 31 | \( ( 1 + \)\(45\!\cdots\!08\)\( T + \)\(62\!\cdots\!86\)\( p T^{2} + \)\(60\!\cdots\!80\)\( p^{2} T^{3} + \)\(57\!\cdots\!45\)\( p^{3} T^{4} + \)\(41\!\cdots\!48\)\( p^{4} T^{5} + \)\(10\!\cdots\!64\)\( p^{6} T^{6} + \)\(41\!\cdots\!48\)\( p^{44} T^{7} + \)\(57\!\cdots\!45\)\( p^{83} T^{8} + \)\(60\!\cdots\!80\)\( p^{122} T^{9} + \)\(62\!\cdots\!86\)\( p^{161} T^{10} + \)\(45\!\cdots\!08\)\( p^{200} T^{11} + p^{240} T^{12} )^{2} \) | |
| 37 | \( ( 1 - \)\(76\!\cdots\!04\)\( p T + \)\(94\!\cdots\!98\)\( p^{2} T^{2} - \)\(47\!\cdots\!36\)\( p^{3} T^{3} + \)\(64\!\cdots\!39\)\( p^{4} T^{4} - \)\(28\!\cdots\!76\)\( p^{6} T^{5} + \)\(28\!\cdots\!04\)\( p^{6} T^{6} - \)\(28\!\cdots\!76\)\( p^{46} T^{7} + \)\(64\!\cdots\!39\)\( p^{84} T^{8} - \)\(47\!\cdots\!36\)\( p^{123} T^{9} + \)\(94\!\cdots\!98\)\( p^{162} T^{10} - \)\(76\!\cdots\!04\)\( p^{201} T^{11} + p^{240} T^{12} )^{2} \) | |
| 41 | \( 1 - \)\(29\!\cdots\!32\)\( T^{2} + \)\(40\!\cdots\!66\)\( T^{4} - \)\(35\!\cdots\!20\)\( T^{6} + \)\(53\!\cdots\!95\)\( p T^{8} - \)\(10\!\cdots\!92\)\( T^{10} + \)\(38\!\cdots\!64\)\( T^{12} - \)\(10\!\cdots\!92\)\( p^{80} T^{14} + \)\(53\!\cdots\!95\)\( p^{161} T^{16} - \)\(35\!\cdots\!20\)\( p^{240} T^{18} + \)\(40\!\cdots\!66\)\( p^{320} T^{20} - \)\(29\!\cdots\!32\)\( p^{400} T^{22} + p^{480} T^{24} \) | |
| 43 | \( ( 1 - \)\(50\!\cdots\!88\)\( T + \)\(73\!\cdots\!42\)\( T^{2} - \)\(23\!\cdots\!08\)\( T^{3} + \)\(17\!\cdots\!59\)\( T^{4} - \)\(34\!\cdots\!84\)\( T^{5} + \)\(25\!\cdots\!36\)\( T^{6} - \)\(34\!\cdots\!84\)\( p^{40} T^{7} + \)\(17\!\cdots\!59\)\( p^{80} T^{8} - \)\(23\!\cdots\!08\)\( p^{120} T^{9} + \)\(73\!\cdots\!42\)\( p^{160} T^{10} - \)\(50\!\cdots\!88\)\( p^{200} T^{11} + p^{240} T^{12} )^{2} \) | |
| 47 | \( 1 - \)\(40\!\cdots\!80\)\( T^{2} + \)\(86\!\cdots\!34\)\( T^{4} - \)\(13\!\cdots\!60\)\( T^{6} + \)\(15\!\cdots\!55\)\( T^{8} - \)\(15\!\cdots\!60\)\( T^{10} + \)\(12\!\cdots\!20\)\( T^{12} - \)\(15\!\cdots\!60\)\( p^{80} T^{14} + \)\(15\!\cdots\!55\)\( p^{160} T^{16} - \)\(13\!\cdots\!60\)\( p^{240} T^{18} + \)\(86\!\cdots\!34\)\( p^{320} T^{20} - \)\(40\!\cdots\!80\)\( p^{400} T^{22} + p^{480} T^{24} \) | |
| 53 | \( 1 - \)\(61\!\cdots\!80\)\( T^{2} + \)\(16\!\cdots\!34\)\( T^{4} - \)\(27\!\cdots\!60\)\( T^{6} + \)\(29\!\cdots\!55\)\( T^{8} - \)\(21\!\cdots\!60\)\( T^{10} + \)\(56\!\cdots\!80\)\( p^{2} T^{12} - \)\(21\!\cdots\!60\)\( p^{80} T^{14} + \)\(29\!\cdots\!55\)\( p^{160} T^{16} - \)\(27\!\cdots\!60\)\( p^{240} T^{18} + \)\(16\!\cdots\!34\)\( p^{320} T^{20} - \)\(61\!\cdots\!80\)\( p^{400} T^{22} + p^{480} T^{24} \) | |
| 59 | \( 1 - \)\(46\!\cdots\!32\)\( T^{2} + \)\(98\!\cdots\!66\)\( T^{4} - \)\(13\!\cdots\!20\)\( T^{6} + \)\(13\!\cdots\!95\)\( T^{8} - \)\(11\!\cdots\!92\)\( T^{10} + \)\(79\!\cdots\!64\)\( T^{12} - \)\(11\!\cdots\!92\)\( p^{80} T^{14} + \)\(13\!\cdots\!95\)\( p^{160} T^{16} - \)\(13\!\cdots\!20\)\( p^{240} T^{18} + \)\(98\!\cdots\!66\)\( p^{320} T^{20} - \)\(46\!\cdots\!32\)\( p^{400} T^{22} + p^{480} T^{24} \) | |
| 61 | \( ( 1 + \)\(24\!\cdots\!48\)\( T + \)\(13\!\cdots\!66\)\( T^{2} + \)\(29\!\cdots\!80\)\( T^{3} + \)\(76\!\cdots\!95\)\( T^{4} + \)\(14\!\cdots\!08\)\( T^{5} + \)\(25\!\cdots\!04\)\( T^{6} + \)\(14\!\cdots\!08\)\( p^{40} T^{7} + \)\(76\!\cdots\!95\)\( p^{80} T^{8} + \)\(29\!\cdots\!80\)\( p^{120} T^{9} + \)\(13\!\cdots\!66\)\( p^{160} T^{10} + \)\(24\!\cdots\!48\)\( p^{200} T^{11} + p^{240} T^{12} )^{2} \) | |
| 67 | \( ( 1 - \)\(27\!\cdots\!88\)\( T + \)\(40\!\cdots\!62\)\( T^{2} - \)\(74\!\cdots\!28\)\( T^{3} + \)\(80\!\cdots\!19\)\( T^{4} - \)\(12\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!16\)\( T^{6} - \)\(12\!\cdots\!44\)\( p^{40} T^{7} + \)\(80\!\cdots\!19\)\( p^{80} T^{8} - \)\(74\!\cdots\!28\)\( p^{120} T^{9} + \)\(40\!\cdots\!62\)\( p^{160} T^{10} - \)\(27\!\cdots\!88\)\( p^{200} T^{11} + p^{240} T^{12} )^{2} \) | |
| 71 | \( 1 - \)\(64\!\cdots\!92\)\( T^{2} + \)\(20\!\cdots\!66\)\( T^{4} - \)\(44\!\cdots\!20\)\( T^{6} + \)\(70\!\cdots\!95\)\( T^{8} - \)\(92\!\cdots\!92\)\( T^{10} + \)\(10\!\cdots\!84\)\( T^{12} - \)\(92\!\cdots\!92\)\( p^{80} T^{14} + \)\(70\!\cdots\!95\)\( p^{160} T^{16} - \)\(44\!\cdots\!20\)\( p^{240} T^{18} + \)\(20\!\cdots\!66\)\( p^{320} T^{20} - \)\(64\!\cdots\!92\)\( p^{400} T^{22} + p^{480} T^{24} \) | |
| 73 | \( ( 1 - \)\(15\!\cdots\!68\)\( T + \)\(13\!\cdots\!82\)\( T^{2} - \)\(27\!\cdots\!88\)\( T^{3} + \)\(84\!\cdots\!59\)\( T^{4} - \)\(19\!\cdots\!24\)\( T^{5} + \)\(33\!\cdots\!56\)\( T^{6} - \)\(19\!\cdots\!24\)\( p^{40} T^{7} + \)\(84\!\cdots\!59\)\( p^{80} T^{8} - \)\(27\!\cdots\!88\)\( p^{120} T^{9} + \)\(13\!\cdots\!82\)\( p^{160} T^{10} - \)\(15\!\cdots\!68\)\( p^{200} T^{11} + p^{240} T^{12} )^{2} \) | |
| 79 | \( ( 1 + \)\(64\!\cdots\!20\)\( T + \)\(36\!\cdots\!34\)\( T^{2} + \)\(21\!\cdots\!40\)\( T^{3} + \)\(63\!\cdots\!55\)\( T^{4} + \)\(30\!\cdots\!40\)\( T^{5} + \)\(65\!\cdots\!20\)\( T^{6} + \)\(30\!\cdots\!40\)\( p^{40} T^{7} + \)\(63\!\cdots\!55\)\( p^{80} T^{8} + \)\(21\!\cdots\!40\)\( p^{120} T^{9} + \)\(36\!\cdots\!34\)\( p^{160} T^{10} + \)\(64\!\cdots\!20\)\( p^{200} T^{11} + p^{240} T^{12} )^{2} \) | |
| 83 | \( 1 - \)\(48\!\cdots\!80\)\( T^{2} + \)\(11\!\cdots\!34\)\( T^{4} - \)\(17\!\cdots\!60\)\( T^{6} + \)\(19\!\cdots\!55\)\( T^{8} - \)\(16\!\cdots\!60\)\( T^{10} + \)\(10\!\cdots\!20\)\( T^{12} - \)\(16\!\cdots\!60\)\( p^{80} T^{14} + \)\(19\!\cdots\!55\)\( p^{160} T^{16} - \)\(17\!\cdots\!60\)\( p^{240} T^{18} + \)\(11\!\cdots\!34\)\( p^{320} T^{20} - \)\(48\!\cdots\!80\)\( p^{400} T^{22} + p^{480} T^{24} \) | |
| 89 | \( 1 - \)\(54\!\cdots\!32\)\( T^{2} + \)\(14\!\cdots\!66\)\( T^{4} - \)\(27\!\cdots\!20\)\( T^{6} + \)\(40\!\cdots\!95\)\( T^{8} - \)\(49\!\cdots\!92\)\( T^{10} + \)\(50\!\cdots\!64\)\( T^{12} - \)\(49\!\cdots\!92\)\( p^{80} T^{14} + \)\(40\!\cdots\!95\)\( p^{160} T^{16} - \)\(27\!\cdots\!20\)\( p^{240} T^{18} + \)\(14\!\cdots\!66\)\( p^{320} T^{20} - \)\(54\!\cdots\!32\)\( p^{400} T^{22} + p^{480} T^{24} \) | |
| 97 | \( ( 1 + \)\(20\!\cdots\!32\)\( T + \)\(30\!\cdots\!02\)\( T^{2} + \)\(30\!\cdots\!92\)\( T^{3} + \)\(26\!\cdots\!19\)\( T^{4} + \)\(17\!\cdots\!16\)\( T^{5} + \)\(10\!\cdots\!36\)\( T^{6} + \)\(17\!\cdots\!16\)\( p^{40} T^{7} + \)\(26\!\cdots\!19\)\( p^{80} T^{8} + \)\(30\!\cdots\!92\)\( p^{120} T^{9} + \)\(30\!\cdots\!02\)\( p^{160} T^{10} + \)\(20\!\cdots\!32\)\( p^{200} T^{11} + p^{240} T^{12} )^{2} \) | |
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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
−3.61433591139661536947797925457, −3.51781992772481911520897984523, −3.41454038003436982608380583253, −3.14428567448025920879624030926, −2.84707087765977643868322808856, −2.81492108555081140016646017508, −2.75961290758052511920585048489, −2.74408361210093665507283437907, −2.71050759055928050243392939116, −2.54433618912655427589325292427, −2.42838845945002052476658950598, −2.14544161488662727445885083400, −1.74479410079259146522761274017, −1.70900290079736835364504565246, −1.67396585384565468210286630756, −1.37897130981789857394562423053, −1.36170011714590443105302010600, −1.33658323555982520019401538057, −1.23675663792823356286318098843, −0.77127377409812884909558647559, −0.71321626587522863612268031498, −0.63907450980936159040154548622, −0.46809136442876418126240435144, −0.21580314364448787219176993925, −0.20484845774615889342933477327, 0.20484845774615889342933477327, 0.21580314364448787219176993925, 0.46809136442876418126240435144, 0.63907450980936159040154548622, 0.71321626587522863612268031498, 0.77127377409812884909558647559, 1.23675663792823356286318098843, 1.33658323555982520019401538057, 1.36170011714590443105302010600, 1.37897130981789857394562423053, 1.67396585384565468210286630756, 1.70900290079736835364504565246, 1.74479410079259146522761274017, 2.14544161488662727445885083400, 2.42838845945002052476658950598, 2.54433618912655427589325292427, 2.71050759055928050243392939116, 2.74408361210093665507283437907, 2.75961290758052511920585048489, 2.81492108555081140016646017508, 2.84707087765977643868322808856, 3.14428567448025920879624030926, 3.41454038003436982608380583253, 3.51781992772481911520897984523, 3.61433591139661536947797925457
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.