Dirichlet series
| L(s) = 1 | − 6.21e3·2-s + 2.69e6·4-s + 5.67e7·5-s + 9.87e9·8-s + 9.42e11·9-s − 3.52e11·10-s − 2.47e13·13-s + 6.32e13·16-s − 1.62e15·17-s − 5.85e15·18-s + 1.53e14·20-s − 2.60e17·25-s + 1.54e17·26-s + 1.56e18·29-s + 9.89e16·32-s + 1.00e19·34-s + 2.54e18·36-s + 1.81e19·37-s + 5.60e17·40-s + 7.50e19·41-s + 5.35e19·45-s + 8.20e20·49-s + 1.62e21·50-s − 6.69e19·52-s − 1.10e21·53-s − 9.72e21·58-s + 7.51e21·61-s + ⋯ |
| L(s) = 1 | − 1.51·2-s + 0.160·4-s + 0.232·5-s + 0.143·8-s + 3.33·9-s − 0.352·10-s − 1.06·13-s + 0.224·16-s − 2.78·17-s − 5.06·18-s + 0.0373·20-s − 4.37·25-s + 1.61·26-s + 4.42·29-s + 0.0858·32-s + 4.23·34-s + 0.536·36-s + 2.75·37-s + 0.0334·40-s + 3.32·41-s + 0.775·45-s + 4.28·49-s + 6.63·50-s − 0.171·52-s − 2.25·53-s − 6.71·58-s + 2.83·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr =\mathstrut & \, \Lambda(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s+12)^{10} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(1048576\) = \(2^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.39681\times 10^{11}\) |
| Root analytic conductor: | \(3.82082\) |
| Motivic weight: | \(24\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 1048576,\ (\ :[12]^{10}),\ 1)\) |
Particular Values
| \(L(\frac{25}{2})\) | \(\approx\) | \(3.851326662\) |
| \(L(\frac12)\) | \(\approx\) | \(3.851326662\) |
| \(L(13)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + 1553 p^{2} T + 560799 p^{6} T^{2} + 47930515 p^{12} T^{3} + 951899933 p^{20} T^{4} + 4493154789 p^{30} T^{5} + 951899933 p^{44} T^{6} + 47930515 p^{60} T^{7} + 560799 p^{78} T^{8} + 1553 p^{98} T^{9} + p^{120} T^{10} \) |
| good | 3 | \( 1 - 314206799854 p T^{2} + \)\(23\!\cdots\!11\)\( p^{7} T^{4} - \)\(36\!\cdots\!92\)\( p^{12} T^{6} + \)\(13\!\cdots\!62\)\( p^{16} T^{8} - \)\(21\!\cdots\!76\)\( p^{27} T^{10} + \)\(13\!\cdots\!62\)\( p^{64} T^{12} - \)\(36\!\cdots\!92\)\( p^{108} T^{14} + \)\(23\!\cdots\!11\)\( p^{151} T^{16} - 314206799854 p^{193} T^{18} + p^{240} T^{20} \) |
| 5 | \( ( 1 - 1135162 p^{2} T + 1052974658196753 p^{3} T^{2} - \)\(10\!\cdots\!88\)\( p^{5} T^{3} + \)\(67\!\cdots\!46\)\( p^{6} T^{4} - \)\(55\!\cdots\!32\)\( p^{10} T^{5} + \)\(67\!\cdots\!46\)\( p^{30} T^{6} - \)\(10\!\cdots\!88\)\( p^{53} T^{7} + 1052974658196753 p^{75} T^{8} - 1135162 p^{98} T^{9} + p^{120} T^{10} )^{2} \) | |
| 7 | \( 1 - \)\(82\!\cdots\!22\)\( T^{2} + \)\(69\!\cdots\!73\)\( p^{2} T^{4} - \)\(55\!\cdots\!96\)\( p^{5} T^{6} + \)\(48\!\cdots\!46\)\( p^{9} T^{8} - \)\(78\!\cdots\!84\)\( p^{15} T^{10} + \)\(48\!\cdots\!46\)\( p^{57} T^{12} - \)\(55\!\cdots\!96\)\( p^{101} T^{14} + \)\(69\!\cdots\!73\)\( p^{146} T^{16} - \)\(82\!\cdots\!22\)\( p^{192} T^{18} + p^{240} T^{20} \) | |
| 11 | \( 1 - \)\(31\!\cdots\!70\)\( p T^{2} + \)\(52\!\cdots\!95\)\( p^{3} T^{4} - \)\(68\!\cdots\!20\)\( p^{5} T^{6} + \)\(60\!\cdots\!70\)\( p^{9} T^{8} - \)\(44\!\cdots\!92\)\( p^{13} T^{10} + \)\(60\!\cdots\!70\)\( p^{57} T^{12} - \)\(68\!\cdots\!20\)\( p^{101} T^{14} + \)\(52\!\cdots\!95\)\( p^{147} T^{16} - \)\(31\!\cdots\!70\)\( p^{193} T^{18} + p^{240} T^{20} \) | |
| 13 | \( ( 1 + 12399032671382 T + \)\(10\!\cdots\!81\)\( T^{2} + \)\(52\!\cdots\!00\)\( p T^{3} + \)\(54\!\cdots\!62\)\( p^{2} T^{4} + \)\(37\!\cdots\!48\)\( p^{3} T^{5} + \)\(54\!\cdots\!62\)\( p^{26} T^{6} + \)\(52\!\cdots\!00\)\( p^{49} T^{7} + \)\(10\!\cdots\!81\)\( p^{72} T^{8} + 12399032671382 p^{96} T^{9} + p^{120} T^{10} )^{2} \) | |
| 17 | \( ( 1 + 812519065310582 T + \)\(55\!\cdots\!53\)\( p T^{2} + \)\(29\!\cdots\!20\)\( p^{2} T^{3} + \)\(12\!\cdots\!26\)\( p^{3} T^{4} + \)\(44\!\cdots\!16\)\( p^{4} T^{5} + \)\(12\!\cdots\!26\)\( p^{27} T^{6} + \)\(29\!\cdots\!20\)\( p^{50} T^{7} + \)\(55\!\cdots\!53\)\( p^{73} T^{8} + 812519065310582 p^{96} T^{9} + p^{120} T^{10} )^{2} \) | |
| 19 | \( 1 - \)\(20\!\cdots\!70\)\( T^{2} + \)\(72\!\cdots\!45\)\( p^{2} T^{4} - \)\(17\!\cdots\!20\)\( p^{4} T^{6} + \)\(33\!\cdots\!70\)\( p^{6} T^{8} - \)\(50\!\cdots\!72\)\( p^{8} T^{10} + \)\(33\!\cdots\!70\)\( p^{54} T^{12} - \)\(17\!\cdots\!20\)\( p^{100} T^{14} + \)\(72\!\cdots\!45\)\( p^{146} T^{16} - \)\(20\!\cdots\!70\)\( p^{192} T^{18} + p^{240} T^{20} \) | |
| 23 | \( 1 - \)\(63\!\cdots\!38\)\( p^{2} T^{2} + \)\(19\!\cdots\!77\)\( p^{4} T^{4} - \)\(39\!\cdots\!48\)\( p^{6} T^{6} + \)\(55\!\cdots\!62\)\( p^{8} T^{8} - \)\(58\!\cdots\!08\)\( p^{10} T^{10} + \)\(55\!\cdots\!62\)\( p^{56} T^{12} - \)\(39\!\cdots\!48\)\( p^{102} T^{14} + \)\(19\!\cdots\!77\)\( p^{148} T^{16} - \)\(63\!\cdots\!38\)\( p^{194} T^{18} + p^{240} T^{20} \) | |
| 29 | \( ( 1 - 782804275329223402 T + \)\(74\!\cdots\!17\)\( T^{2} - \)\(36\!\cdots\!52\)\( T^{3} + \)\(19\!\cdots\!62\)\( T^{4} - \)\(67\!\cdots\!32\)\( T^{5} + \)\(19\!\cdots\!62\)\( p^{24} T^{6} - \)\(36\!\cdots\!52\)\( p^{48} T^{7} + \)\(74\!\cdots\!17\)\( p^{72} T^{8} - 782804275329223402 p^{96} T^{9} + p^{120} T^{10} )^{2} \) | |
| 31 | \( 1 - \)\(29\!\cdots\!70\)\( T^{2} + \)\(39\!\cdots\!45\)\( T^{4} - \)\(30\!\cdots\!20\)\( T^{6} + \)\(15\!\cdots\!70\)\( T^{8} - \)\(74\!\cdots\!52\)\( T^{10} + \)\(15\!\cdots\!70\)\( p^{48} T^{12} - \)\(30\!\cdots\!20\)\( p^{96} T^{14} + \)\(39\!\cdots\!45\)\( p^{144} T^{16} - \)\(29\!\cdots\!70\)\( p^{192} T^{18} + p^{240} T^{20} \) | |
| 37 | \( ( 1 - 9054474706503062698 T + \)\(15\!\cdots\!01\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!18\)\( T^{4} - \)\(72\!\cdots\!84\)\( T^{5} + \)\(10\!\cdots\!18\)\( p^{24} T^{6} - \)\(12\!\cdots\!00\)\( p^{48} T^{7} + \)\(15\!\cdots\!01\)\( p^{72} T^{8} - 9054474706503062698 p^{96} T^{9} + p^{120} T^{10} )^{2} \) | |
| 41 | \( ( 1 - 37540714406792671690 T + \)\(27\!\cdots\!65\)\( T^{2} - \)\(75\!\cdots\!40\)\( T^{3} + \)\(28\!\cdots\!90\)\( T^{4} - \)\(57\!\cdots\!52\)\( T^{5} + \)\(28\!\cdots\!90\)\( p^{24} T^{6} - \)\(75\!\cdots\!40\)\( p^{48} T^{7} + \)\(27\!\cdots\!65\)\( p^{72} T^{8} - 37540714406792671690 p^{96} T^{9} + p^{120} T^{10} )^{2} \) | |
| 43 | \( 1 - \)\(81\!\cdots\!82\)\( T^{2} + \)\(38\!\cdots\!17\)\( T^{4} - \)\(12\!\cdots\!12\)\( T^{6} + \)\(28\!\cdots\!82\)\( T^{8} - \)\(51\!\cdots\!12\)\( T^{10} + \)\(28\!\cdots\!82\)\( p^{48} T^{12} - \)\(12\!\cdots\!12\)\( p^{96} T^{14} + \)\(38\!\cdots\!17\)\( p^{144} T^{16} - \)\(81\!\cdots\!82\)\( p^{192} T^{18} + p^{240} T^{20} \) | |
| 47 | \( 1 - \)\(91\!\cdots\!82\)\( T^{2} + \)\(40\!\cdots\!57\)\( T^{4} - \)\(11\!\cdots\!92\)\( T^{6} + \)\(11\!\cdots\!78\)\( p^{2} T^{8} - \)\(77\!\cdots\!72\)\( p^{4} T^{10} + \)\(11\!\cdots\!78\)\( p^{50} T^{12} - \)\(11\!\cdots\!92\)\( p^{96} T^{14} + \)\(40\!\cdots\!57\)\( p^{144} T^{16} - \)\(91\!\cdots\!82\)\( p^{192} T^{18} + p^{240} T^{20} \) | |
| 53 | \( ( 1 + \)\(55\!\cdots\!22\)\( T + \)\(81\!\cdots\!61\)\( T^{2} + \)\(29\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!98\)\( T^{4} + \)\(91\!\cdots\!16\)\( T^{5} + \)\(30\!\cdots\!98\)\( p^{24} T^{6} + \)\(29\!\cdots\!00\)\( p^{48} T^{7} + \)\(81\!\cdots\!61\)\( p^{72} T^{8} + \)\(55\!\cdots\!22\)\( p^{96} T^{9} + p^{120} T^{10} )^{2} \) | |
| 59 | \( 1 - \)\(15\!\cdots\!30\)\( T^{2} + \)\(11\!\cdots\!25\)\( T^{4} - \)\(51\!\cdots\!00\)\( T^{6} + \)\(17\!\cdots\!30\)\( T^{8} - \)\(55\!\cdots\!52\)\( T^{10} + \)\(17\!\cdots\!30\)\( p^{48} T^{12} - \)\(51\!\cdots\!00\)\( p^{96} T^{14} + \)\(11\!\cdots\!25\)\( p^{144} T^{16} - \)\(15\!\cdots\!30\)\( p^{192} T^{18} + p^{240} T^{20} \) | |
| 61 | \( ( 1 - \)\(37\!\cdots\!90\)\( T + \)\(27\!\cdots\!65\)\( T^{2} - \)\(57\!\cdots\!40\)\( T^{3} + \)\(27\!\cdots\!90\)\( T^{4} - \)\(41\!\cdots\!52\)\( T^{5} + \)\(27\!\cdots\!90\)\( p^{24} T^{6} - \)\(57\!\cdots\!40\)\( p^{48} T^{7} + \)\(27\!\cdots\!65\)\( p^{72} T^{8} - \)\(37\!\cdots\!90\)\( p^{96} T^{9} + p^{120} T^{10} )^{2} \) | |
| 67 | \( 1 - \)\(33\!\cdots\!62\)\( T^{2} + \)\(57\!\cdots\!97\)\( T^{4} - \)\(67\!\cdots\!92\)\( T^{6} + \)\(60\!\cdots\!42\)\( T^{8} - \)\(44\!\cdots\!12\)\( T^{10} + \)\(60\!\cdots\!42\)\( p^{48} T^{12} - \)\(67\!\cdots\!92\)\( p^{96} T^{14} + \)\(57\!\cdots\!97\)\( p^{144} T^{16} - \)\(33\!\cdots\!62\)\( p^{192} T^{18} + p^{240} T^{20} \) | |
| 71 | \( 1 - \)\(15\!\cdots\!30\)\( T^{2} + \)\(12\!\cdots\!25\)\( T^{4} - \)\(69\!\cdots\!00\)\( T^{6} + \)\(28\!\cdots\!30\)\( T^{8} - \)\(17\!\cdots\!72\)\( p^{2} T^{10} + \)\(28\!\cdots\!30\)\( p^{48} T^{12} - \)\(69\!\cdots\!00\)\( p^{96} T^{14} + \)\(12\!\cdots\!25\)\( p^{144} T^{16} - \)\(15\!\cdots\!30\)\( p^{192} T^{18} + p^{240} T^{20} \) | |
| 73 | \( ( 1 - \)\(25\!\cdots\!98\)\( T + \)\(20\!\cdots\!41\)\( T^{2} - \)\(40\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!78\)\( T^{4} - \)\(28\!\cdots\!24\)\( T^{5} + \)\(19\!\cdots\!78\)\( p^{24} T^{6} - \)\(40\!\cdots\!00\)\( p^{48} T^{7} + \)\(20\!\cdots\!41\)\( p^{72} T^{8} - \)\(25\!\cdots\!98\)\( p^{96} T^{9} + p^{120} T^{10} )^{2} \) | |
| 79 | \( 1 - \)\(18\!\cdots\!70\)\( T^{2} + \)\(18\!\cdots\!45\)\( T^{4} - \)\(12\!\cdots\!20\)\( T^{6} + \)\(64\!\cdots\!70\)\( T^{8} - \)\(25\!\cdots\!52\)\( T^{10} + \)\(64\!\cdots\!70\)\( p^{48} T^{12} - \)\(12\!\cdots\!20\)\( p^{96} T^{14} + \)\(18\!\cdots\!45\)\( p^{144} T^{16} - \)\(18\!\cdots\!70\)\( p^{192} T^{18} + p^{240} T^{20} \) | |
| 83 | \( 1 - \)\(64\!\cdots\!42\)\( T^{2} + \)\(19\!\cdots\!77\)\( T^{4} - \)\(39\!\cdots\!32\)\( T^{6} + \)\(60\!\cdots\!62\)\( T^{8} - \)\(76\!\cdots\!72\)\( T^{10} + \)\(60\!\cdots\!62\)\( p^{48} T^{12} - \)\(39\!\cdots\!32\)\( p^{96} T^{14} + \)\(19\!\cdots\!77\)\( p^{144} T^{16} - \)\(64\!\cdots\!42\)\( p^{192} T^{18} + p^{240} T^{20} \) | |
| 89 | \( ( 1 + \)\(40\!\cdots\!78\)\( T + \)\(26\!\cdots\!97\)\( T^{2} + \)\(84\!\cdots\!68\)\( T^{3} + \)\(31\!\cdots\!42\)\( T^{4} + \)\(72\!\cdots\!88\)\( T^{5} + \)\(31\!\cdots\!42\)\( p^{24} T^{6} + \)\(84\!\cdots\!68\)\( p^{48} T^{7} + \)\(26\!\cdots\!97\)\( p^{72} T^{8} + \)\(40\!\cdots\!78\)\( p^{96} T^{9} + p^{120} T^{10} )^{2} \) | |
| 97 | \( ( 1 - \)\(11\!\cdots\!18\)\( T + \)\(28\!\cdots\!01\)\( T^{2} - \)\(22\!\cdots\!20\)\( T^{3} + \)\(30\!\cdots\!58\)\( T^{4} - \)\(16\!\cdots\!44\)\( T^{5} + \)\(30\!\cdots\!58\)\( p^{24} T^{6} - \)\(22\!\cdots\!20\)\( p^{48} T^{7} + \)\(28\!\cdots\!01\)\( p^{72} T^{8} - \)\(11\!\cdots\!18\)\( p^{96} T^{9} + p^{120} T^{10} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.70562525773032876064202740275, −5.36608163981823162885724962328, −5.12345127493947565280807956498, −4.68423086521198200578766161801, −4.45492195267057643694696303718, −4.42657688355296903364093554124, −4.40886659059171445985877298995, −4.23280228050894004430887988579, −3.89484506544319212878189775075, −3.70977083464857759476211855639, −3.59359233130208033489994128394, −2.88943589840874789155650490069, −2.69644239970067479368353127460, −2.44729407344652149033615218598, −2.37114287227049341764265823183, −2.11605029434647920423339990168, −1.96214249419511736445029329185, −1.94534571192499017262710902496, −1.23425324513571243094582148926, −1.17918331452048432111698713399, −0.918563957108077587433765121413, −0.834497535102378126832878795640, −0.55153169317243363075981047306, −0.32647697656875805528133763271, −0.30211461224610157728769112829, 0.30211461224610157728769112829, 0.32647697656875805528133763271, 0.55153169317243363075981047306, 0.834497535102378126832878795640, 0.918563957108077587433765121413, 1.17918331452048432111698713399, 1.23425324513571243094582148926, 1.94534571192499017262710902496, 1.96214249419511736445029329185, 2.11605029434647920423339990168, 2.37114287227049341764265823183, 2.44729407344652149033615218598, 2.69644239970067479368353127460, 2.88943589840874789155650490069, 3.59359233130208033489994128394, 3.70977083464857759476211855639, 3.89484506544319212878189775075, 4.23280228050894004430887988579, 4.40886659059171445985877298995, 4.42657688355296903364093554124, 4.45492195267057643694696303718, 4.68423086521198200578766161801, 5.12345127493947565280807956498, 5.36608163981823162885724962328, 5.70562525773032876064202740275
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.