Dirichlet series
| L(s) = 1 | − 2·2-s + 5·4-s − 4·5-s + 5·7-s − 8·8-s + 8·10-s − 4·11-s + 16·13-s − 10·14-s + 15·16-s − 12·17-s + 19-s − 20·20-s + 8·22-s − 3·23-s + 20·25-s − 32·26-s + 25·28-s + 14·29-s − 3·31-s − 20·32-s + 24·34-s − 20·35-s − 2·38-s + 32·40-s + 10·41-s + 14·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 5/2·4-s − 1.78·5-s + 1.88·7-s − 2.82·8-s + 2.52·10-s − 1.20·11-s + 4.43·13-s − 2.67·14-s + 15/4·16-s − 2.91·17-s + 0.229·19-s − 4.47·20-s + 1.70·22-s − 0.625·23-s + 4·25-s − 6.27·26-s + 4.72·28-s + 2.59·29-s − 0.538·31-s − 3.53·32-s + 4.11·34-s − 3.38·35-s − 0.324·38-s + 5.05·40-s + 1.56·41-s + 2.13·43-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{40} \cdot 7^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.61910\times 10^{6}\) |
| Root analytic conductor: | \(2.12779\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{40} \cdot 7^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(3.476365238\) |
| \(L(\frac12)\) | \(\approx\) | \(3.476365238\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 5 T + 3 p T^{2} - 43 T^{3} + 83 T^{4} - 87 T^{5} + 83 p T^{6} - 43 p^{2} T^{7} + 3 p^{4} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 2 | \( 1 + p T - T^{2} - p^{2} T^{3} - p T^{4} - p T^{5} + 3 p T^{6} + 21 T^{7} + 3 p T^{8} - 13 T^{9} - 5 T^{10} - 13 p T^{11} + 3 p^{3} T^{12} + 21 p^{3} T^{13} + 3 p^{5} T^{14} - p^{6} T^{15} - p^{7} T^{16} - p^{9} T^{17} - p^{8} T^{18} + p^{10} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 + 4 T - 4 T^{2} - 44 T^{3} - 41 T^{4} + 119 T^{5} + 222 T^{6} + 456 T^{7} + 1623 T^{8} - 2021 T^{9} - 16541 T^{10} - 2021 p T^{11} + 1623 p^{2} T^{12} + 456 p^{3} T^{13} + 222 p^{4} T^{14} + 119 p^{5} T^{15} - 41 p^{6} T^{16} - 44 p^{7} T^{17} - 4 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 + 4 T - 31 T^{2} - 134 T^{3} + 607 T^{4} + 2492 T^{5} - 8385 T^{6} - 27495 T^{7} + 98940 T^{8} + 135733 T^{9} - 1043873 T^{10} + 135733 p T^{11} + 98940 p^{2} T^{12} - 27495 p^{3} T^{13} - 8385 p^{4} T^{14} + 2492 p^{5} T^{15} + 607 p^{6} T^{16} - 134 p^{7} T^{17} - 31 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( ( 1 - 8 T + 6 p T^{2} - 31 p T^{3} + 2174 T^{4} - 7779 T^{5} + 2174 p T^{6} - 31 p^{3} T^{7} + 6 p^{4} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 17 | \( 1 + 12 T + 14 T^{2} - 192 T^{3} + 1185 T^{4} + 11847 T^{5} - 6180 T^{6} - 65736 T^{7} + 1002861 T^{8} + 2436261 T^{9} - 7749777 T^{10} + 2436261 p T^{11} + 1002861 p^{2} T^{12} - 65736 p^{3} T^{13} - 6180 p^{4} T^{14} + 11847 p^{5} T^{15} + 1185 p^{6} T^{16} - 192 p^{7} T^{17} + 14 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - T - 53 T^{2} + 10 p T^{3} + 1262 T^{4} - 7007 T^{5} - 13111 T^{6} + 116110 T^{7} + 67964 T^{8} - 721616 T^{9} - 440023 T^{10} - 721616 p T^{11} + 67964 p^{2} T^{12} + 116110 p^{3} T^{13} - 13111 p^{4} T^{14} - 7007 p^{5} T^{15} + 1262 p^{6} T^{16} + 10 p^{8} T^{17} - 53 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 + 3 T - 43 T^{2} - 294 T^{3} + 6 T^{4} + 5127 T^{5} + 21792 T^{6} + 135027 T^{7} + 502362 T^{8} - 3271749 T^{9} - 33095343 T^{10} - 3271749 p T^{11} + 502362 p^{2} T^{12} + 135027 p^{3} T^{13} + 21792 p^{4} T^{14} + 5127 p^{5} T^{15} + 6 p^{6} T^{16} - 294 p^{7} T^{17} - 43 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( ( 1 - 7 T + 125 T^{2} - 647 T^{3} + 6535 T^{4} - 25761 T^{5} + 6535 p T^{6} - 647 p^{2} T^{7} + 125 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 31 | \( 1 + 3 T - 125 T^{2} - 214 T^{3} + 9282 T^{4} + 8387 T^{5} - 503981 T^{6} - 245082 T^{7} + 21459514 T^{8} + 116758 p T^{9} - 734820027 T^{10} + 116758 p^{2} T^{11} + 21459514 p^{2} T^{12} - 245082 p^{3} T^{13} - 503981 p^{4} T^{14} + 8387 p^{5} T^{15} + 9282 p^{6} T^{16} - 214 p^{7} T^{17} - 125 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 89 T^{2} + 560 T^{3} + 4503 T^{4} - 45352 T^{5} + 27130 T^{6} + 2296536 T^{7} - 9801827 T^{8} - 33131096 T^{9} + 610977105 T^{10} - 33131096 p T^{11} - 9801827 p^{2} T^{12} + 2296536 p^{3} T^{13} + 27130 p^{4} T^{14} - 45352 p^{5} T^{15} + 4503 p^{6} T^{16} + 560 p^{7} T^{17} - 89 p^{8} T^{18} + p^{10} T^{20} \) | |
| 41 | \( ( 1 - 5 T + 161 T^{2} - 769 T^{3} + 11569 T^{4} - 46293 T^{5} + 11569 p T^{6} - 769 p^{2} T^{7} + 161 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 43 | \( ( 1 - 7 T + 126 T^{2} - 474 T^{3} + 5544 T^{4} - 14049 T^{5} + 5544 p T^{6} - 474 p^{2} T^{7} + 126 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 47 | \( 1 + 27 T + 281 T^{2} + 1758 T^{3} + 13050 T^{4} + 78783 T^{5} - 25248 T^{6} - 1518381 T^{7} + 9454350 T^{8} + 53043051 T^{9} - 242331903 T^{10} + 53043051 p T^{11} + 9454350 p^{2} T^{12} - 1518381 p^{3} T^{13} - 25248 p^{4} T^{14} + 78783 p^{5} T^{15} + 13050 p^{6} T^{16} + 1758 p^{7} T^{17} + 281 p^{8} T^{18} + 27 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 - 21 T + 41 T^{2} + 924 T^{3} + 12966 T^{4} - 177027 T^{5} - 601755 T^{6} + 3783942 T^{7} + 110973258 T^{8} - 340111866 T^{9} - 4044436041 T^{10} - 340111866 p T^{11} + 110973258 p^{2} T^{12} + 3783942 p^{3} T^{13} - 601755 p^{4} T^{14} - 177027 p^{5} T^{15} + 12966 p^{6} T^{16} + 924 p^{7} T^{17} + 41 p^{8} T^{18} - 21 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 30 T + 299 T^{2} + 1644 T^{3} + 26547 T^{4} + 5838 p T^{5} + 1635267 T^{6} + 9620487 T^{7} + 170035344 T^{8} + 1056366303 T^{9} + 3109579647 T^{10} + 1056366303 p T^{11} + 170035344 p^{2} T^{12} + 9620487 p^{3} T^{13} + 1635267 p^{4} T^{14} + 5838 p^{6} T^{15} + 26547 p^{6} T^{16} + 1644 p^{7} T^{17} + 299 p^{8} T^{18} + 30 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 14 T - 143 T^{2} - 2072 T^{3} + 23777 T^{4} + 251656 T^{5} - 2164351 T^{6} - 13562879 T^{7} + 202896254 T^{8} + 466067647 T^{9} - 12461386219 T^{10} + 466067647 p T^{11} + 202896254 p^{2} T^{12} - 13562879 p^{3} T^{13} - 2164351 p^{4} T^{14} + 251656 p^{5} T^{15} + 23777 p^{6} T^{16} - 2072 p^{7} T^{17} - 143 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 2 T - 128 T^{2} - 128 T^{3} + 6161 T^{4} - 2183 T^{5} + 29300 T^{6} + 394018 T^{7} - 17169907 T^{8} - 2850929 T^{9} + 1197895103 T^{10} - 2850929 p T^{11} - 17169907 p^{2} T^{12} + 394018 p^{3} T^{13} + 29300 p^{4} T^{14} - 2183 p^{5} T^{15} + 6161 p^{6} T^{16} - 128 p^{7} T^{17} - 128 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( ( 1 - 3 T + 187 T^{2} - 285 T^{3} + 15679 T^{4} - 10143 T^{5} + 15679 p T^{6} - 285 p^{2} T^{7} + 187 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 73 | \( 1 - 15 T - 134 T^{2} + 2501 T^{3} + 16563 T^{4} - 235276 T^{5} - 2002535 T^{6} + 9021201 T^{7} + 288508378 T^{8} - 238799411 T^{9} - 25271949561 T^{10} - 238799411 p T^{11} + 288508378 p^{2} T^{12} + 9021201 p^{3} T^{13} - 2002535 p^{4} T^{14} - 235276 p^{5} T^{15} + 16563 p^{6} T^{16} + 2501 p^{7} T^{17} - 134 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 4 T - 284 T^{2} - 1776 T^{3} + 44175 T^{4} + 312399 T^{5} - 4187754 T^{6} - 29772300 T^{7} + 295992489 T^{8} + 1067553919 T^{9} - 20151634301 T^{10} + 1067553919 p T^{11} + 295992489 p^{2} T^{12} - 29772300 p^{3} T^{13} - 4187754 p^{4} T^{14} + 312399 p^{5} T^{15} + 44175 p^{6} T^{16} - 1776 p^{7} T^{17} - 284 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( ( 1 - 9 T + 229 T^{2} - 882 T^{3} + 19849 T^{4} - 37179 T^{5} + 19849 p T^{6} - 882 p^{2} T^{7} + 229 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 89 | \( 1 + 28 T + 104 T^{2} - 1736 T^{3} + 31273 T^{4} + 611939 T^{5} - 1780638 T^{6} - 18973932 T^{7} + 740914101 T^{8} + 3271180573 T^{9} - 40614588329 T^{10} + 3271180573 p T^{11} + 740914101 p^{2} T^{12} - 18973932 p^{3} T^{13} - 1780638 p^{4} T^{14} + 611939 p^{5} T^{15} + 31273 p^{6} T^{16} - 1736 p^{7} T^{17} + 104 p^{8} T^{18} + 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( ( 1 - 12 T + 341 T^{2} - 29 p T^{3} + 54712 T^{4} - 367945 T^{5} + 54712 p T^{6} - 29 p^{3} T^{7} + 341 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
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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
−3.79900579645261041706688193941, −3.77687819881981192170125181738, −3.75363418926082418244506043854, −3.66166160728175994835616965071, −3.62539170091085997319565481116, −3.37310206428250416129392449125, −3.36027428857783327435270249649, −3.15499169639933630134750846514, −2.97085593764786714275405364153, −2.81521984906619434725858539246, −2.77482759207447622561329809513, −2.64068758623870610240380326289, −2.50976659935337113580309744003, −2.49451512629000778338469837957, −2.13027730405071359139973465030, −1.90942043892350860101929773542, −1.83007745641334021022329031318, −1.78573938528367162337928328664, −1.60979761625620545671991949265, −1.24386171141654837471150465614, −1.21230099400409468048425297378, −1.14197204049237352751689208855, −0.799637964423740662669410121731, −0.67038782869256459032301254888, −0.30289335633676641798923121586, 0.30289335633676641798923121586, 0.67038782869256459032301254888, 0.799637964423740662669410121731, 1.14197204049237352751689208855, 1.21230099400409468048425297378, 1.24386171141654837471150465614, 1.60979761625620545671991949265, 1.78573938528367162337928328664, 1.83007745641334021022329031318, 1.90942043892350860101929773542, 2.13027730405071359139973465030, 2.49451512629000778338469837957, 2.50976659935337113580309744003, 2.64068758623870610240380326289, 2.77482759207447622561329809513, 2.81521984906619434725858539246, 2.97085593764786714275405364153, 3.15499169639933630134750846514, 3.36027428857783327435270249649, 3.37310206428250416129392449125, 3.62539170091085997319565481116, 3.66166160728175994835616965071, 3.75363418926082418244506043854, 3.77687819881981192170125181738, 3.79900579645261041706688193941
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.