Dirichlet series
| L(s) = 1 | − 5·2-s + 10·4-s + 5-s + 5·7-s − 5·8-s − 5·10-s + 11·11-s + 6·13-s − 25·14-s − 20·16-s − 2·17-s − 6·19-s + 10·20-s − 55·22-s + 5·23-s + 13·25-s − 30·26-s + 50·28-s + 8·29-s + 4·31-s + 49·32-s + 10·34-s + 5·35-s − 28·37-s + 30·38-s − 5·40-s + 24·41-s + ⋯ |
| L(s) = 1 | − 3.53·2-s + 5·4-s + 0.447·5-s + 1.88·7-s − 1.76·8-s − 1.58·10-s + 3.31·11-s + 1.66·13-s − 6.68·14-s − 5·16-s − 0.485·17-s − 1.37·19-s + 2.23·20-s − 11.7·22-s + 1.04·23-s + 13/5·25-s − 5.88·26-s + 9.44·28-s + 1.48·29-s + 0.718·31-s + 8.66·32-s + 1.71·34-s + 0.845·35-s − 4.60·37-s + 4.86·38-s − 0.790·40-s + 3.74·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{30} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{30} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 3^{30} \cdot 23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.20422\times 10^{9}\) |
| Root analytic conductor: | \(3.14919\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 3^{30} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(8.212480197\) |
| \(L(\frac12)\) | \(\approx\) | \(8.212480197\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{5} \) |
| 3 | \( 1 \) | |
| 23 | \( ( 1 - T + T^{2} )^{5} \) | |
| good | 5 | \( 1 - T - 12 T^{2} + 17 T^{3} + 59 T^{4} - 96 T^{5} - 151 T^{6} + 91 T^{7} + 122 p T^{8} + 71 p T^{9} - 3969 T^{10} + 71 p^{2} T^{11} + 122 p^{3} T^{12} + 91 p^{3} T^{13} - 151 p^{4} T^{14} - 96 p^{5} T^{15} + 59 p^{6} T^{16} + 17 p^{7} T^{17} - 12 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) |
| 7 | \( 1 - 5 T + T^{2} + 36 T^{3} - 60 T^{4} - 3 p T^{5} + 579 T^{6} - 2388 T^{7} + 1674 T^{8} + 1420 p T^{9} - 4649 p T^{10} + 1420 p^{2} T^{11} + 1674 p^{2} T^{12} - 2388 p^{3} T^{13} + 579 p^{4} T^{14} - 3 p^{6} T^{15} - 60 p^{6} T^{16} + 36 p^{7} T^{17} + p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - p T + 51 T^{2} - 158 T^{3} + 404 T^{4} - 339 T^{5} - 2173 T^{6} + 9524 T^{7} - 28172 T^{8} + 42302 T^{9} + 16173 T^{10} + 42302 p T^{11} - 28172 p^{2} T^{12} + 9524 p^{3} T^{13} - 2173 p^{4} T^{14} - 339 p^{5} T^{15} + 404 p^{6} T^{16} - 158 p^{7} T^{17} + 51 p^{8} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - 6 T + 5 T^{2} - 20 T^{3} + 45 T^{4} + 1202 T^{5} - 2647 T^{6} - 4383 T^{7} + 268 p T^{8} - 117441 T^{9} + 927573 T^{10} - 117441 p T^{11} + 268 p^{3} T^{12} - 4383 p^{3} T^{13} - 2647 p^{4} T^{14} + 1202 p^{5} T^{15} + 45 p^{6} T^{16} - 20 p^{7} T^{17} + 5 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( ( 1 + T + 10 T^{2} - 6 T^{3} + 356 T^{4} + 1255 T^{5} + 356 p T^{6} - 6 p^{2} T^{7} + 10 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 19 | \( ( 1 + 3 T + 46 T^{2} + 185 T^{3} + 1519 T^{4} + 4007 T^{5} + 1519 p T^{6} + 185 p^{2} T^{7} + 46 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 29 | \( 1 - 8 T - 24 T^{2} + 238 T^{3} + 404 T^{4} - 1089 T^{5} - 19663 T^{6} + 48341 T^{7} - 74519 T^{8} - 3115828 T^{9} + 37379931 T^{10} - 3115828 p T^{11} - 74519 p^{2} T^{12} + 48341 p^{3} T^{13} - 19663 p^{4} T^{14} - 1089 p^{5} T^{15} + 404 p^{6} T^{16} + 238 p^{7} T^{17} - 24 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 4 T - 55 T^{2} - 234 T^{3} + 3233 T^{4} + 16236 T^{5} + 2641 T^{6} - 895791 T^{7} - 2656330 T^{8} + 7293825 T^{9} + 184807975 T^{10} + 7293825 p T^{11} - 2656330 p^{2} T^{12} - 895791 p^{3} T^{13} + 2641 p^{4} T^{14} + 16236 p^{5} T^{15} + 3233 p^{6} T^{16} - 234 p^{7} T^{17} - 55 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( ( 1 + 14 T + 132 T^{2} + 921 T^{3} + 7524 T^{4} + 49011 T^{5} + 7524 p T^{6} + 921 p^{2} T^{7} + 132 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 41 | \( 1 - 24 T + 209 T^{2} - 882 T^{3} + 5649 T^{4} - 63918 T^{5} + 322905 T^{6} + 218475 T^{7} - 8289378 T^{8} + 32890059 T^{9} - 110377551 T^{10} + 32890059 p T^{11} - 8289378 p^{2} T^{12} + 218475 p^{3} T^{13} + 322905 p^{4} T^{14} - 63918 p^{5} T^{15} + 5649 p^{6} T^{16} - 882 p^{7} T^{17} + 209 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 27 T + 290 T^{2} - 1583 T^{3} + 7305 T^{4} - 69718 T^{5} + 648713 T^{6} - 4490445 T^{7} + 30954166 T^{8} - 224355969 T^{9} + 1527879303 T^{10} - 224355969 p T^{11} + 30954166 p^{2} T^{12} - 4490445 p^{3} T^{13} + 648713 p^{4} T^{14} - 69718 p^{5} T^{15} + 7305 p^{6} T^{16} - 1583 p^{7} T^{17} + 290 p^{8} T^{18} - 27 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 9 T + 20 T^{2} - 585 T^{3} + 3402 T^{4} + 14085 T^{5} + 55158 T^{6} - 683181 T^{7} - 8274507 T^{8} + 34529310 T^{9} + 155202156 T^{10} + 34529310 p T^{11} - 8274507 p^{2} T^{12} - 683181 p^{3} T^{13} + 55158 p^{4} T^{14} + 14085 p^{5} T^{15} + 3402 p^{6} T^{16} - 585 p^{7} T^{17} + 20 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( ( 1 - 13 T + 232 T^{2} - 2139 T^{3} + 22997 T^{4} - 153913 T^{5} + 22997 p T^{6} - 2139 p^{2} T^{7} + 232 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 59 | \( 1 - 9 T - 85 T^{2} - 468 T^{3} + 16164 T^{4} + 747 p T^{5} + 3498 T^{6} - 11276145 T^{7} - 18667932 T^{8} + 74287899 T^{9} + 6051201705 T^{10} + 74287899 p T^{11} - 18667932 p^{2} T^{12} - 11276145 p^{3} T^{13} + 3498 p^{4} T^{14} + 747 p^{6} T^{15} + 16164 p^{6} T^{16} - 468 p^{7} T^{17} - 85 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 3 T - 175 T^{2} - 500 T^{3} + 19086 T^{4} + 100685 T^{5} - 858277 T^{6} - 9540030 T^{7} + 10218070 T^{8} + 241004658 T^{9} + 1530344991 T^{10} + 241004658 p T^{11} + 10218070 p^{2} T^{12} - 9540030 p^{3} T^{13} - 858277 p^{4} T^{14} + 100685 p^{5} T^{15} + 19086 p^{6} T^{16} - 500 p^{7} T^{17} - 175 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 5 T - 299 T^{2} + 936 T^{3} + 55920 T^{4} - 113781 T^{5} - 7105101 T^{6} + 7329912 T^{7} + 10337562 p T^{8} - 229515980 T^{9} - 52036763723 T^{10} - 229515980 p T^{11} + 10337562 p^{3} T^{12} + 7329912 p^{3} T^{13} - 7105101 p^{4} T^{14} - 113781 p^{5} T^{15} + 55920 p^{6} T^{16} + 936 p^{7} T^{17} - 299 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( ( 1 + 27 T + 505 T^{2} + 6759 T^{3} + 72487 T^{4} + 666207 T^{5} + 72487 p T^{6} + 6759 p^{2} T^{7} + 505 p^{3} T^{8} + 27 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 73 | \( ( 1 - 17 T + 380 T^{2} - 4145 T^{3} + 53551 T^{4} - 424281 T^{5} + 53551 p T^{6} - 4145 p^{2} T^{7} + 380 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 79 | \( 1 + 11 T - 124 T^{2} - 87 T^{3} + 22175 T^{4} - 61854 T^{5} - 1054601 T^{6} + 13861761 T^{7} + 12254294 T^{8} - 250303683 T^{9} + 7884566791 T^{10} - 250303683 p T^{11} + 12254294 p^{2} T^{12} + 13861761 p^{3} T^{13} - 1054601 p^{4} T^{14} - 61854 p^{5} T^{15} + 22175 p^{6} T^{16} - 87 p^{7} T^{17} - 124 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 23 T + 18 T^{2} + 1561 T^{3} + 22715 T^{4} - 255954 T^{5} - 3000457 T^{6} + 18411491 T^{7} + 297205762 T^{8} - 57558973 T^{9} - 34600311825 T^{10} - 57558973 p T^{11} + 297205762 p^{2} T^{12} + 18411491 p^{3} T^{13} - 3000457 p^{4} T^{14} - 255954 p^{5} T^{15} + 22715 p^{6} T^{16} + 1561 p^{7} T^{17} + 18 p^{8} T^{18} - 23 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( ( 1 + 39 T + 928 T^{2} + 15117 T^{3} + 194869 T^{4} + 2010321 T^{5} + 194869 p T^{6} + 15117 p^{2} T^{7} + 928 p^{3} T^{8} + 39 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 97 | \( 1 - 28 T + 68 T^{2} + 2214 T^{3} + 42620 T^{4} - 729225 T^{5} - 4353971 T^{6} + 33695709 T^{7} + 1171263797 T^{8} - 4174326396 T^{9} - 90283522193 T^{10} - 4174326396 p T^{11} + 1171263797 p^{2} T^{12} + 33695709 p^{3} T^{13} - 4353971 p^{4} T^{14} - 729225 p^{5} T^{15} + 42620 p^{6} T^{16} + 2214 p^{7} T^{17} + 68 p^{8} T^{18} - 28 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−3.53034198944142164696860073793, −3.39045021340696780411795838630, −3.22565980988562059487186633053, −3.21067155659491487049685055697, −3.15239848904030975788836759723, −2.94628125023888361044358996084, −2.56912783823698972617323174867, −2.52273095373550513675993846150, −2.44313077797932870240210343365, −2.35680974899961803076414212275, −2.25174261159652806885922666383, −2.16120724175179638111611892472, −2.08838721148899656832733481514, −2.01163114114572784312108759508, −1.56202528797424962748422988605, −1.55364019796347634012140356101, −1.42734816929960346421228838946, −1.39716194407234968445176107659, −1.15112008761602440674510232431, −1.09029803298247973131339811537, −0.846718264705156951271421727500, −0.843562838153737774318966616727, −0.69330760754172318773167632367, −0.61082118582300499825005826886, −0.45574893137317086555767727631, 0.45574893137317086555767727631, 0.61082118582300499825005826886, 0.69330760754172318773167632367, 0.843562838153737774318966616727, 0.846718264705156951271421727500, 1.09029803298247973131339811537, 1.15112008761602440674510232431, 1.39716194407234968445176107659, 1.42734816929960346421228838946, 1.55364019796347634012140356101, 1.56202528797424962748422988605, 2.01163114114572784312108759508, 2.08838721148899656832733481514, 2.16120724175179638111611892472, 2.25174261159652806885922666383, 2.35680974899961803076414212275, 2.44313077797932870240210343365, 2.52273095373550513675993846150, 2.56912783823698972617323174867, 2.94628125023888361044358996084, 3.15239848904030975788836759723, 3.21067155659491487049685055697, 3.22565980988562059487186633053, 3.39045021340696780411795838630, 3.53034198944142164696860073793
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.