Dirichlet series
| L(s) = 1 | + 5·2-s − 3·3-s + 10·4-s − 5-s − 15·6-s + 5·7-s + 5·8-s + 3·9-s − 5·10-s − 11·11-s − 30·12-s + 6·13-s + 25·14-s + 3·15-s − 20·16-s + 2·17-s + 15·18-s − 6·19-s − 10·20-s − 15·21-s − 55·22-s − 5·23-s − 15·24-s + 13·25-s + 30·26-s + 9·27-s + 50·28-s + ⋯ |
| L(s) = 1 | + 3.53·2-s − 1.73·3-s + 5·4-s − 0.447·5-s − 6.12·6-s + 1.88·7-s + 1.76·8-s + 9-s − 1.58·10-s − 3.31·11-s − 8.66·12-s + 1.66·13-s + 6.68·14-s + 0.774·15-s − 5·16-s + 0.485·17-s + 3.53·18-s − 1.37·19-s − 2.23·20-s − 3.27·21-s − 11.7·22-s − 1.04·23-s − 3.06·24-s + 13/5·25-s + 5.88·26-s + 1.73·27-s + 9.44·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 3^{20} \cdot 23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(155874.\) |
| Root analytic conductor: | \(1.81818\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 3^{20} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(4.369518575\) |
| \(L(\frac12)\) | \(\approx\) | \(4.369518575\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 + p T + 2 p T^{2} - 8 p T^{4} - 7 p^{2} T^{5} - 8 p^{2} T^{6} + 2 p^{4} T^{8} + p^{5} T^{9} + p^{5} T^{10} \) | |
| 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
−4.31614438339273500131975055632, −4.14920497446164269268180306456, −4.03036429513644087672201381908, −3.97536383064532456504524971833, −3.81810949445497966069421010249, −3.77957005975174960343429160679, −3.57808359849548183076835507827, −3.50132512760822309809313677123, −3.26105249934799425870477991597, −3.21740010107713385842097365574, −3.17329589530296970099717931838, −3.02358258991302223239838101484, −2.97270502645439724782294087870, −2.72178752541589522511375795264, −2.32930500956132890068337299037, −2.23221189192443564046680540563, −2.13764624967416130220729954907, −2.12409645534624956083425432119, −1.88895554314995649490598330062, −1.88551091773091280102064044679, −1.41347394058441208997029240118, −1.03606227550314843133327818850, −0.868000747050115453922118955093, −0.59457158024240716549585809484, −0.27213714686340533917418810016, 0.27213714686340533917418810016, 0.59457158024240716549585809484, 0.868000747050115453922118955093, 1.03606227550314843133327818850, 1.41347394058441208997029240118, 1.88551091773091280102064044679, 1.88895554314995649490598330062, 2.12409645534624956083425432119, 2.13764624967416130220729954907, 2.23221189192443564046680540563, 2.32930500956132890068337299037, 2.72178752541589522511375795264, 2.97270502645439724782294087870, 3.02358258991302223239838101484, 3.17329589530296970099717931838, 3.21740010107713385842097365574, 3.26105249934799425870477991597, 3.50132512760822309809313677123, 3.57808359849548183076835507827, 3.77957005975174960343429160679, 3.81810949445497966069421010249, 3.97536383064532456504524971833, 4.03036429513644087672201381908, 4.14920497446164269268180306456, 4.31614438339273500131975055632
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.