Dirichlet series
| L(s) = 1 | − 6·5-s + 4·7-s − 8·11-s − 2·13-s − 6·17-s − 20·23-s + 5·25-s − 8·29-s − 4·31-s − 24·35-s − 4·37-s + 14·41-s + 20·43-s − 48·47-s − 28·49-s − 22·53-s + 48·55-s − 2·59-s − 8·61-s + 12·65-s − 6·67-s − 20·71-s + 20·73-s − 32·77-s − 4·79-s − 46·83-s + 36·85-s + ⋯ |
| L(s) = 1 | − 2.68·5-s + 1.51·7-s − 2.41·11-s − 0.554·13-s − 1.45·17-s − 4.17·23-s + 25-s − 1.48·29-s − 0.718·31-s − 4.05·35-s − 0.657·37-s + 2.18·41-s + 3.04·43-s − 7.00·47-s − 4·49-s − 3.02·53-s + 6.47·55-s − 0.260·59-s − 1.02·61-s + 1.48·65-s − 0.733·67-s − 2.37·71-s + 2.34·73-s − 3.64·77-s − 0.450·79-s − 5.04·83-s + 3.90·85-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 167^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 167^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{20} \cdot 3^{20} \cdot 167^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.50070\times 10^{16}\) |
| Root analytic conductor: | \(6.92864\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(10\) |
| Selberg data: | \((20,\ 2^{20} \cdot 3^{20} \cdot 167^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 167 | \( ( 1 - T )^{10} \) | |
| good | 5 | \( 1 + 6 T + 31 T^{2} + 118 T^{3} + 84 p T^{4} + 1262 T^{5} + 3754 T^{6} + 10022 T^{7} + 26368 T^{8} + 63276 T^{9} + 148879 T^{10} + 63276 p T^{11} + 26368 p^{2} T^{12} + 10022 p^{3} T^{13} + 3754 p^{4} T^{14} + 1262 p^{5} T^{15} + 84 p^{7} T^{16} + 118 p^{7} T^{17} + 31 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) |
| 7 | \( 1 - 4 T + 44 T^{2} - 170 T^{3} + 960 T^{4} - 3378 T^{5} + 13757 T^{6} - 42578 T^{7} + 143706 T^{8} - 55448 p T^{9} + 1144593 T^{10} - 55448 p^{2} T^{11} + 143706 p^{2} T^{12} - 42578 p^{3} T^{13} + 13757 p^{4} T^{14} - 3378 p^{5} T^{15} + 960 p^{6} T^{16} - 170 p^{7} T^{17} + 44 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 + 8 T + 73 T^{2} + 368 T^{3} + 2114 T^{4} + 9084 T^{5} + 43602 T^{6} + 170180 T^{7} + 687924 T^{8} + 2358932 T^{9} + 8343787 T^{10} + 2358932 p T^{11} + 687924 p^{2} T^{12} + 170180 p^{3} T^{13} + 43602 p^{4} T^{14} + 9084 p^{5} T^{15} + 2114 p^{6} T^{16} + 368 p^{7} T^{17} + 73 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + 2 T + 72 T^{2} + 186 T^{3} + 2796 T^{4} + 7554 T^{5} + 74088 T^{6} + 192782 T^{7} + 111081 p T^{8} + 3441600 T^{9} + 21428349 T^{10} + 3441600 p T^{11} + 111081 p^{3} T^{12} + 192782 p^{3} T^{13} + 74088 p^{4} T^{14} + 7554 p^{5} T^{15} + 2796 p^{6} T^{16} + 186 p^{7} T^{17} + 72 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 6 T + 73 T^{2} + 278 T^{3} + 2415 T^{4} + 5388 T^{5} + 43190 T^{6} + 6396 T^{7} + 466109 T^{8} - 1601566 T^{9} + 4603487 T^{10} - 1601566 p T^{11} + 466109 p^{2} T^{12} + 6396 p^{3} T^{13} + 43190 p^{4} T^{14} + 5388 p^{5} T^{15} + 2415 p^{6} T^{16} + 278 p^{7} T^{17} + 73 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 72 T^{2} + 2510 T^{4} + 642 T^{5} + 57562 T^{6} + 39308 T^{7} + 1006725 T^{8} + 1125458 T^{9} + 17251013 T^{10} + 1125458 p T^{11} + 1006725 p^{2} T^{12} + 39308 p^{3} T^{13} + 57562 p^{4} T^{14} + 642 p^{5} T^{15} + 2510 p^{6} T^{16} + 72 p^{8} T^{18} + p^{10} T^{20} \) | |
| 23 | \( 1 + 20 T + 327 T^{2} + 3770 T^{3} + 37980 T^{4} + 319026 T^{5} + 2419166 T^{6} + 16113194 T^{7} + 98396320 T^{8} + 538676186 T^{9} + 2720812683 T^{10} + 538676186 p T^{11} + 98396320 p^{2} T^{12} + 16113194 p^{3} T^{13} + 2419166 p^{4} T^{14} + 319026 p^{5} T^{15} + 37980 p^{6} T^{16} + 3770 p^{7} T^{17} + 327 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 8 T + 191 T^{2} + 1228 T^{3} + 16872 T^{4} + 92576 T^{5} + 948244 T^{6} + 4588668 T^{7} + 38877162 T^{8} + 168989592 T^{9} + 1253952287 T^{10} + 168989592 p T^{11} + 38877162 p^{2} T^{12} + 4588668 p^{3} T^{13} + 948244 p^{4} T^{14} + 92576 p^{5} T^{15} + 16872 p^{6} T^{16} + 1228 p^{7} T^{17} + 191 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 4 T + 142 T^{2} + 458 T^{3} + 355 p T^{4} + 30384 T^{5} + 601849 T^{6} + 1434880 T^{7} + 25562280 T^{8} + 54307126 T^{9} + 878289431 T^{10} + 54307126 p T^{11} + 25562280 p^{2} T^{12} + 1434880 p^{3} T^{13} + 601849 p^{4} T^{14} + 30384 p^{5} T^{15} + 355 p^{7} T^{16} + 458 p^{7} T^{17} + 142 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 4 T + 203 T^{2} + 854 T^{3} + 22808 T^{4} + 91856 T^{5} + 1727746 T^{6} + 6489918 T^{7} + 96059522 T^{8} + 326243728 T^{9} + 4058751253 T^{10} + 326243728 p T^{11} + 96059522 p^{2} T^{12} + 6489918 p^{3} T^{13} + 1727746 p^{4} T^{14} + 91856 p^{5} T^{15} + 22808 p^{6} T^{16} + 854 p^{7} T^{17} + 203 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 14 T + 290 T^{2} - 2856 T^{3} + 35413 T^{4} - 279742 T^{5} + 2727407 T^{6} - 18824774 T^{7} + 159158784 T^{8} - 986229368 T^{9} + 7356881549 T^{10} - 986229368 p T^{11} + 159158784 p^{2} T^{12} - 18824774 p^{3} T^{13} + 2727407 p^{4} T^{14} - 279742 p^{5} T^{15} + 35413 p^{6} T^{16} - 2856 p^{7} T^{17} + 290 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 20 T + 424 T^{2} - 5856 T^{3} + 77619 T^{4} - 832522 T^{5} + 8433959 T^{6} - 74018054 T^{7} + 612099082 T^{8} - 4500726094 T^{9} + 31190623967 T^{10} - 4500726094 p T^{11} + 612099082 p^{2} T^{12} - 74018054 p^{3} T^{13} + 8433959 p^{4} T^{14} - 832522 p^{5} T^{15} + 77619 p^{6} T^{16} - 5856 p^{7} T^{17} + 424 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 + 48 T + 1383 T^{2} + 28760 T^{3} + 477360 T^{4} + 6596704 T^{5} + 78236101 T^{6} + 809345700 T^{7} + 7394658901 T^{8} + 60054409196 T^{9} + 435533508747 T^{10} + 60054409196 p T^{11} + 7394658901 p^{2} T^{12} + 809345700 p^{3} T^{13} + 78236101 p^{4} T^{14} + 6596704 p^{5} T^{15} + 477360 p^{6} T^{16} + 28760 p^{7} T^{17} + 1383 p^{8} T^{18} + 48 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 22 T + 551 T^{2} + 8384 T^{3} + 126720 T^{4} + 1499280 T^{5} + 17111347 T^{6} + 165696154 T^{7} + 28978505 p T^{8} + 12456885392 T^{9} + 96538460331 T^{10} + 12456885392 p T^{11} + 28978505 p^{3} T^{12} + 165696154 p^{3} T^{13} + 17111347 p^{4} T^{14} + 1499280 p^{5} T^{15} + 126720 p^{6} T^{16} + 8384 p^{7} T^{17} + 551 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 2 T + 315 T^{2} + 360 T^{3} + 44416 T^{4} + 12072 T^{5} + 3832523 T^{6} - 2142498 T^{7} + 245635973 T^{8} - 288960572 T^{9} + 14343462495 T^{10} - 288960572 p T^{11} + 245635973 p^{2} T^{12} - 2142498 p^{3} T^{13} + 3832523 p^{4} T^{14} + 12072 p^{5} T^{15} + 44416 p^{6} T^{16} + 360 p^{7} T^{17} + 315 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 8 T + 256 T^{2} + 2016 T^{3} + 36630 T^{4} + 291138 T^{5} + 3934246 T^{6} + 29798532 T^{7} + 333755353 T^{8} + 2290787898 T^{9} + 22604219961 T^{10} + 2290787898 p T^{11} + 333755353 p^{2} T^{12} + 29798532 p^{3} T^{13} + 3934246 p^{4} T^{14} + 291138 p^{5} T^{15} + 36630 p^{6} T^{16} + 2016 p^{7} T^{17} + 256 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 6 T + 356 T^{2} + 1810 T^{3} + 68636 T^{4} + 318454 T^{5} + 9047396 T^{6} + 37814198 T^{7} + 885832585 T^{8} + 3336093832 T^{9} + 67206009417 T^{10} + 3336093832 p T^{11} + 885832585 p^{2} T^{12} + 37814198 p^{3} T^{13} + 9047396 p^{4} T^{14} + 318454 p^{5} T^{15} + 68636 p^{6} T^{16} + 1810 p^{7} T^{17} + 356 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 20 T + 413 T^{2} + 5416 T^{3} + 78441 T^{4} + 862990 T^{5} + 10116693 T^{6} + 96430862 T^{7} + 987629625 T^{8} + 8457342282 T^{9} + 77624229705 T^{10} + 8457342282 p T^{11} + 987629625 p^{2} T^{12} + 96430862 p^{3} T^{13} + 10116693 p^{4} T^{14} + 862990 p^{5} T^{15} + 78441 p^{6} T^{16} + 5416 p^{7} T^{17} + 413 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 20 T + 652 T^{2} - 130 p T^{3} + 180120 T^{4} - 28484 p T^{5} + 29480734 T^{6} - 283788426 T^{7} + 3311081589 T^{8} - 27471524666 T^{9} + 276476108261 T^{10} - 27471524666 p T^{11} + 3311081589 p^{2} T^{12} - 283788426 p^{3} T^{13} + 29480734 p^{4} T^{14} - 28484 p^{6} T^{15} + 180120 p^{6} T^{16} - 130 p^{8} T^{17} + 652 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 4 T + 684 T^{2} + 2424 T^{3} + 2741 p T^{4} + 676826 T^{5} + 41907971 T^{6} + 114663634 T^{7} + 5502155218 T^{8} + 13018598954 T^{9} + 512639976643 T^{10} + 13018598954 p T^{11} + 5502155218 p^{2} T^{12} + 114663634 p^{3} T^{13} + 41907971 p^{4} T^{14} + 676826 p^{5} T^{15} + 2741 p^{7} T^{16} + 2424 p^{7} T^{17} + 684 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 46 T + 1627 T^{2} + 39946 T^{3} + 835088 T^{4} + 14345466 T^{5} + 218449939 T^{6} + 2885544604 T^{7} + 34445382845 T^{8} + 364612104940 T^{9} + 3516567220763 T^{10} + 364612104940 p T^{11} + 34445382845 p^{2} T^{12} + 2885544604 p^{3} T^{13} + 218449939 p^{4} T^{14} + 14345466 p^{5} T^{15} + 835088 p^{6} T^{16} + 39946 p^{7} T^{17} + 1627 p^{8} T^{18} + 46 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 8 T + 428 T^{2} - 3052 T^{3} + 101630 T^{4} - 660200 T^{5} + 17030276 T^{6} - 100257152 T^{7} + 2164606323 T^{8} - 11514510644 T^{9} + 216271050931 T^{10} - 11514510644 p T^{11} + 2164606323 p^{2} T^{12} - 100257152 p^{3} T^{13} + 17030276 p^{4} T^{14} - 660200 p^{5} T^{15} + 101630 p^{6} T^{16} - 3052 p^{7} T^{17} + 428 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 + 34 T + 919 T^{2} + 17748 T^{3} + 313381 T^{4} + 4683554 T^{5} + 65950679 T^{6} + 827454592 T^{7} + 9878651509 T^{8} + 106806253562 T^{9} + 1101981063735 T^{10} + 106806253562 p T^{11} + 9878651509 p^{2} T^{12} + 827454592 p^{3} T^{13} + 65950679 p^{4} T^{14} + 4683554 p^{5} T^{15} + 313381 p^{6} T^{16} + 17748 p^{7} T^{17} + 919 p^{8} T^{18} + 34 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−3.24696305624484658833165159758, −3.15499478225021175836969380515, −3.07797988259832198863905387427, −2.95254979856778706298408541367, −2.92276269156101057065035514182, −2.64363656165203948921997191142, −2.62609187304312314001214548426, −2.50227902652188366619890343642, −2.45848247633705908922965176903, −2.34186075606049088019405765483, −2.30557341048299568776556791392, −2.25262196055315800254741644641, −2.18135146191781478054938135504, −2.08438008271688525957447165210, −1.96621258993475261290530329663, −1.65276502533495315237937018816, −1.59665424577840920866175202719, −1.55365766687917556057077543065, −1.54540101434027545578433791205, −1.46753509605523472442472427598, −1.28322454468950760554651346196, −1.26076542313847149828147183530, −1.09388233856811213329273638772, −1.05128439162928493098650646306, −1.00935737261585278472614730034, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.00935737261585278472614730034, 1.05128439162928493098650646306, 1.09388233856811213329273638772, 1.26076542313847149828147183530, 1.28322454468950760554651346196, 1.46753509605523472442472427598, 1.54540101434027545578433791205, 1.55365766687917556057077543065, 1.59665424577840920866175202719, 1.65276502533495315237937018816, 1.96621258993475261290530329663, 2.08438008271688525957447165210, 2.18135146191781478054938135504, 2.25262196055315800254741644641, 2.30557341048299568776556791392, 2.34186075606049088019405765483, 2.45848247633705908922965176903, 2.50227902652188366619890343642, 2.62609187304312314001214548426, 2.64363656165203948921997191142, 2.92276269156101057065035514182, 2.95254979856778706298408541367, 3.07797988259832198863905387427, 3.15499478225021175836969380515, 3.24696305624484658833165159758
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.