Dirichlet series
| L(s) = 1 | − 4·2-s + 2·4-s + 9·5-s + 7·7-s + 11·8-s − 36·10-s + 12·11-s + 15·13-s − 28·14-s − 11·16-s − 17-s − 13·19-s + 18·20-s − 48·22-s + 7·23-s + 49·25-s − 60·26-s + 14·28-s + 24·29-s − 31-s − 11·32-s + 4·34-s + 63·35-s + 2·37-s + 52·38-s + 99·40-s + 7·41-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4-s + 4.02·5-s + 2.64·7-s + 3.88·8-s − 11.3·10-s + 3.61·11-s + 4.16·13-s − 7.48·14-s − 2.75·16-s − 0.242·17-s − 2.98·19-s + 4.02·20-s − 10.2·22-s + 1.45·23-s + 49/5·25-s − 11.7·26-s + 2.64·28-s + 4.45·29-s − 0.179·31-s − 1.94·32-s + 0.685·34-s + 10.6·35-s + 0.328·37-s + 8.43·38-s + 15.6·40-s + 1.09·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 67^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 67^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{20} \cdot 67^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.69798\times 10^{6}\) |
| Root analytic conductor: | \(2.19430\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{20} \cdot 67^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(7.774419815\) |
| \(L(\frac12)\) | \(\approx\) | \(7.774419815\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 67 | \( 1 - 32 T + 474 T^{2} - 5103 T^{3} + 51404 T^{4} - 463759 T^{5} + 51404 p T^{6} - 5103 p^{2} T^{7} + 474 p^{3} T^{8} - 32 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 2 | \( 1 + p^{2} T + 7 p T^{2} + 37 T^{3} + 87 T^{4} + 175 T^{5} + 41 p^{3} T^{6} + 555 T^{7} + 893 T^{8} + 335 p^{2} T^{9} + 1957 T^{10} + 335 p^{3} T^{11} + 893 p^{2} T^{12} + 555 p^{3} T^{13} + 41 p^{7} T^{14} + 175 p^{5} T^{15} + 87 p^{6} T^{16} + 37 p^{7} T^{17} + 7 p^{9} T^{18} + p^{11} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 - 9 T + 32 T^{2} - 56 T^{3} + 58 T^{4} - 66 T^{5} + 8 p T^{6} + 16 p T^{7} + p^{3} T^{8} - 41 p T^{9} - 1024 T^{10} - 41 p^{2} T^{11} + p^{5} T^{12} + 16 p^{4} T^{13} + 8 p^{5} T^{14} - 66 p^{5} T^{15} + 58 p^{6} T^{16} - 56 p^{7} T^{17} + 32 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 7 | \( 1 - p T + 20 T^{2} - 36 T^{3} + 5 p T^{4} + 139 T^{5} - 866 T^{6} + 2042 T^{7} - 928 T^{8} - 3662 T^{9} + 6357 T^{10} - 3662 p T^{11} - 928 p^{2} T^{12} + 2042 p^{3} T^{13} - 866 p^{4} T^{14} + 139 p^{5} T^{15} + 5 p^{7} T^{16} - 36 p^{7} T^{17} + 20 p^{8} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 12 T + 67 T^{2} - 210 T^{3} + 331 T^{4} + 54 T^{5} - 1253 T^{6} + 5862 T^{7} - 54911 T^{8} + 380676 T^{9} - 1604459 T^{10} + 380676 p T^{11} - 54911 p^{2} T^{12} + 5862 p^{3} T^{13} - 1253 p^{4} T^{14} + 54 p^{5} T^{15} + 331 p^{6} T^{16} - 210 p^{7} T^{17} + 67 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - 15 T + 124 T^{2} - 785 T^{3} + 3937 T^{4} - 15036 T^{5} + 39059 T^{6} - 24447 T^{7} - 441032 T^{8} + 3210869 T^{9} - 13871441 T^{10} + 3210869 p T^{11} - 441032 p^{2} T^{12} - 24447 p^{3} T^{13} + 39059 p^{4} T^{14} - 15036 p^{5} T^{15} + 3937 p^{6} T^{16} - 785 p^{7} T^{17} + 124 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + T - 16 T^{2} - 110 T^{3} + 162 T^{4} + 2989 T^{5} + 6032 T^{6} - 2629 p T^{7} - 232113 T^{8} + 394161 T^{9} + 4546981 T^{10} + 394161 p T^{11} - 232113 p^{2} T^{12} - 2629 p^{4} T^{13} + 6032 p^{4} T^{14} + 2989 p^{5} T^{15} + 162 p^{6} T^{16} - 110 p^{7} T^{17} - 16 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 13 T + 106 T^{2} + 713 T^{3} + 207 p T^{4} + 18596 T^{5} + 87623 T^{6} + 423897 T^{7} + 2059028 T^{8} + 10075747 T^{9} + 46433641 T^{10} + 10075747 p T^{11} + 2059028 p^{2} T^{12} + 423897 p^{3} T^{13} + 87623 p^{4} T^{14} + 18596 p^{5} T^{15} + 207 p^{7} T^{16} + 713 p^{7} T^{17} + 106 p^{8} T^{18} + 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - 7 T + 26 T^{2} - 98 T^{3} + 792 T^{4} - 3477 T^{5} + 3318 T^{6} + 73861 T^{7} - 660375 T^{8} + 2981913 T^{9} - 13802689 T^{10} + 2981913 p T^{11} - 660375 p^{2} T^{12} + 73861 p^{3} T^{13} + 3318 p^{4} T^{14} - 3477 p^{5} T^{15} + 792 p^{6} T^{16} - 98 p^{7} T^{17} + 26 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( ( 1 - 12 T + 6 p T^{2} - 1345 T^{3} + 10738 T^{4} - 57695 T^{5} + 10738 p T^{6} - 1345 p^{2} T^{7} + 6 p^{4} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 31 | \( 1 + T - 96 T^{2} + 38 T^{3} + 4554 T^{4} - 9659 T^{5} - 160986 T^{6} + 399792 T^{7} + 4471660 T^{8} - 5189602 T^{9} - 118020297 T^{10} - 5189602 p T^{11} + 4471660 p^{2} T^{12} + 399792 p^{3} T^{13} - 160986 p^{4} T^{14} - 9659 p^{5} T^{15} + 4554 p^{6} T^{16} + 38 p^{7} T^{17} - 96 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( ( 1 - T + 115 T^{2} - 6 p T^{3} + 7045 T^{4} - 11183 T^{5} + 7045 p T^{6} - 6 p^{3} T^{7} + 115 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 41 | \( 1 - 7 T + 96 T^{2} - 704 T^{3} + 8175 T^{4} - 55751 T^{5} + 446660 T^{6} - 3422298 T^{7} + 24649420 T^{8} - 173770812 T^{9} + 1031860521 T^{10} - 173770812 p T^{11} + 24649420 p^{2} T^{12} - 3422298 p^{3} T^{13} + 446660 p^{4} T^{14} - 55751 p^{5} T^{15} + 8175 p^{6} T^{16} - 704 p^{7} T^{17} + 96 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 2 T - 6 T^{2} - 582 T^{3} - 5636 T^{4} + 4580 T^{5} + 240189 T^{6} + 2814538 T^{7} + 7041898 T^{8} - 89993528 T^{9} - 849756491 T^{10} - 89993528 p T^{11} + 7041898 p^{2} T^{12} + 2814538 p^{3} T^{13} + 240189 p^{4} T^{14} + 4580 p^{5} T^{15} - 5636 p^{6} T^{16} - 582 p^{7} T^{17} - 6 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 + 33 T + 514 T^{2} + 5335 T^{3} + 49047 T^{4} + 480304 T^{5} + 4578613 T^{6} + 38268043 T^{7} + 290594374 T^{8} + 2144844383 T^{9} + 15216567057 T^{10} + 2144844383 p T^{11} + 290594374 p^{2} T^{12} + 38268043 p^{3} T^{13} + 4578613 p^{4} T^{14} + 480304 p^{5} T^{15} + 49047 p^{6} T^{16} + 5335 p^{7} T^{17} + 514 p^{8} T^{18} + 33 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 21 T + 223 T^{2} + 1689 T^{3} + 5764 T^{4} - 30711 T^{5} - 472154 T^{6} - 2166249 T^{7} + 18603217 T^{8} + 404884458 T^{9} + 3651520423 T^{10} + 404884458 p T^{11} + 18603217 p^{2} T^{12} - 2166249 p^{3} T^{13} - 472154 p^{4} T^{14} - 30711 p^{5} T^{15} + 5764 p^{6} T^{16} + 1689 p^{7} T^{17} + 223 p^{8} T^{18} + 21 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 38 T + 780 T^{2} - 11580 T^{3} + 134805 T^{4} - 1271546 T^{5} + 9754596 T^{6} - 58954873 T^{7} + 257266380 T^{8} - 626546195 T^{9} + 693923825 T^{10} - 626546195 p T^{11} + 257266380 p^{2} T^{12} - 58954873 p^{3} T^{13} + 9754596 p^{4} T^{14} - 1271546 p^{5} T^{15} + 134805 p^{6} T^{16} - 11580 p^{7} T^{17} + 780 p^{8} T^{18} - 38 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 50 T + 1361 T^{2} + 27006 T^{3} + 436427 T^{4} + 6041668 T^{5} + 73537037 T^{6} + 800267862 T^{7} + 7871676041 T^{8} + 1154716662 p T^{9} + 575435713711 T^{10} + 1154716662 p^{2} T^{11} + 7871676041 p^{2} T^{12} + 800267862 p^{3} T^{13} + 73537037 p^{4} T^{14} + 6041668 p^{5} T^{15} + 436427 p^{6} T^{16} + 27006 p^{7} T^{17} + 1361 p^{8} T^{18} + 50 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 16 T + 229 T^{2} - 2880 T^{3} + 34067 T^{4} - 281918 T^{5} + 2410381 T^{6} - 16485108 T^{7} + 113568589 T^{8} - 557632108 T^{9} + 4892733385 T^{10} - 557632108 p T^{11} + 113568589 p^{2} T^{12} - 16485108 p^{3} T^{13} + 2410381 p^{4} T^{14} - 281918 p^{5} T^{15} + 34067 p^{6} T^{16} - 2880 p^{7} T^{17} + 229 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 3 T - 64 T^{2} - 975 T^{3} + 7674 T^{4} + 99215 T^{5} + 253197 T^{6} - 9519450 T^{7} - 42925319 T^{8} + 69163963 T^{9} + 7652930627 T^{10} + 69163963 p T^{11} - 42925319 p^{2} T^{12} - 9519450 p^{3} T^{13} + 253197 p^{4} T^{14} + 99215 p^{5} T^{15} + 7674 p^{6} T^{16} - 975 p^{7} T^{17} - 64 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 19 T + 62 T^{2} - 609 T^{3} - 6613 T^{4} - 122592 T^{5} - 483895 T^{6} + 8282371 T^{7} + 12917888 T^{8} - 330909161 T^{9} + 129979873 T^{10} - 330909161 p T^{11} + 12917888 p^{2} T^{12} + 8282371 p^{3} T^{13} - 483895 p^{4} T^{14} - 122592 p^{5} T^{15} - 6613 p^{6} T^{16} - 609 p^{7} T^{17} + 62 p^{8} T^{18} + 19 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 5 T - 278 T^{2} - 2190 T^{3} + 34190 T^{4} + 279669 T^{5} - 3046052 T^{6} - 13271839 T^{7} + 361100265 T^{8} + 134358741 T^{9} - 37550170305 T^{10} + 134358741 p T^{11} + 361100265 p^{2} T^{12} - 13271839 p^{3} T^{13} - 3046052 p^{4} T^{14} + 279669 p^{5} T^{15} + 34190 p^{6} T^{16} - 2190 p^{7} T^{17} - 278 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 7 T + 92 T^{2} - 309 T^{3} - 8327 T^{4} - 33802 T^{5} - 207607 T^{6} + 9191021 T^{7} + 57717494 T^{8} + 55533401 T^{9} - 837860429 T^{10} + 55533401 p T^{11} + 57717494 p^{2} T^{12} + 9191021 p^{3} T^{13} - 207607 p^{4} T^{14} - 33802 p^{5} T^{15} - 8327 p^{6} T^{16} - 309 p^{7} T^{17} + 92 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( ( 1 - 27 T + 605 T^{2} - 9441 T^{3} + 126309 T^{4} - 1321885 T^{5} + 126309 p T^{6} - 9441 p^{2} T^{7} + 605 p^{3} T^{8} - 27 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
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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
−4.05613939958161265381642932925, −3.96908525323752465326385900047, −3.92723431732745209371151891812, −3.58470208251434575435242562049, −3.31876986725617136272038431779, −3.31242985769767016146511747823, −3.25226509064300584966177298576, −3.18374799624901267342019444582, −2.93788994053725545507424728846, −2.80378663631025801848607307313, −2.74783520993624819418453132830, −2.45003648855993839461565855809, −2.42010780508427374014069007294, −2.01525937042674858074279799792, −1.95332921878164039525851026526, −1.84937235323535571910740086949, −1.61232979622665992908402488292, −1.54714980545896343288089740145, −1.44510642243594983477890508102, −1.44003947377979930805874847537, −1.35228526848397167815632241940, −0.798764712563969764476399839761, −0.77140219114721654860101484413, −0.74103690231089633391414768100, −0.71296569238839075747785582248, 0.71296569238839075747785582248, 0.74103690231089633391414768100, 0.77140219114721654860101484413, 0.798764712563969764476399839761, 1.35228526848397167815632241940, 1.44003947377979930805874847537, 1.44510642243594983477890508102, 1.54714980545896343288089740145, 1.61232979622665992908402488292, 1.84937235323535571910740086949, 1.95332921878164039525851026526, 2.01525937042674858074279799792, 2.42010780508427374014069007294, 2.45003648855993839461565855809, 2.74783520993624819418453132830, 2.80378663631025801848607307313, 2.93788994053725545507424728846, 3.18374799624901267342019444582, 3.25226509064300584966177298576, 3.31242985769767016146511747823, 3.31876986725617136272038431779, 3.58470208251434575435242562049, 3.92723431732745209371151891812, 3.96908525323752465326385900047, 4.05613939958161265381642932925
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.