Dirichlet series
| L(s) = 1 | − 3·2-s + 9·3-s − 21·4-s + 8·5-s − 27·6-s − 24·7-s + 72·8-s − 191·9-s − 24·10-s + 50·11-s − 189·12-s − 624·13-s + 72·14-s + 72·15-s + 401·16-s − 292·17-s + 573·18-s + 305·19-s − 168·20-s − 216·21-s − 150·22-s + 98·23-s + 648·24-s + 1.25e3·25-s + 1.87e3·26-s − 1.96e3·27-s + 504·28-s + ⋯ |
| L(s) = 1 | − 3/4·2-s + 3-s − 1.31·4-s + 8/25·5-s − 3/4·6-s − 0.489·7-s + 9/8·8-s − 2.35·9-s − 0.239·10-s + 0.413·11-s − 1.31·12-s − 3.69·13-s + 0.367·14-s + 8/25·15-s + 1.56·16-s − 1.01·17-s + 1.76·18-s + 0.844·19-s − 0.419·20-s − 0.489·21-s − 0.309·22-s + 0.185·23-s + 9/8·24-s + 2.00·25-s + 2.76·26-s − 2.69·27-s + 9/14·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(19^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(19^{10}\right)^{s/2} \, \Gamma_{\C}(s+2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(19^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(854.042\) |
| Root analytic conductor: | \(1.40143\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 19^{10} ,\ ( \ : [2]^{10} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(0.2494351677\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2494351677\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 19 | \( 1 - 305 T + 452 p T^{2} + 75247 p^{2} T^{3} + 141391 p^{4} T^{4} - 272184 p^{6} T^{5} + 141391 p^{8} T^{6} + 75247 p^{10} T^{7} + 452 p^{13} T^{8} - 305 p^{16} T^{9} + p^{20} T^{10} \) |
| good | 2 | \( 1 + 3 T + 15 p T^{2} + 81 T^{3} + p^{8} T^{4} + 801 T^{5} - 2257 T^{6} - 8019 p T^{7} - 44527 p T^{8} - 128583 p^{2} T^{9} - 278413 p^{2} T^{10} - 128583 p^{6} T^{11} - 44527 p^{9} T^{12} - 8019 p^{13} T^{13} - 2257 p^{16} T^{14} + 801 p^{20} T^{15} + p^{32} T^{16} + 81 p^{28} T^{17} + 15 p^{33} T^{18} + 3 p^{36} T^{19} + p^{40} T^{20} \) |
| 3 | \( 1 - p^{2} T + 272 T^{2} - 245 p^{2} T^{3} + 34042 T^{4} - 34819 p^{2} T^{5} + 402158 p^{2} T^{6} - 405275 p^{4} T^{7} + 4395145 p^{4} T^{8} - 3413812 p^{6} T^{9} + 13701316 p^{7} T^{10} - 3413812 p^{10} T^{11} + 4395145 p^{12} T^{12} - 405275 p^{16} T^{13} + 402158 p^{18} T^{14} - 34819 p^{22} T^{15} + 34042 p^{24} T^{16} - 245 p^{30} T^{17} + 272 p^{32} T^{18} - p^{38} T^{19} + p^{40} T^{20} \) | |
| 5 | \( 1 - 8 T - 238 p T^{2} + 2984 T^{3} + 810407 T^{4} - 1252366 T^{5} + 3897452 T^{6} + 452366072 T^{7} - 346785177907 T^{8} - 116131646486 p T^{9} + 350870811287666 T^{10} - 116131646486 p^{5} T^{11} - 346785177907 p^{8} T^{12} + 452366072 p^{12} T^{13} + 3897452 p^{16} T^{14} - 1252366 p^{20} T^{15} + 810407 p^{24} T^{16} + 2984 p^{28} T^{17} - 238 p^{33} T^{18} - 8 p^{36} T^{19} + p^{40} T^{20} \) | |
| 7 | \( ( 1 + 12 T + 10531 T^{2} + 119450 T^{3} + 47282308 T^{4} + 435174356 T^{5} + 47282308 p^{4} T^{6} + 119450 p^{8} T^{7} + 10531 p^{12} T^{8} + 12 p^{16} T^{9} + p^{20} T^{10} )^{2} \) | |
| 11 | \( ( 1 - 25 T + 33138 T^{2} - 599855 T^{3} + 55946395 p T^{4} - 15493519680 T^{5} + 55946395 p^{5} T^{6} - 599855 p^{8} T^{7} + 33138 p^{12} T^{8} - 25 p^{16} T^{9} + p^{20} T^{10} )^{2} \) | |
| 13 | \( 1 + 48 p T + 19848 p T^{2} + 473472 p^{2} T^{3} + 20425002001 T^{4} + 4644470515548 T^{5} + 981278202402182 T^{6} + 199734541503733716 T^{7} + 39153228768638107369 T^{8} + \)\(73\!\cdots\!04\)\( T^{9} + \)\(12\!\cdots\!54\)\( T^{10} + \)\(73\!\cdots\!04\)\( p^{4} T^{11} + 39153228768638107369 p^{8} T^{12} + 199734541503733716 p^{12} T^{13} + 981278202402182 p^{16} T^{14} + 4644470515548 p^{20} T^{15} + 20425002001 p^{24} T^{16} + 473472 p^{30} T^{17} + 19848 p^{33} T^{18} + 48 p^{37} T^{19} + p^{40} T^{20} \) | |
| 17 | \( 1 + 292 T - 142942 T^{2} - 21554148 T^{3} + 14963481323 T^{4} - 724024051716 T^{5} - 521787570429224 T^{6} + 430071178574051904 T^{7} + 23665089421696713365 T^{8} - \)\(16\!\cdots\!44\)\( T^{9} + \)\(11\!\cdots\!34\)\( T^{10} - \)\(16\!\cdots\!44\)\( p^{4} T^{11} + 23665089421696713365 p^{8} T^{12} + 430071178574051904 p^{12} T^{13} - 521787570429224 p^{16} T^{14} - 724024051716 p^{20} T^{15} + 14963481323 p^{24} T^{16} - 21554148 p^{28} T^{17} - 142942 p^{32} T^{18} + 292 p^{36} T^{19} + p^{40} T^{20} \) | |
| 23 | \( 1 - 98 T - 654482 T^{2} + 254591240 T^{3} + 5269923199 p T^{4} - 90906560806810 T^{5} - 10554279761729116 T^{6} - 10535052682387961542 T^{7} + \)\(16\!\cdots\!93\)\( T^{8} + \)\(51\!\cdots\!74\)\( T^{9} - \)\(83\!\cdots\!34\)\( T^{10} + \)\(51\!\cdots\!74\)\( p^{4} T^{11} + \)\(16\!\cdots\!93\)\( p^{8} T^{12} - 10535052682387961542 p^{12} T^{13} - 10554279761729116 p^{16} T^{14} - 90906560806810 p^{20} T^{15} + 5269923199 p^{25} T^{16} + 254591240 p^{28} T^{17} - 654482 p^{32} T^{18} - 98 p^{36} T^{19} + p^{40} T^{20} \) | |
| 29 | \( 1 - 2598 T + 5532396 T^{2} - 8528007744 T^{3} + 11428057214413 T^{4} - 12942493118958438 T^{5} + 13368578443656929546 T^{6} - \)\(12\!\cdots\!82\)\( T^{7} + \)\(10\!\cdots\!41\)\( T^{8} - \)\(91\!\cdots\!46\)\( T^{9} + \)\(78\!\cdots\!70\)\( T^{10} - \)\(91\!\cdots\!46\)\( p^{4} T^{11} + \)\(10\!\cdots\!41\)\( p^{8} T^{12} - \)\(12\!\cdots\!82\)\( p^{12} T^{13} + 13368578443656929546 p^{16} T^{14} - 12942493118958438 p^{20} T^{15} + 11428057214413 p^{24} T^{16} - 8528007744 p^{28} T^{17} + 5532396 p^{32} T^{18} - 2598 p^{36} T^{19} + p^{40} T^{20} \) | |
| 31 | \( 1 - 2250354 T^{2} + 2629559369209 T^{4} - 2791387459145146036 T^{6} + \)\(29\!\cdots\!62\)\( T^{8} - \)\(27\!\cdots\!04\)\( T^{10} + \)\(29\!\cdots\!62\)\( p^{8} T^{12} - 2791387459145146036 p^{16} T^{14} + 2629559369209 p^{24} T^{16} - 2250354 p^{32} T^{18} + p^{40} T^{20} \) | |
| 37 | \( 1 - 11260114 T^{2} + 61238975018925 T^{4} - \)\(22\!\cdots\!96\)\( T^{6} + \)\(59\!\cdots\!70\)\( T^{8} - \)\(12\!\cdots\!84\)\( T^{10} + \)\(59\!\cdots\!70\)\( p^{8} T^{12} - \)\(22\!\cdots\!96\)\( p^{16} T^{14} + 61238975018925 p^{24} T^{16} - 11260114 p^{32} T^{18} + p^{40} T^{20} \) | |
| 41 | \( 1 + 1407 T + 8940738 T^{2} + 11651162985 T^{3} + 35511811820374 T^{4} + 864519392779137 p T^{5} + 2554780880139108580 p T^{6} + \)\(95\!\cdots\!09\)\( T^{7} + \)\(34\!\cdots\!93\)\( T^{8} + \)\(43\!\cdots\!70\)\( T^{9} + \)\(11\!\cdots\!64\)\( T^{10} + \)\(43\!\cdots\!70\)\( p^{4} T^{11} + \)\(34\!\cdots\!93\)\( p^{8} T^{12} + \)\(95\!\cdots\!09\)\( p^{12} T^{13} + 2554780880139108580 p^{17} T^{14} + 864519392779137 p^{21} T^{15} + 35511811820374 p^{24} T^{16} + 11651162985 p^{28} T^{17} + 8940738 p^{32} T^{18} + 1407 p^{36} T^{19} + p^{40} T^{20} \) | |
| 43 | \( 1 - 5424 T + 9131348 T^{2} - 9239968076 T^{3} + 32702040864717 T^{4} - 73618008932454436 T^{5} + 58288330477558445126 T^{6} + \)\(83\!\cdots\!44\)\( T^{7} - \)\(47\!\cdots\!95\)\( T^{8} + \)\(82\!\cdots\!92\)\( T^{9} - \)\(98\!\cdots\!94\)\( T^{10} + \)\(82\!\cdots\!92\)\( p^{4} T^{11} - \)\(47\!\cdots\!95\)\( p^{8} T^{12} + \)\(83\!\cdots\!44\)\( p^{12} T^{13} + 58288330477558445126 p^{16} T^{14} - 73618008932454436 p^{20} T^{15} + 32702040864717 p^{24} T^{16} - 9239968076 p^{28} T^{17} + 9131348 p^{32} T^{18} - 5424 p^{36} T^{19} + p^{40} T^{20} \) | |
| 47 | \( 1 + 2416 T - 14937134 T^{2} - 28947972448 T^{3} + 149005338150881 T^{4} + 193759491213085406 T^{5} - \)\(11\!\cdots\!00\)\( T^{6} - \)\(88\!\cdots\!96\)\( T^{7} + \)\(72\!\cdots\!29\)\( T^{8} + \)\(19\!\cdots\!98\)\( T^{9} - \)\(37\!\cdots\!22\)\( T^{10} + \)\(19\!\cdots\!98\)\( p^{4} T^{11} + \)\(72\!\cdots\!29\)\( p^{8} T^{12} - \)\(88\!\cdots\!96\)\( p^{12} T^{13} - \)\(11\!\cdots\!00\)\( p^{16} T^{14} + 193759491213085406 p^{20} T^{15} + 149005338150881 p^{24} T^{16} - 28947972448 p^{28} T^{17} - 14937134 p^{32} T^{18} + 2416 p^{36} T^{19} + p^{40} T^{20} \) | |
| 53 | \( 1 - 1122 T + 24343004 T^{2} - 26842027872 T^{3} + 276570796560429 T^{4} - 370666941236012460 T^{5} + \)\(26\!\cdots\!90\)\( T^{6} - \)\(45\!\cdots\!98\)\( T^{7} + \)\(26\!\cdots\!41\)\( T^{8} - \)\(47\!\cdots\!76\)\( T^{9} + \)\(23\!\cdots\!46\)\( T^{10} - \)\(47\!\cdots\!76\)\( p^{4} T^{11} + \)\(26\!\cdots\!41\)\( p^{8} T^{12} - \)\(45\!\cdots\!98\)\( p^{12} T^{13} + \)\(26\!\cdots\!90\)\( p^{16} T^{14} - 370666941236012460 p^{20} T^{15} + 276570796560429 p^{24} T^{16} - 26842027872 p^{28} T^{17} + 24343004 p^{32} T^{18} - 1122 p^{36} T^{19} + p^{40} T^{20} \) | |
| 59 | \( 1 - 15387 T + 135334734 T^{2} - 868054696857 T^{3} + 4328482760780464 T^{4} - 16621616883446023761 T^{5} + \)\(45\!\cdots\!62\)\( T^{6} - \)\(79\!\cdots\!59\)\( p T^{7} - \)\(33\!\cdots\!69\)\( T^{8} + \)\(26\!\cdots\!74\)\( T^{9} - \)\(11\!\cdots\!28\)\( T^{10} + \)\(26\!\cdots\!74\)\( p^{4} T^{11} - \)\(33\!\cdots\!69\)\( p^{8} T^{12} - \)\(79\!\cdots\!59\)\( p^{13} T^{13} + \)\(45\!\cdots\!62\)\( p^{16} T^{14} - 16621616883446023761 p^{20} T^{15} + 4328482760780464 p^{24} T^{16} - 868054696857 p^{28} T^{17} + 135334734 p^{32} T^{18} - 15387 p^{36} T^{19} + p^{40} T^{20} \) | |
| 61 | \( 1 - 860 T - 48579422 T^{2} - 6966946920 T^{3} + 1254219865003767 T^{4} + 842831961196105530 T^{5} - \)\(25\!\cdots\!80\)\( T^{6} - \)\(99\!\cdots\!80\)\( T^{7} + \)\(42\!\cdots\!93\)\( T^{8} + \)\(24\!\cdots\!30\)\( T^{9} - \)\(63\!\cdots\!22\)\( T^{10} + \)\(24\!\cdots\!30\)\( p^{4} T^{11} + \)\(42\!\cdots\!93\)\( p^{8} T^{12} - \)\(99\!\cdots\!80\)\( p^{12} T^{13} - \)\(25\!\cdots\!80\)\( p^{16} T^{14} + 842831961196105530 p^{20} T^{15} + 1254219865003767 p^{24} T^{16} - 6966946920 p^{28} T^{17} - 48579422 p^{32} T^{18} - 860 p^{36} T^{19} + p^{40} T^{20} \) | |
| 67 | \( 1 - 14763 T + 137313390 T^{2} - 954644478921 T^{3} + 5624107415594716 T^{4} - 26659270464046183521 T^{5} + \)\(10\!\cdots\!98\)\( T^{6} - \)\(29\!\cdots\!57\)\( T^{7} + \)\(46\!\cdots\!71\)\( T^{8} + \)\(12\!\cdots\!42\)\( T^{9} - \)\(10\!\cdots\!52\)\( T^{10} + \)\(12\!\cdots\!42\)\( p^{4} T^{11} + \)\(46\!\cdots\!71\)\( p^{8} T^{12} - \)\(29\!\cdots\!57\)\( p^{12} T^{13} + \)\(10\!\cdots\!98\)\( p^{16} T^{14} - 26659270464046183521 p^{20} T^{15} + 5624107415594716 p^{24} T^{16} - 954644478921 p^{28} T^{17} + 137313390 p^{32} T^{18} - 14763 p^{36} T^{19} + p^{40} T^{20} \) | |
| 71 | \( 1 + 384 p T + 434405882 T^{2} + 71666169600 p T^{3} + 47996922695454903 T^{4} + \)\(38\!\cdots\!56\)\( T^{5} + \)\(27\!\cdots\!60\)\( T^{6} + \)\(18\!\cdots\!84\)\( T^{7} + \)\(11\!\cdots\!77\)\( T^{8} + \)\(62\!\cdots\!80\)\( T^{9} + \)\(32\!\cdots\!74\)\( T^{10} + \)\(62\!\cdots\!80\)\( p^{4} T^{11} + \)\(11\!\cdots\!77\)\( p^{8} T^{12} + \)\(18\!\cdots\!84\)\( p^{12} T^{13} + \)\(27\!\cdots\!60\)\( p^{16} T^{14} + \)\(38\!\cdots\!56\)\( p^{20} T^{15} + 47996922695454903 p^{24} T^{16} + 71666169600 p^{29} T^{17} + 434405882 p^{32} T^{18} + 384 p^{37} T^{19} + p^{40} T^{20} \) | |
| 73 | \( 1 - 1561 T - 45649594 T^{2} - 99028916631 T^{3} + 426851714674718 T^{4} + 6700779130973164353 T^{5} + \)\(17\!\cdots\!52\)\( T^{6} - \)\(69\!\cdots\!27\)\( T^{7} - \)\(10\!\cdots\!15\)\( T^{8} - \)\(17\!\cdots\!38\)\( T^{9} - \)\(17\!\cdots\!92\)\( T^{10} - \)\(17\!\cdots\!38\)\( p^{4} T^{11} - \)\(10\!\cdots\!15\)\( p^{8} T^{12} - \)\(69\!\cdots\!27\)\( p^{12} T^{13} + \)\(17\!\cdots\!52\)\( p^{16} T^{14} + 6700779130973164353 p^{20} T^{15} + 426851714674718 p^{24} T^{16} - 99028916631 p^{28} T^{17} - 45649594 p^{32} T^{18} - 1561 p^{36} T^{19} + p^{40} T^{20} \) | |
| 79 | \( 1 - 24750 T + 442508766 T^{2} - 5898451333500 T^{3} + 67059328128970723 T^{4} - \)\(65\!\cdots\!88\)\( T^{5} + \)\(58\!\cdots\!60\)\( T^{6} - \)\(46\!\cdots\!34\)\( T^{7} + \)\(35\!\cdots\!01\)\( T^{8} - \)\(24\!\cdots\!40\)\( T^{9} + \)\(15\!\cdots\!42\)\( T^{10} - \)\(24\!\cdots\!40\)\( p^{4} T^{11} + \)\(35\!\cdots\!01\)\( p^{8} T^{12} - \)\(46\!\cdots\!34\)\( p^{12} T^{13} + \)\(58\!\cdots\!60\)\( p^{16} T^{14} - \)\(65\!\cdots\!88\)\( p^{20} T^{15} + 67059328128970723 p^{24} T^{16} - 5898451333500 p^{28} T^{17} + 442508766 p^{32} T^{18} - 24750 p^{36} T^{19} + p^{40} T^{20} \) | |
| 83 | \( ( 1 - 3001 T + 118891044 T^{2} - 237543422849 T^{3} + 6928297699658807 T^{4} - 8322652838171067576 T^{5} + 6928297699658807 p^{4} T^{6} - 237543422849 p^{8} T^{7} + 118891044 p^{12} T^{8} - 3001 p^{16} T^{9} + p^{20} T^{10} )^{2} \) | |
| 89 | \( 1 + 22566 T + 494506844 T^{2} + 7328655835872 T^{3} + 103015369026888237 T^{4} + \)\(11\!\cdots\!20\)\( T^{5} + \)\(13\!\cdots\!30\)\( T^{6} + \)\(12\!\cdots\!34\)\( T^{7} + \)\(11\!\cdots\!25\)\( T^{8} + \)\(98\!\cdots\!48\)\( T^{9} + \)\(82\!\cdots\!66\)\( T^{10} + \)\(98\!\cdots\!48\)\( p^{4} T^{11} + \)\(11\!\cdots\!25\)\( p^{8} T^{12} + \)\(12\!\cdots\!34\)\( p^{12} T^{13} + \)\(13\!\cdots\!30\)\( p^{16} T^{14} + \)\(11\!\cdots\!20\)\( p^{20} T^{15} + 103015369026888237 p^{24} T^{16} + 7328655835872 p^{28} T^{17} + 494506844 p^{32} T^{18} + 22566 p^{36} T^{19} + p^{40} T^{20} \) | |
| 97 | \( 1 - 46287 T + 1409011986 T^{2} - 32162515608681 T^{3} + 616592367251865778 T^{4} - \)\(10\!\cdots\!57\)\( T^{5} + \)\(14\!\cdots\!16\)\( T^{6} - \)\(19\!\cdots\!33\)\( T^{7} + \)\(22\!\cdots\!81\)\( T^{8} - \)\(24\!\cdots\!74\)\( T^{9} + \)\(24\!\cdots\!20\)\( T^{10} - \)\(24\!\cdots\!74\)\( p^{4} T^{11} + \)\(22\!\cdots\!81\)\( p^{8} T^{12} - \)\(19\!\cdots\!33\)\( p^{12} T^{13} + \)\(14\!\cdots\!16\)\( p^{16} T^{14} - \)\(10\!\cdots\!57\)\( p^{20} T^{15} + 616592367251865778 p^{24} T^{16} - 32162515608681 p^{28} T^{17} + 1409011986 p^{32} T^{18} - 46287 p^{36} T^{19} + p^{40} T^{20} \) | |
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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
−7.69003474028073789197735169257, −7.56626140776977057432150252377, −6.86048522745591039438143555157, −6.83304536315025630997585177445, −6.76638963313086023014787926262, −6.53907450750488461148970737899, −6.52234829806753048460194540270, −6.11561328951823225836511385998, −5.89986264064007621598236466243, −5.43849830909099737289700156126, −5.15355865318842038159412029035, −5.15063494567016896416000281776, −4.90362465149445099359028329392, −4.84409882738780592324317336628, −4.82955360170490391756841062855, −4.09011585580201676911616834321, −3.93003981229310913139853285790, −3.49749324714296198308031145215, −2.98389855025897535345658538559, −2.92284834520781993355028211720, −2.72191012136388483351886521091, −2.57798562571163687268247860353, −1.84200640226682905148493091293, −0.825526933680763760019763622280, −0.21847194291957140268237958099, 0.21847194291957140268237958099, 0.825526933680763760019763622280, 1.84200640226682905148493091293, 2.57798562571163687268247860353, 2.72191012136388483351886521091, 2.92284834520781993355028211720, 2.98389855025897535345658538559, 3.49749324714296198308031145215, 3.93003981229310913139853285790, 4.09011585580201676911616834321, 4.82955360170490391756841062855, 4.84409882738780592324317336628, 4.90362465149445099359028329392, 5.15063494567016896416000281776, 5.15355865318842038159412029035, 5.43849830909099737289700156126, 5.89986264064007621598236466243, 6.11561328951823225836511385998, 6.52234829806753048460194540270, 6.53907450750488461148970737899, 6.76638963313086023014787926262, 6.83304536315025630997585177445, 6.86048522745591039438143555157, 7.56626140776977057432150252377, 7.69003474028073789197735169257
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.