Dirichlet series
| L(s) = 1 | + 10·2-s − 3-s + 55·4-s − 10·5-s − 10·6-s + 3·7-s + 220·8-s − 8·9-s − 100·10-s − 10·11-s − 55·12-s + 6·13-s + 30·14-s + 10·15-s + 715·16-s + 3·17-s − 80·18-s + 3·19-s − 550·20-s − 3·21-s − 100·22-s − 6·23-s − 220·24-s + 55·25-s + 60·26-s + 12·27-s + 165·28-s + ⋯ |
| L(s) = 1 | + 7.07·2-s − 0.577·3-s + 55/2·4-s − 4.47·5-s − 4.08·6-s + 1.13·7-s + 77.7·8-s − 8/3·9-s − 31.6·10-s − 3.01·11-s − 15.8·12-s + 1.66·13-s + 8.01·14-s + 2.58·15-s + 178.·16-s + 0.727·17-s − 18.8·18-s + 0.688·19-s − 122.·20-s − 0.654·21-s − 21.3·22-s − 1.25·23-s − 44.9·24-s + 11·25-s + 11.7·26-s + 2.30·27-s + 31.1·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 5^{10} \cdot 11^{10} \cdot 43^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 5^{10} \cdot 11^{10} \cdot 43^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 5^{10} \cdot 11^{10} \cdot 43^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.90721\times 10^{15}\) |
| Root analytic conductor: | \(6.14566\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 5^{10} \cdot 11^{10} \cdot 43^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(3314.666515\) |
| \(L(\frac12)\) | \(\approx\) | \(3314.666515\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{10} \) |
| 5 | \( ( 1 + T )^{10} \) | |
| 11 | \( ( 1 + T )^{10} \) | |
| 43 | \( ( 1 + T )^{10} \) | |
| good | 3 | \( 1 + T + p^{2} T^{2} + 5 T^{3} + 13 p T^{4} + 16 T^{5} + 47 p T^{6} + 31 p T^{7} + 493 T^{8} + 457 T^{9} + 1570 T^{10} + 457 p T^{11} + 493 p^{2} T^{12} + 31 p^{4} T^{13} + 47 p^{5} T^{14} + 16 p^{5} T^{15} + 13 p^{7} T^{16} + 5 p^{7} T^{17} + p^{10} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) |
| 7 | \( 1 - 3 T + 23 T^{2} - 55 T^{3} + 377 T^{4} - 832 T^{5} + 4295 T^{6} - 8587 T^{7} + 41029 T^{8} - 74659 T^{9} + 310406 T^{10} - 74659 p T^{11} + 41029 p^{2} T^{12} - 8587 p^{3} T^{13} + 4295 p^{4} T^{14} - 832 p^{5} T^{15} + 377 p^{6} T^{16} - 55 p^{7} T^{17} + 23 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - 6 T + 84 T^{2} - 418 T^{3} + 3205 T^{4} - 13516 T^{5} + 75728 T^{6} - 277812 T^{7} + 1302530 T^{8} - 4304704 T^{9} + 1407320 p T^{10} - 4304704 p T^{11} + 1302530 p^{2} T^{12} - 277812 p^{3} T^{13} + 75728 p^{4} T^{14} - 13516 p^{5} T^{15} + 3205 p^{6} T^{16} - 418 p^{7} T^{17} + 84 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 3 T + 79 T^{2} - 55 T^{3} + 2667 T^{4} + 3370 T^{5} + 63817 T^{6} + 183251 T^{7} + 1240693 T^{8} + 5099123 T^{9} + 21063318 T^{10} + 5099123 p T^{11} + 1240693 p^{2} T^{12} + 183251 p^{3} T^{13} + 63817 p^{4} T^{14} + 3370 p^{5} T^{15} + 2667 p^{6} T^{16} - 55 p^{7} T^{17} + 79 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 3 T + 115 T^{2} - 13 p T^{3} + 5989 T^{4} - 9726 T^{5} + 197455 T^{6} - 260451 T^{7} + 254683 p T^{8} - 5570165 T^{9} + 98435590 T^{10} - 5570165 p T^{11} + 254683 p^{3} T^{12} - 260451 p^{3} T^{13} + 197455 p^{4} T^{14} - 9726 p^{5} T^{15} + 5989 p^{6} T^{16} - 13 p^{8} T^{17} + 115 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 + 6 T + 88 T^{2} + 366 T^{3} + 3093 T^{4} + 6756 T^{5} + 58336 T^{6} - 19340 T^{7} + 857690 T^{8} - 2683544 T^{9} + 16598352 T^{10} - 2683544 p T^{11} + 857690 p^{2} T^{12} - 19340 p^{3} T^{13} + 58336 p^{4} T^{14} + 6756 p^{5} T^{15} + 3093 p^{6} T^{16} + 366 p^{7} T^{17} + 88 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 14 T + 227 T^{2} - 2556 T^{3} + 25513 T^{4} - 7862 p T^{5} + 1788476 T^{6} - 13009482 T^{7} + 85714862 T^{8} - 520127830 T^{9} + 2927629890 T^{10} - 520127830 p T^{11} + 85714862 p^{2} T^{12} - 13009482 p^{3} T^{13} + 1788476 p^{4} T^{14} - 7862 p^{6} T^{15} + 25513 p^{6} T^{16} - 2556 p^{7} T^{17} + 227 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 6 T + 195 T^{2} - 946 T^{3} + 16765 T^{4} - 66768 T^{5} + 868580 T^{6} - 2906624 T^{7} + 32424322 T^{8} - 96817692 T^{9} + 1036858162 T^{10} - 96817692 p T^{11} + 32424322 p^{2} T^{12} - 2906624 p^{3} T^{13} + 868580 p^{4} T^{14} - 66768 p^{5} T^{15} + 16765 p^{6} T^{16} - 946 p^{7} T^{17} + 195 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 10 T + 219 T^{2} - 2156 T^{3} + 25425 T^{4} - 6018 p T^{5} + 1980084 T^{6} - 15028254 T^{7} + 111064470 T^{8} - 739726538 T^{9} + 4690259362 T^{10} - 739726538 p T^{11} + 111064470 p^{2} T^{12} - 15028254 p^{3} T^{13} + 1980084 p^{4} T^{14} - 6018 p^{6} T^{15} + 25425 p^{6} T^{16} - 2156 p^{7} T^{17} + 219 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 26 T + 568 T^{2} - 8126 T^{3} + 102901 T^{4} - 1031276 T^{5} + 9512368 T^{6} - 74663940 T^{7} + 561940858 T^{8} - 3778852616 T^{9} + 25305902704 T^{10} - 3778852616 p T^{11} + 561940858 p^{2} T^{12} - 74663940 p^{3} T^{13} + 9512368 p^{4} T^{14} - 1031276 p^{5} T^{15} + 102901 p^{6} T^{16} - 8126 p^{7} T^{17} + 568 p^{8} T^{18} - 26 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 3 T + 216 T^{2} - 462 T^{3} + 21393 T^{4} + 2986 T^{5} + 1198157 T^{6} + 5291774 T^{7} + 39956426 T^{8} + 513458025 T^{9} + 1253746126 T^{10} + 513458025 p T^{11} + 39956426 p^{2} T^{12} + 5291774 p^{3} T^{13} + 1198157 p^{4} T^{14} + 2986 p^{5} T^{15} + 21393 p^{6} T^{16} - 462 p^{7} T^{17} + 216 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 9 T + 335 T^{2} + 3167 T^{3} + 57403 T^{4} + 528218 T^{5} + 6462509 T^{6} + 55521489 T^{7} + 524250913 T^{8} + 76688193 p T^{9} + 31902942054 T^{10} + 76688193 p^{2} T^{11} + 524250913 p^{2} T^{12} + 55521489 p^{3} T^{13} + 6462509 p^{4} T^{14} + 528218 p^{5} T^{15} + 57403 p^{6} T^{16} + 3167 p^{7} T^{17} + 335 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 9 T + 352 T^{2} + 3036 T^{3} + 65353 T^{4} + 514134 T^{5} + 8047243 T^{6} + 57124084 T^{7} + 718893956 T^{8} + 4545182105 T^{9} + 48543806878 T^{10} + 4545182105 p T^{11} + 718893956 p^{2} T^{12} + 57124084 p^{3} T^{13} + 8047243 p^{4} T^{14} + 514134 p^{5} T^{15} + 65353 p^{6} T^{16} + 3036 p^{7} T^{17} + 352 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 22 T + 527 T^{2} - 7520 T^{3} + 108761 T^{4} - 1210666 T^{5} + 13519012 T^{6} - 127170094 T^{7} + 1196665118 T^{8} - 9876003226 T^{9} + 81789096666 T^{10} - 9876003226 p T^{11} + 1196665118 p^{2} T^{12} - 127170094 p^{3} T^{13} + 13519012 p^{4} T^{14} - 1210666 p^{5} T^{15} + 108761 p^{6} T^{16} - 7520 p^{7} T^{17} + 527 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 18 T + 678 T^{2} - 9582 T^{3} + 200581 T^{4} - 2318640 T^{5} + 34851944 T^{6} - 336934560 T^{7} + 3995410818 T^{8} - 32591015076 T^{9} + 318514284452 T^{10} - 32591015076 p T^{11} + 3995410818 p^{2} T^{12} - 336934560 p^{3} T^{13} + 34851944 p^{4} T^{14} - 2318640 p^{5} T^{15} + 200581 p^{6} T^{16} - 9582 p^{7} T^{17} + 678 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 27 T + 769 T^{2} - 12677 T^{3} + 211109 T^{4} - 2561290 T^{5} + 31908719 T^{6} - 314370001 T^{7} + 3272553983 T^{8} - 27991906775 T^{9} + 258616728138 T^{10} - 27991906775 p T^{11} + 3272553983 p^{2} T^{12} - 314370001 p^{3} T^{13} + 31908719 p^{4} T^{14} - 2561290 p^{5} T^{15} + 211109 p^{6} T^{16} - 12677 p^{7} T^{17} + 769 p^{8} T^{18} - 27 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 28 T + 870 T^{2} - 16480 T^{3} + 305597 T^{4} - 4432908 T^{5} + 61081368 T^{6} - 715012580 T^{7} + 7878073170 T^{8} - 76194275636 T^{9} + 691714590212 T^{10} - 76194275636 p T^{11} + 7878073170 p^{2} T^{12} - 715012580 p^{3} T^{13} + 61081368 p^{4} T^{14} - 4432908 p^{5} T^{15} + 305597 p^{6} T^{16} - 16480 p^{7} T^{17} + 870 p^{8} T^{18} - 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 3 T + 332 T^{2} + 102 T^{3} + 54215 T^{4} + 198396 T^{5} + 6151961 T^{6} + 42581034 T^{7} + 552365978 T^{8} + 5220090999 T^{9} + 44251998226 T^{10} + 5220090999 p T^{11} + 552365978 p^{2} T^{12} + 42581034 p^{3} T^{13} + 6151961 p^{4} T^{14} + 198396 p^{5} T^{15} + 54215 p^{6} T^{16} + 102 p^{7} T^{17} + 332 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 11 T + 516 T^{2} + 3520 T^{3} + 109357 T^{4} + 440698 T^{5} + 14706827 T^{6} + 38564244 T^{7} + 1640221632 T^{8} + 3806657943 T^{9} + 153598851254 T^{10} + 3806657943 p T^{11} + 1640221632 p^{2} T^{12} + 38564244 p^{3} T^{13} + 14706827 p^{4} T^{14} + 440698 p^{5} T^{15} + 109357 p^{6} T^{16} + 3520 p^{7} T^{17} + 516 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 16 T + 415 T^{2} - 4952 T^{3} + 82237 T^{4} - 845196 T^{5} + 11783236 T^{6} - 109548380 T^{7} + 1365146978 T^{8} - 11587294544 T^{9} + 131722501274 T^{10} - 11587294544 p T^{11} + 1365146978 p^{2} T^{12} - 109548380 p^{3} T^{13} + 11783236 p^{4} T^{14} - 845196 p^{5} T^{15} + 82237 p^{6} T^{16} - 4952 p^{7} T^{17} + 415 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 22 T + 850 T^{2} - 12746 T^{3} + 279525 T^{4} - 3077796 T^{5} + 50380344 T^{6} - 427306108 T^{7} + 6104174938 T^{8} - 43486839104 T^{9} + 613073662444 T^{10} - 43486839104 p T^{11} + 6104174938 p^{2} T^{12} - 427306108 p^{3} T^{13} + 50380344 p^{4} T^{14} - 3077796 p^{5} T^{15} + 279525 p^{6} T^{16} - 12746 p^{7} T^{17} + 850 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.99364611529322708622934590902, −2.94870657129799008701240821276, −2.94270826654474208996458831500, −2.93995508654952596498528510399, −2.84163240477513889268976239163, −2.45531136439002544821725532921, −2.37223917917479785828202579168, −2.25375615549270477340642250345, −2.15362430647123287055301557513, −2.14633448267772755638939565777, −2.12513604461184602846713409886, −2.12453420681203751349600614994, −1.88586196244509739656662222296, −1.84575018183817353561672362919, −1.81176686429952790985402768113, −1.36788756072564474072621428665, −1.16188425406718480466706930005, −1.10335318960383970024485482023, −1.00115717208234175147185070746, −0.789917684526382694573337447832, −0.72858875437598586134367119665, −0.68553006152812577210547906685, −0.56074957273132449981928138712, −0.52797237425800727479532924107, −0.32218396180818617210881542341, 0.32218396180818617210881542341, 0.52797237425800727479532924107, 0.56074957273132449981928138712, 0.68553006152812577210547906685, 0.72858875437598586134367119665, 0.789917684526382694573337447832, 1.00115717208234175147185070746, 1.10335318960383970024485482023, 1.16188425406718480466706930005, 1.36788756072564474072621428665, 1.81176686429952790985402768113, 1.84575018183817353561672362919, 1.88586196244509739656662222296, 2.12453420681203751349600614994, 2.12513604461184602846713409886, 2.14633448267772755638939565777, 2.15362430647123287055301557513, 2.25375615549270477340642250345, 2.37223917917479785828202579168, 2.45531136439002544821725532921, 2.84163240477513889268976239163, 2.93995508654952596498528510399, 2.94270826654474208996458831500, 2.94870657129799008701240821276, 2.99364611529322708622934590902
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.