Dirichlet series
| L(s) = 1 | + 10·2-s − 10·3-s + 55·4-s + 8·5-s − 100·6-s + 10·7-s + 220·8-s + 55·9-s + 80·10-s + 4·11-s − 550·12-s + 6·13-s + 100·14-s − 80·15-s + 715·16-s + 5·17-s + 550·18-s + 15·19-s + 440·20-s − 100·21-s + 40·22-s + 12·23-s − 2.20e3·24-s + 6·25-s + 60·26-s − 220·27-s + 550·28-s + ⋯ |
| L(s) = 1 | + 7.07·2-s − 5.77·3-s + 55/2·4-s + 3.57·5-s − 40.8·6-s + 3.77·7-s + 77.7·8-s + 55/3·9-s + 25.2·10-s + 1.20·11-s − 158.·12-s + 1.66·13-s + 26.7·14-s − 20.6·15-s + 178.·16-s + 1.21·17-s + 129.·18-s + 3.44·19-s + 98.3·20-s − 21.8·21-s + 8.52·22-s + 2.50·23-s − 449.·24-s + 6/5·25-s + 11.7·26-s − 42.3·27-s + 103.·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 7^{10} \cdot 191^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 7^{10} \cdot 191^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 3^{10} \cdot 7^{10} \cdot 191^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.16304\times 10^{18}\) |
| Root analytic conductor: | \(8.00349\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 3^{10} \cdot 7^{10} \cdot 191^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(94559.82378\) |
| \(L(\frac12)\) | \(\approx\) | \(94559.82378\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{10} \) |
| 3 | \( ( 1 + T )^{10} \) | |
| 7 | \( ( 1 - T )^{10} \) | |
| 191 | \( ( 1 - T )^{10} \) | |
| good | 5 | \( 1 - 8 T + 58 T^{2} - 284 T^{3} + 253 p T^{4} - 4638 T^{5} + 15686 T^{6} - 46619 T^{7} + 129149 T^{8} - 64609 p T^{9} + 757226 T^{10} - 64609 p^{2} T^{11} + 129149 p^{2} T^{12} - 46619 p^{3} T^{13} + 15686 p^{4} T^{14} - 4638 p^{5} T^{15} + 253 p^{7} T^{16} - 284 p^{7} T^{17} + 58 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) |
| 11 | \( 1 - 4 T + 64 T^{2} - 203 T^{3} + 1834 T^{4} - 4809 T^{5} + 31909 T^{6} - 71671 T^{7} + 401854 T^{8} - 828741 T^{9} + 4449224 T^{10} - 828741 p T^{11} + 401854 p^{2} T^{12} - 71671 p^{3} T^{13} + 31909 p^{4} T^{14} - 4809 p^{5} T^{15} + 1834 p^{6} T^{16} - 203 p^{7} T^{17} + 64 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - 6 T + 84 T^{2} - 335 T^{3} + 2726 T^{4} - 7045 T^{5} + 47319 T^{6} - 63827 T^{7} + 545432 T^{8} - 138641 T^{9} + 6108156 T^{10} - 138641 p T^{11} + 545432 p^{2} T^{12} - 63827 p^{3} T^{13} + 47319 p^{4} T^{14} - 7045 p^{5} T^{15} + 2726 p^{6} T^{16} - 335 p^{7} T^{17} + 84 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 5 T + 122 T^{2} - 516 T^{3} + 6867 T^{4} - 25615 T^{5} + 243464 T^{6} - 819362 T^{7} + 6186373 T^{8} - 18759853 T^{9} + 119551405 T^{10} - 18759853 p T^{11} + 6186373 p^{2} T^{12} - 819362 p^{3} T^{13} + 243464 p^{4} T^{14} - 25615 p^{5} T^{15} + 6867 p^{6} T^{16} - 516 p^{7} T^{17} + 122 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 15 T + 237 T^{2} - 2240 T^{3} + 20858 T^{4} - 7736 p T^{5} + 1014959 T^{6} - 5755643 T^{7} + 32136252 T^{8} - 152692922 T^{9} + 717658454 T^{10} - 152692922 p T^{11} + 32136252 p^{2} T^{12} - 5755643 p^{3} T^{13} + 1014959 p^{4} T^{14} - 7736 p^{6} T^{15} + 20858 p^{6} T^{16} - 2240 p^{7} T^{17} + 237 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - 12 T + 193 T^{2} - 1430 T^{3} + 13488 T^{4} - 72559 T^{5} + 539083 T^{6} - 2374815 T^{7} + 16112459 T^{8} - 63491122 T^{9} + 403892128 T^{10} - 63491122 p T^{11} + 16112459 p^{2} T^{12} - 2374815 p^{3} T^{13} + 539083 p^{4} T^{14} - 72559 p^{5} T^{15} + 13488 p^{6} T^{16} - 1430 p^{7} T^{17} + 193 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 7 T + 225 T^{2} - 1396 T^{3} + 24058 T^{4} - 132978 T^{5} + 1612655 T^{6} - 7906645 T^{7} + 75101692 T^{8} - 322594882 T^{9} + 2539671030 T^{10} - 322594882 p T^{11} + 75101692 p^{2} T^{12} - 7906645 p^{3} T^{13} + 1612655 p^{4} T^{14} - 132978 p^{5} T^{15} + 24058 p^{6} T^{16} - 1396 p^{7} T^{17} + 225 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 28 T + 604 T^{2} - 9062 T^{3} + 115686 T^{4} - 1216433 T^{5} + 365361 p T^{6} - 91648270 T^{7} + 669065887 T^{8} - 4336050606 T^{9} + 25547969297 T^{10} - 4336050606 p T^{11} + 669065887 p^{2} T^{12} - 91648270 p^{3} T^{13} + 365361 p^{5} T^{14} - 1216433 p^{5} T^{15} + 115686 p^{6} T^{16} - 9062 p^{7} T^{17} + 604 p^{8} T^{18} - 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 3 T + 204 T^{2} + 730 T^{3} + 21033 T^{4} + 86043 T^{5} + 1435605 T^{6} + 6408903 T^{7} + 73371569 T^{8} + 330844839 T^{9} + 2999091872 T^{10} + 330844839 p T^{11} + 73371569 p^{2} T^{12} + 6408903 p^{3} T^{13} + 1435605 p^{4} T^{14} + 86043 p^{5} T^{15} + 21033 p^{6} T^{16} + 730 p^{7} T^{17} + 204 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 24 T + 430 T^{2} - 5596 T^{3} + 63997 T^{4} - 626617 T^{5} + 5666355 T^{6} - 46302249 T^{7} + 355679202 T^{8} - 2511522098 T^{9} + 16740121806 T^{10} - 2511522098 p T^{11} + 355679202 p^{2} T^{12} - 46302249 p^{3} T^{13} + 5666355 p^{4} T^{14} - 626617 p^{5} T^{15} + 63997 p^{6} T^{16} - 5596 p^{7} T^{17} + 430 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 4 T + 161 T^{2} - 472 T^{3} + 12487 T^{4} - 17995 T^{5} + 597702 T^{6} + 740140 T^{7} + 19922087 T^{8} + 107490773 T^{9} + 683597200 T^{10} + 107490773 p T^{11} + 19922087 p^{2} T^{12} + 740140 p^{3} T^{13} + 597702 p^{4} T^{14} - 17995 p^{5} T^{15} + 12487 p^{6} T^{16} - 472 p^{7} T^{17} + 161 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 16 T + 485 T^{2} - 5860 T^{3} + 100894 T^{4} - 20885 p T^{5} + 12295661 T^{6} - 99380423 T^{7} + 989154609 T^{8} - 6726541224 T^{9} + 55333717948 T^{10} - 6726541224 p T^{11} + 989154609 p^{2} T^{12} - 99380423 p^{3} T^{13} + 12295661 p^{4} T^{14} - 20885 p^{6} T^{15} + 100894 p^{6} T^{16} - 5860 p^{7} T^{17} + 485 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 - 5 T + 233 T^{2} - 1553 T^{3} + 33170 T^{4} - 218326 T^{5} + 3355825 T^{6} - 21202972 T^{7} + 255097441 T^{8} - 1485829372 T^{9} + 15294167860 T^{10} - 1485829372 p T^{11} + 255097441 p^{2} T^{12} - 21202972 p^{3} T^{13} + 3355825 p^{4} T^{14} - 218326 p^{5} T^{15} + 33170 p^{6} T^{16} - 1553 p^{7} T^{17} + 233 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 17 T + 408 T^{2} - 4582 T^{3} + 70931 T^{4} - 654682 T^{5} + 8177542 T^{6} - 65029177 T^{7} + 694168569 T^{8} - 4867260348 T^{9} + 45956705970 T^{10} - 4867260348 p T^{11} + 694168569 p^{2} T^{12} - 65029177 p^{3} T^{13} + 8177542 p^{4} T^{14} - 654682 p^{5} T^{15} + 70931 p^{6} T^{16} - 4582 p^{7} T^{17} + 408 p^{8} T^{18} - 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 15 T + 506 T^{2} - 6501 T^{3} + 118059 T^{4} - 1317404 T^{5} + 16922778 T^{6} - 164894898 T^{7} + 1666751189 T^{8} - 14126263266 T^{9} + 118675938574 T^{10} - 14126263266 p T^{11} + 1666751189 p^{2} T^{12} - 164894898 p^{3} T^{13} + 16922778 p^{4} T^{14} - 1317404 p^{5} T^{15} + 118059 p^{6} T^{16} - 6501 p^{7} T^{17} + 506 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 6 T + 425 T^{2} - 2378 T^{3} + 84376 T^{4} - 431467 T^{5} + 10645990 T^{6} - 48780615 T^{7} + 985447984 T^{8} - 4039004968 T^{9} + 72796485384 T^{10} - 4039004968 p T^{11} + 985447984 p^{2} T^{12} - 48780615 p^{3} T^{13} + 10645990 p^{4} T^{14} - 431467 p^{5} T^{15} + 84376 p^{6} T^{16} - 2378 p^{7} T^{17} + 425 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 5 T + 203 T^{2} - 1597 T^{3} + 30523 T^{4} - 211048 T^{5} + 3363252 T^{6} - 22874171 T^{7} + 301261848 T^{8} - 1885892563 T^{9} + 23629742275 T^{10} - 1885892563 p T^{11} + 301261848 p^{2} T^{12} - 22874171 p^{3} T^{13} + 3363252 p^{4} T^{14} - 211048 p^{5} T^{15} + 30523 p^{6} T^{16} - 1597 p^{7} T^{17} + 203 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 16 T + 352 T^{2} - 3304 T^{3} + 49442 T^{4} - 325353 T^{5} + 4033225 T^{6} - 12049730 T^{7} + 180682257 T^{8} + 335953188 T^{9} + 8396876563 T^{10} + 335953188 p T^{11} + 180682257 p^{2} T^{12} - 12049730 p^{3} T^{13} + 4033225 p^{4} T^{14} - 325353 p^{5} T^{15} + 49442 p^{6} T^{16} - 3304 p^{7} T^{17} + 352 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 5 T + 465 T^{2} + 2426 T^{3} + 94235 T^{4} + 555423 T^{5} + 10912430 T^{6} + 80276238 T^{7} + 851647576 T^{8} + 8321261877 T^{9} + 61210513967 T^{10} + 8321261877 p T^{11} + 851647576 p^{2} T^{12} + 80276238 p^{3} T^{13} + 10912430 p^{4} T^{14} + 555423 p^{5} T^{15} + 94235 p^{6} T^{16} + 2426 p^{7} T^{17} + 465 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 24 T + 618 T^{2} - 9221 T^{3} + 136518 T^{4} - 1518297 T^{5} + 16747877 T^{6} - 156709671 T^{7} + 1480990420 T^{8} - 13087429851 T^{9} + 120143119748 T^{10} - 13087429851 p T^{11} + 1480990420 p^{2} T^{12} - 156709671 p^{3} T^{13} + 16747877 p^{4} T^{14} - 1518297 p^{5} T^{15} + 136518 p^{6} T^{16} - 9221 p^{7} T^{17} + 618 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 17 T + 560 T^{2} - 8711 T^{3} + 167261 T^{4} - 2244070 T^{5} + 32797309 T^{6} - 378419448 T^{7} + 4578584118 T^{8} - 45568887554 T^{9} + 472130537246 T^{10} - 45568887554 p T^{11} + 4578584118 p^{2} T^{12} - 378419448 p^{3} T^{13} + 32797309 p^{4} T^{14} - 2244070 p^{5} T^{15} + 167261 p^{6} T^{16} - 8711 p^{7} T^{17} + 560 p^{8} T^{18} - 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 20 T + 723 T^{2} - 9755 T^{3} + 204095 T^{4} - 1956908 T^{5} + 31317745 T^{6} - 210880236 T^{7} + 3163205292 T^{8} - 15562139561 T^{9} + 285935861312 T^{10} - 15562139561 p T^{11} + 3163205292 p^{2} T^{12} - 210880236 p^{3} T^{13} + 31317745 p^{4} T^{14} - 1956908 p^{5} T^{15} + 204095 p^{6} T^{16} - 9755 p^{7} T^{17} + 723 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 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
−2.65801840093709720570395001832, −2.58627943341852392239902965113, −2.55932830961783511450056927429, −2.54122160888114030468743014914, −2.53553666679167904250256178548, −2.02648461152261454167979727793, −2.00623840416989159561290895380, −1.93976781235299917290616413145, −1.86832998536364974878228458730, −1.85142842600178347087552943523, −1.81298303554747445911603822169, −1.75869682044288954972415157732, −1.74820936834990252256178942070, −1.56421246354036235099225584053, −1.53877891433945387161578938209, −1.09228707455740338601919994042, −0.979809292569977367186490493329, −0.976711580531831621534106850786, −0.975517606149034635738372995451, −0.967369020715271251186657375136, −0.961033605185082854866244639685, −0.837118504607028838326339251542, −0.73427981102896019959799659275, −0.64680918052593808542111536290, −0.52823939222707856295108239753, 0.52823939222707856295108239753, 0.64680918052593808542111536290, 0.73427981102896019959799659275, 0.837118504607028838326339251542, 0.961033605185082854866244639685, 0.967369020715271251186657375136, 0.975517606149034635738372995451, 0.976711580531831621534106850786, 0.979809292569977367186490493329, 1.09228707455740338601919994042, 1.53877891433945387161578938209, 1.56421246354036235099225584053, 1.74820936834990252256178942070, 1.75869682044288954972415157732, 1.81298303554747445911603822169, 1.85142842600178347087552943523, 1.86832998536364974878228458730, 1.93976781235299917290616413145, 2.00623840416989159561290895380, 2.02648461152261454167979727793, 2.53553666679167904250256178548, 2.54122160888114030468743014914, 2.55932830961783511450056927429, 2.58627943341852392239902965113, 2.65801840093709720570395001832
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.