Dirichlet series
| L(s) = 1 | + 2·2-s + 2·3-s − 3·4-s − 3·5-s + 4·6-s + 3·7-s − 6·8-s − 8·9-s − 6·10-s + 3·11-s − 6·12-s + 13-s + 6·14-s − 6·15-s + 4·16-s + 16·17-s − 16·18-s + 4·19-s + 9·20-s + 6·21-s + 6·22-s + 11·23-s − 12·24-s − 22·25-s + 2·26-s − 12·27-s − 9·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s − 3/2·4-s − 1.34·5-s + 1.63·6-s + 1.13·7-s − 2.12·8-s − 8/3·9-s − 1.89·10-s + 0.904·11-s − 1.73·12-s + 0.277·13-s + 1.60·14-s − 1.54·15-s + 16-s + 3.88·17-s − 3.77·18-s + 0.917·19-s + 2.01·20-s + 1.30·21-s + 1.27·22-s + 2.29·23-s − 2.44·24-s − 4.39·25-s + 0.392·26-s − 2.30·27-s − 1.70·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(71^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(71^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(71^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.11670\times 10^{16}\) |
| Root analytic conductor: | \(6.34449\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 71^{20} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(21.29214836\) |
| \(L(\frac12)\) | \(\approx\) | \(21.29214836\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 71 | \( 1 \) |
| good | 2 | \( 1 - p T + 7 T^{2} - 7 p T^{3} + 33 T^{4} - 15 p^{2} T^{5} + 111 T^{6} - 183 T^{7} + 305 T^{8} - 111 p^{2} T^{9} + 669 T^{10} - 111 p^{3} T^{11} + 305 p^{2} T^{12} - 183 p^{3} T^{13} + 111 p^{4} T^{14} - 15 p^{7} T^{15} + 33 p^{6} T^{16} - 7 p^{8} T^{17} + 7 p^{8} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) |
| 3 | \( 1 - 2 T + 4 p T^{2} - 28 T^{3} + 28 p T^{4} - 61 p T^{5} + 433 T^{6} - 832 T^{7} + 1732 T^{8} - 3023 T^{9} + 5677 T^{10} - 3023 p T^{11} + 1732 p^{2} T^{12} - 832 p^{3} T^{13} + 433 p^{4} T^{14} - 61 p^{6} T^{15} + 28 p^{7} T^{16} - 28 p^{7} T^{17} + 4 p^{9} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 5 | \( 1 + 3 T + 31 T^{2} + 89 T^{3} + 488 T^{4} + 1268 T^{5} + 5034 T^{6} + 11654 T^{7} + 37621 T^{8} + 15437 p T^{9} + 42803 p T^{10} + 15437 p^{2} T^{11} + 37621 p^{2} T^{12} + 11654 p^{3} T^{13} + 5034 p^{4} T^{14} + 1268 p^{5} T^{15} + 488 p^{6} T^{16} + 89 p^{7} T^{17} + 31 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 7 | \( 1 - 3 T + 39 T^{2} - 12 p T^{3} + 631 T^{4} - 771 T^{5} + 5238 T^{6} + 824 T^{7} + 22348 T^{8} + 68815 T^{9} + 78889 T^{10} + 68815 p T^{11} + 22348 p^{2} T^{12} + 824 p^{3} T^{13} + 5238 p^{4} T^{14} - 771 p^{5} T^{15} + 631 p^{6} T^{16} - 12 p^{8} T^{17} + 39 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 3 T + 70 T^{2} - 241 T^{3} + 2593 T^{4} - 8641 T^{5} + 63687 T^{6} - 193046 T^{7} + 1109001 T^{8} - 2976499 T^{9} + 14169235 T^{10} - 2976499 p T^{11} + 1109001 p^{2} T^{12} - 193046 p^{3} T^{13} + 63687 p^{4} T^{14} - 8641 p^{5} T^{15} + 2593 p^{6} T^{16} - 241 p^{7} T^{17} + 70 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - T + 49 T^{2} + 10 T^{3} + 1535 T^{4} + 50 p T^{5} + 35340 T^{6} + 24910 T^{7} + 622405 T^{8} + 457919 T^{9} + 9092219 T^{10} + 457919 p T^{11} + 622405 p^{2} T^{12} + 24910 p^{3} T^{13} + 35340 p^{4} T^{14} + 50 p^{6} T^{15} + 1535 p^{6} T^{16} + 10 p^{7} T^{17} + 49 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 16 T + 198 T^{2} - 1646 T^{3} + 11839 T^{4} - 70169 T^{5} + 389957 T^{6} - 1972784 T^{7} + 9712787 T^{8} - 44196994 T^{9} + 191050673 T^{10} - 44196994 p T^{11} + 9712787 p^{2} T^{12} - 1972784 p^{3} T^{13} + 389957 p^{4} T^{14} - 70169 p^{5} T^{15} + 11839 p^{6} T^{16} - 1646 p^{7} T^{17} + 198 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 4 T + 118 T^{2} - 409 T^{3} + 6459 T^{4} - 20283 T^{5} + 224242 T^{6} - 656784 T^{7} + 5730393 T^{8} - 15791134 T^{9} + 118672589 T^{10} - 15791134 p T^{11} + 5730393 p^{2} T^{12} - 656784 p^{3} T^{13} + 224242 p^{4} T^{14} - 20283 p^{5} T^{15} + 6459 p^{6} T^{16} - 409 p^{7} T^{17} + 118 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - 11 T + 153 T^{2} - 1216 T^{3} + 10822 T^{4} - 73003 T^{5} + 510320 T^{6} - 2980216 T^{7} + 17486120 T^{8} - 89829009 T^{9} + 458008117 T^{10} - 89829009 p T^{11} + 17486120 p^{2} T^{12} - 2980216 p^{3} T^{13} + 510320 p^{4} T^{14} - 73003 p^{5} T^{15} + 10822 p^{6} T^{16} - 1216 p^{7} T^{17} + 153 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 8 T + 215 T^{2} + 1495 T^{3} + 21222 T^{4} + 133413 T^{5} + 1308129 T^{6} + 7539049 T^{7} + 57189438 T^{8} + 299111343 T^{9} + 1888763399 T^{10} + 299111343 p T^{11} + 57189438 p^{2} T^{12} + 7539049 p^{3} T^{13} + 1308129 p^{4} T^{14} + 133413 p^{5} T^{15} + 21222 p^{6} T^{16} + 1495 p^{7} T^{17} + 215 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 12 T + 271 T^{2} - 2621 T^{3} + 33760 T^{4} - 269603 T^{5} + 2553317 T^{6} - 17186587 T^{7} + 130821452 T^{8} - 749676247 T^{9} + 4771666959 T^{10} - 749676247 p T^{11} + 130821452 p^{2} T^{12} - 17186587 p^{3} T^{13} + 2553317 p^{4} T^{14} - 269603 p^{5} T^{15} + 33760 p^{6} T^{16} - 2621 p^{7} T^{17} + 271 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 3 T + 188 T^{2} + 321 T^{3} + 17836 T^{4} + 16807 T^{5} + 1168998 T^{6} + 561835 T^{7} + 58645484 T^{8} + 13072191 T^{9} + 2381920755 T^{10} + 13072191 p T^{11} + 58645484 p^{2} T^{12} + 561835 p^{3} T^{13} + 1168998 p^{4} T^{14} + 16807 p^{5} T^{15} + 17836 p^{6} T^{16} + 321 p^{7} T^{17} + 188 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 30 T + 749 T^{2} - 12695 T^{3} + 187908 T^{4} - 2262813 T^{5} + 24430067 T^{6} - 227137601 T^{7} + 1919190616 T^{8} - 14280949101 T^{9} + 2369271569 p T^{10} - 14280949101 p T^{11} + 1919190616 p^{2} T^{12} - 227137601 p^{3} T^{13} + 24430067 p^{4} T^{14} - 2262813 p^{5} T^{15} + 187908 p^{6} T^{16} - 12695 p^{7} T^{17} + 749 p^{8} T^{18} - 30 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 4 T + 321 T^{2} - 1097 T^{3} + 48661 T^{4} - 143497 T^{5} + 108158 p T^{6} - 11935508 T^{7} + 313055918 T^{8} - 700588704 T^{9} + 15561082739 T^{10} - 700588704 p T^{11} + 313055918 p^{2} T^{12} - 11935508 p^{3} T^{13} + 108158 p^{5} T^{14} - 143497 p^{5} T^{15} + 48661 p^{6} T^{16} - 1097 p^{7} T^{17} + 321 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 48 T + 1429 T^{2} - 30606 T^{3} + 523921 T^{4} - 7433009 T^{5} + 90207546 T^{6} - 949898684 T^{7} + 8794280943 T^{8} - 71972860468 T^{9} + 523428078151 T^{10} - 71972860468 p T^{11} + 8794280943 p^{2} T^{12} - 949898684 p^{3} T^{13} + 90207546 p^{4} T^{14} - 7433009 p^{5} T^{15} + 523921 p^{6} T^{16} - 30606 p^{7} T^{17} + 1429 p^{8} T^{18} - 48 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 - 29 T + 671 T^{2} - 11515 T^{3} + 167669 T^{4} - 2106935 T^{5} + 23512717 T^{6} - 235599865 T^{7} + 2145954910 T^{8} - 17807550591 T^{9} + 135547752225 T^{10} - 17807550591 p T^{11} + 2145954910 p^{2} T^{12} - 235599865 p^{3} T^{13} + 23512717 p^{4} T^{14} - 2106935 p^{5} T^{15} + 167669 p^{6} T^{16} - 11515 p^{7} T^{17} + 671 p^{8} T^{18} - 29 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 17 T + 393 T^{2} - 4393 T^{3} + 56905 T^{4} - 445787 T^{5} + 4090452 T^{6} - 21327085 T^{7} + 160902161 T^{8} - 463161899 T^{9} + 5784200865 T^{10} - 463161899 p T^{11} + 160902161 p^{2} T^{12} - 21327085 p^{3} T^{13} + 4090452 p^{4} T^{14} - 445787 p^{5} T^{15} + 56905 p^{6} T^{16} - 4393 p^{7} T^{17} + 393 p^{8} T^{18} - 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 16 T + 632 T^{2} - 8152 T^{3} + 176178 T^{4} - 1880575 T^{5} + 28823183 T^{6} - 257837290 T^{7} + 3081845811 T^{8} - 23154768720 T^{9} + 225649248425 T^{10} - 23154768720 p T^{11} + 3081845811 p^{2} T^{12} - 257837290 p^{3} T^{13} + 28823183 p^{4} T^{14} - 1880575 p^{5} T^{15} + 176178 p^{6} T^{16} - 8152 p^{7} T^{17} + 632 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 36 T + 906 T^{2} + 16867 T^{3} + 265374 T^{4} + 3560667 T^{5} + 42612227 T^{6} + 457925390 T^{7} + 4524190841 T^{8} + 41156706993 T^{9} + 349465459579 T^{10} + 41156706993 p T^{11} + 4524190841 p^{2} T^{12} + 457925390 p^{3} T^{13} + 42612227 p^{4} T^{14} + 3560667 p^{5} T^{15} + 265374 p^{6} T^{16} + 16867 p^{7} T^{17} + 906 p^{8} T^{18} + 36 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 17 T + 435 T^{2} - 5614 T^{3} + 94161 T^{4} - 1031950 T^{5} + 13714238 T^{6} - 130614542 T^{7} + 1471563301 T^{8} - 12401055463 T^{9} + 121862110441 T^{10} - 12401055463 p T^{11} + 1471563301 p^{2} T^{12} - 130614542 p^{3} T^{13} + 13714238 p^{4} T^{14} - 1031950 p^{5} T^{15} + 94161 p^{6} T^{16} - 5614 p^{7} T^{17} + 435 p^{8} T^{18} - 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 38 T + 1244 T^{2} - 345 p T^{3} + 534332 T^{4} - 8471936 T^{5} + 122875905 T^{6} - 1531457355 T^{7} + 17663672729 T^{8} - 179330849338 T^{9} + 1693260658783 T^{10} - 179330849338 p T^{11} + 17663672729 p^{2} T^{12} - 1531457355 p^{3} T^{13} + 122875905 p^{4} T^{14} - 8471936 p^{5} T^{15} + 534332 p^{6} T^{16} - 345 p^{8} T^{17} + 1244 p^{8} T^{18} - 38 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 10 T + 577 T^{2} + 5111 T^{3} + 160882 T^{4} + 1271138 T^{5} + 28797983 T^{6} + 202725594 T^{7} + 3675897327 T^{8} + 22885374549 T^{9} + 350515453657 T^{10} + 22885374549 p T^{11} + 3675897327 p^{2} T^{12} + 202725594 p^{3} T^{13} + 28797983 p^{4} T^{14} + 1271138 p^{5} T^{15} + 160882 p^{6} T^{16} + 5111 p^{7} T^{17} + 577 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 17 T + 739 T^{2} + 10217 T^{3} + 254531 T^{4} + 2973233 T^{5} + 54201973 T^{6} + 542262829 T^{7} + 7909012458 T^{8} + 67927029231 T^{9} + 826550621907 T^{10} + 67927029231 p T^{11} + 7909012458 p^{2} T^{12} + 542262829 p^{3} T^{13} + 54201973 p^{4} T^{14} + 2973233 p^{5} T^{15} + 254531 p^{6} T^{16} + 10217 p^{7} T^{17} + 739 p^{8} T^{18} + 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 29 T + 753 T^{2} - 13598 T^{3} + 237606 T^{4} - 3447077 T^{5} + 48289004 T^{6} - 592388194 T^{7} + 7044759241 T^{8} - 75193057556 T^{9} + 779454822431 T^{10} - 75193057556 p T^{11} + 7044759241 p^{2} T^{12} - 592388194 p^{3} T^{13} + 48289004 p^{4} T^{14} - 3447077 p^{5} T^{15} + 237606 p^{6} T^{16} - 13598 p^{7} T^{17} + 753 p^{8} T^{18} - 29 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−2.82039238503304244697547266723, −2.73692464197226973460129336421, −2.70353570041568836074486642176, −2.67304140763418810127559847490, −2.54644484557885200876802839569, −2.38163075337382827313564880585, −2.32015286837690032768974957244, −2.26500626528144778904418275158, −2.23897494773470032632465461899, −2.10365964473357979475081457799, −1.86140565357492220789683599403, −1.73242959554996908714509172453, −1.66891293137663583248837591745, −1.61970018782820123568325757112, −1.32304860690948155861722505748, −1.19501290413681888606598498608, −1.13754427090540893742578827570, −0.981080170518012664757948266539, −0.941219136907534419502829109331, −0.804578825235133267530066933764, −0.802746879552744505058419899031, −0.58952476174460801902390370904, −0.58516076084415726358442495152, −0.22200632173105166958679231613, −0.19603835543064196100116384975, 0.19603835543064196100116384975, 0.22200632173105166958679231613, 0.58516076084415726358442495152, 0.58952476174460801902390370904, 0.802746879552744505058419899031, 0.804578825235133267530066933764, 0.941219136907534419502829109331, 0.981080170518012664757948266539, 1.13754427090540893742578827570, 1.19501290413681888606598498608, 1.32304860690948155861722505748, 1.61970018782820123568325757112, 1.66891293137663583248837591745, 1.73242959554996908714509172453, 1.86140565357492220789683599403, 2.10365964473357979475081457799, 2.23897494773470032632465461899, 2.26500626528144778904418275158, 2.32015286837690032768974957244, 2.38163075337382827313564880585, 2.54644484557885200876802839569, 2.67304140763418810127559847490, 2.70353570041568836074486642176, 2.73692464197226973460129336421, 2.82039238503304244697547266723
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.