Dirichlet series
| L(s) = 1 | + 3·3-s − 2·5-s − 11·7-s − 5·9-s + 11·11-s + 11·13-s − 6·15-s + 9·17-s + 20·19-s − 33·21-s + 12·23-s − 19·25-s − 22·27-s + 8·29-s + 7·31-s + 33·33-s + 22·35-s − 10·37-s + 33·39-s − 2·41-s + 24·43-s + 10·45-s + 2·47-s + 66·49-s + 27·51-s + 3·53-s − 22·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.894·5-s − 4.15·7-s − 5/3·9-s + 3.31·11-s + 3.05·13-s − 1.54·15-s + 2.18·17-s + 4.58·19-s − 7.20·21-s + 2.50·23-s − 3.79·25-s − 4.23·27-s + 1.48·29-s + 1.25·31-s + 5.74·33-s + 3.71·35-s − 1.64·37-s + 5.28·39-s − 0.312·41-s + 3.65·43-s + 1.49·45-s + 0.291·47-s + 66/7·49-s + 3.78·51-s + 0.412·53-s − 2.96·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{33} \cdot 7^{11} \cdot 11^{11} \cdot 13^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{33} \cdot 7^{11} \cdot 11^{11} \cdot 13^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(22\) |
| Conductor: | \(2^{33} \cdot 7^{11} \cdot 11^{11} \cdot 13^{11}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.30824\times 10^{19}\) |
| Root analytic conductor: | \(7.99651\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((22,\ 2^{33} \cdot 7^{11} \cdot 11^{11} \cdot 13^{11} ,\ ( \ : [1/2]^{11} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(487.6447671\) |
| \(L(\frac12)\) | \(\approx\) | \(487.6447671\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + T )^{11} \) | |
| 11 | \( ( 1 - T )^{11} \) | |
| 13 | \( ( 1 - T )^{11} \) | |
| good | 3 | \( 1 - p T + 14 T^{2} - 35 T^{3} + 110 T^{4} - 256 T^{5} + 644 T^{6} - 1346 T^{7} + 2900 T^{8} - 5551 T^{9} + 3539 p T^{10} - 18530 T^{11} + 3539 p^{2} T^{12} - 5551 p^{2} T^{13} + 2900 p^{3} T^{14} - 1346 p^{4} T^{15} + 644 p^{5} T^{16} - 256 p^{6} T^{17} + 110 p^{7} T^{18} - 35 p^{8} T^{19} + 14 p^{9} T^{20} - p^{11} T^{21} + p^{11} T^{22} \) |
| 5 | \( 1 + 2 T + 23 T^{2} + 2 p^{2} T^{3} + 298 T^{4} + 659 T^{5} + 548 p T^{6} + 6083 T^{7} + 3918 p T^{8} + 42828 T^{9} + 115828 T^{10} + 238548 T^{11} + 115828 p T^{12} + 42828 p^{2} T^{13} + 3918 p^{4} T^{14} + 6083 p^{4} T^{15} + 548 p^{6} T^{16} + 659 p^{6} T^{17} + 298 p^{7} T^{18} + 2 p^{10} T^{19} + 23 p^{9} T^{20} + 2 p^{10} T^{21} + p^{11} T^{22} \) | |
| 17 | \( 1 - 9 T + 134 T^{2} - 851 T^{3} + 7210 T^{4} - 35742 T^{5} + 228876 T^{6} - 978070 T^{7} + 5442480 T^{8} - 1272673 p T^{9} + 109691371 T^{10} - 404886510 T^{11} + 109691371 p T^{12} - 1272673 p^{3} T^{13} + 5442480 p^{3} T^{14} - 978070 p^{4} T^{15} + 228876 p^{5} T^{16} - 35742 p^{6} T^{17} + 7210 p^{7} T^{18} - 851 p^{8} T^{19} + 134 p^{9} T^{20} - 9 p^{10} T^{21} + p^{11} T^{22} \) | |
| 19 | \( 1 - 20 T + 317 T^{2} - 3546 T^{3} + 34648 T^{4} - 282731 T^{5} + 2085336 T^{6} - 13545407 T^{7} + 80854454 T^{8} - 434652258 T^{9} + 2164177438 T^{10} - 9790215924 T^{11} + 2164177438 p T^{12} - 434652258 p^{2} T^{13} + 80854454 p^{3} T^{14} - 13545407 p^{4} T^{15} + 2085336 p^{5} T^{16} - 282731 p^{6} T^{17} + 34648 p^{7} T^{18} - 3546 p^{8} T^{19} + 317 p^{9} T^{20} - 20 p^{10} T^{21} + p^{11} T^{22} \) | |
| 23 | \( 1 - 12 T + 192 T^{2} - 1500 T^{3} + 14405 T^{4} - 84603 T^{5} + 622928 T^{6} - 2921524 T^{7} + 18550871 T^{8} - 73654597 T^{9} + 448012645 T^{10} - 1670444328 T^{11} + 448012645 p T^{12} - 73654597 p^{2} T^{13} + 18550871 p^{3} T^{14} - 2921524 p^{4} T^{15} + 622928 p^{5} T^{16} - 84603 p^{6} T^{17} + 14405 p^{7} T^{18} - 1500 p^{8} T^{19} + 192 p^{9} T^{20} - 12 p^{10} T^{21} + p^{11} T^{22} \) | |
| 29 | \( 1 - 8 T + 156 T^{2} - 642 T^{3} + 9613 T^{4} - 22149 T^{5} + 463232 T^{6} - 864816 T^{7} + 20159887 T^{8} - 29244127 T^{9} + 676855499 T^{10} - 744028548 T^{11} + 676855499 p T^{12} - 29244127 p^{2} T^{13} + 20159887 p^{3} T^{14} - 864816 p^{4} T^{15} + 463232 p^{5} T^{16} - 22149 p^{6} T^{17} + 9613 p^{7} T^{18} - 642 p^{8} T^{19} + 156 p^{9} T^{20} - 8 p^{10} T^{21} + p^{11} T^{22} \) | |
| 31 | \( 1 - 7 T + 249 T^{2} - 1248 T^{3} + 26392 T^{4} - 87502 T^{5} + 1595246 T^{6} - 2689786 T^{7} + 63810197 T^{8} - 10013439 T^{9} + 65107945 p T^{10} + 1239762012 T^{11} + 65107945 p^{2} T^{12} - 10013439 p^{2} T^{13} + 63810197 p^{3} T^{14} - 2689786 p^{4} T^{15} + 1595246 p^{5} T^{16} - 87502 p^{6} T^{17} + 26392 p^{7} T^{18} - 1248 p^{8} T^{19} + 249 p^{9} T^{20} - 7 p^{10} T^{21} + p^{11} T^{22} \) | |
| 37 | \( 1 + 10 T + 319 T^{2} + 2837 T^{3} + 47082 T^{4} + 374492 T^{5} + 4295164 T^{6} + 30718176 T^{7} + 273936311 T^{8} + 1762949114 T^{9} + 13082515083 T^{10} + 75043044598 T^{11} + 13082515083 p T^{12} + 1762949114 p^{2} T^{13} + 273936311 p^{3} T^{14} + 30718176 p^{4} T^{15} + 4295164 p^{5} T^{16} + 374492 p^{6} T^{17} + 47082 p^{7} T^{18} + 2837 p^{8} T^{19} + 319 p^{9} T^{20} + 10 p^{10} T^{21} + p^{11} T^{22} \) | |
| 41 | \( 1 + 2 T + 235 T^{2} + 329 T^{3} + 26186 T^{4} + 12410 T^{5} + 1884556 T^{6} - 1260320 T^{7} + 101198111 T^{8} - 172222516 T^{9} + 4539153071 T^{10} - 9570361186 T^{11} + 4539153071 p T^{12} - 172222516 p^{2} T^{13} + 101198111 p^{3} T^{14} - 1260320 p^{4} T^{15} + 1884556 p^{5} T^{16} + 12410 p^{6} T^{17} + 26186 p^{7} T^{18} + 329 p^{8} T^{19} + 235 p^{9} T^{20} + 2 p^{10} T^{21} + p^{11} T^{22} \) | |
| 43 | \( 1 - 24 T + 570 T^{2} - 8624 T^{3} + 124347 T^{4} - 1424297 T^{5} + 15555778 T^{6} - 145133755 T^{7} + 1294429263 T^{8} - 10163934916 T^{9} + 76425625783 T^{10} - 511951909880 T^{11} + 76425625783 p T^{12} - 10163934916 p^{2} T^{13} + 1294429263 p^{3} T^{14} - 145133755 p^{4} T^{15} + 15555778 p^{5} T^{16} - 1424297 p^{6} T^{17} + 124347 p^{7} T^{18} - 8624 p^{8} T^{19} + 570 p^{9} T^{20} - 24 p^{10} T^{21} + p^{11} T^{22} \) | |
| 47 | \( 1 - 2 T + 212 T^{2} - 828 T^{3} + 21145 T^{4} - 135423 T^{5} + 1494800 T^{6} - 12228020 T^{7} + 94899263 T^{8} - 744129267 T^{9} + 5452191061 T^{10} - 36881557944 T^{11} + 5452191061 p T^{12} - 744129267 p^{2} T^{13} + 94899263 p^{3} T^{14} - 12228020 p^{4} T^{15} + 1494800 p^{5} T^{16} - 135423 p^{6} T^{17} + 21145 p^{7} T^{18} - 828 p^{8} T^{19} + 212 p^{9} T^{20} - 2 p^{10} T^{21} + p^{11} T^{22} \) | |
| 53 | \( 1 - 3 T + 387 T^{2} - 1280 T^{3} + 73310 T^{4} - 252172 T^{5} + 8990200 T^{6} - 30805260 T^{7} + 797190694 T^{8} - 2608887812 T^{9} + 53993953024 T^{10} - 160882747794 T^{11} + 53993953024 p T^{12} - 2608887812 p^{2} T^{13} + 797190694 p^{3} T^{14} - 30805260 p^{4} T^{15} + 8990200 p^{5} T^{16} - 252172 p^{6} T^{17} + 73310 p^{7} T^{18} - 1280 p^{8} T^{19} + 387 p^{9} T^{20} - 3 p^{10} T^{21} + p^{11} T^{22} \) | |
| 59 | \( 1 - T + 224 T^{2} + 133 T^{3} + 36141 T^{4} + 15757 T^{5} + 4070190 T^{6} + 3199184 T^{7} + 364899481 T^{8} + 236484468 T^{9} + 26261474493 T^{10} + 18909200678 T^{11} + 26261474493 p T^{12} + 236484468 p^{2} T^{13} + 364899481 p^{3} T^{14} + 3199184 p^{4} T^{15} + 4070190 p^{5} T^{16} + 15757 p^{6} T^{17} + 36141 p^{7} T^{18} + 133 p^{8} T^{19} + 224 p^{9} T^{20} - p^{10} T^{21} + p^{11} T^{22} \) | |
| 61 | \( 1 + 22 T + 618 T^{2} + 9937 T^{3} + 169986 T^{4} + 2182629 T^{5} + 28409784 T^{6} + 304289221 T^{7} + 3247908158 T^{8} + 29662032931 T^{9} + 268209689585 T^{10} + 2107055396584 T^{11} + 268209689585 p T^{12} + 29662032931 p^{2} T^{13} + 3247908158 p^{3} T^{14} + 304289221 p^{4} T^{15} + 28409784 p^{5} T^{16} + 2182629 p^{6} T^{17} + 169986 p^{7} T^{18} + 9937 p^{8} T^{19} + 618 p^{9} T^{20} + 22 p^{10} T^{21} + p^{11} T^{22} \) | |
| 67 | \( 1 - 14 T + 424 T^{2} - 5390 T^{3} + 93554 T^{4} - 1043318 T^{5} + 13734392 T^{6} - 134567702 T^{7} + 1484214336 T^{8} - 12925275678 T^{9} + 124679534891 T^{10} - 971445857060 T^{11} + 124679534891 p T^{12} - 12925275678 p^{2} T^{13} + 1484214336 p^{3} T^{14} - 134567702 p^{4} T^{15} + 13734392 p^{5} T^{16} - 1043318 p^{6} T^{17} + 93554 p^{7} T^{18} - 5390 p^{8} T^{19} + 424 p^{9} T^{20} - 14 p^{10} T^{21} + p^{11} T^{22} \) | |
| 71 | \( 1 - 6 T + 608 T^{2} - 4098 T^{3} + 175185 T^{4} - 1254282 T^{5} + 31750168 T^{6} - 228972208 T^{7} + 4038464595 T^{8} - 27776745808 T^{9} + 379629912533 T^{10} - 2347019342620 T^{11} + 379629912533 p T^{12} - 27776745808 p^{2} T^{13} + 4038464595 p^{3} T^{14} - 228972208 p^{4} T^{15} + 31750168 p^{5} T^{16} - 1254282 p^{6} T^{17} + 175185 p^{7} T^{18} - 4098 p^{8} T^{19} + 608 p^{9} T^{20} - 6 p^{10} T^{21} + p^{11} T^{22} \) | |
| 73 | \( 1 - 3 T + 417 T^{2} - 1931 T^{3} + 76749 T^{4} - 469436 T^{5} + 8556635 T^{6} - 60136020 T^{7} + 704611696 T^{8} - 4924805341 T^{9} + 51542750918 T^{10} - 346096909546 T^{11} + 51542750918 p T^{12} - 4924805341 p^{2} T^{13} + 704611696 p^{3} T^{14} - 60136020 p^{4} T^{15} + 8556635 p^{5} T^{16} - 469436 p^{6} T^{17} + 76749 p^{7} T^{18} - 1931 p^{8} T^{19} + 417 p^{9} T^{20} - 3 p^{10} T^{21} + p^{11} T^{22} \) | |
| 79 | \( 1 - 8 T + 522 T^{2} - 2838 T^{3} + 124107 T^{4} - 440869 T^{5} + 19198670 T^{6} - 46106323 T^{7} + 2302081539 T^{8} - 4421151020 T^{9} + 225768765741 T^{10} - 385980877516 T^{11} + 225768765741 p T^{12} - 4421151020 p^{2} T^{13} + 2302081539 p^{3} T^{14} - 46106323 p^{4} T^{15} + 19198670 p^{5} T^{16} - 440869 p^{6} T^{17} + 124107 p^{7} T^{18} - 2838 p^{8} T^{19} + 522 p^{9} T^{20} - 8 p^{10} T^{21} + p^{11} T^{22} \) | |
| 83 | \( 1 - 29 T + 1022 T^{2} - 21484 T^{3} + 447233 T^{4} - 7344360 T^{5} + 114115596 T^{6} - 1525731728 T^{7} + 19142277847 T^{8} - 213330073578 T^{9} + 2232579781879 T^{10} - 20979351912130 T^{11} + 2232579781879 p T^{12} - 213330073578 p^{2} T^{13} + 19142277847 p^{3} T^{14} - 1525731728 p^{4} T^{15} + 114115596 p^{5} T^{16} - 7344360 p^{6} T^{17} + 447233 p^{7} T^{18} - 21484 p^{8} T^{19} + 1022 p^{9} T^{20} - 29 p^{10} T^{21} + p^{11} T^{22} \) | |
| 89 | \( 1 - 20 T + 591 T^{2} - 8534 T^{3} + 147340 T^{4} - 1724615 T^{5} + 23172868 T^{6} - 245128371 T^{7} + 2937871924 T^{8} - 29454684006 T^{9} + 321616971802 T^{10} - 2943714830264 T^{11} + 321616971802 p T^{12} - 29454684006 p^{2} T^{13} + 2937871924 p^{3} T^{14} - 245128371 p^{4} T^{15} + 23172868 p^{5} T^{16} - 1724615 p^{6} T^{17} + 147340 p^{7} T^{18} - 8534 p^{8} T^{19} + 591 p^{9} T^{20} - 20 p^{10} T^{21} + p^{11} T^{22} \) | |
| 97 | \( 1 + 25 T + 835 T^{2} + 16799 T^{3} + 339933 T^{4} + 5591948 T^{5} + 86844721 T^{6} + 1201983042 T^{7} + 15427260042 T^{8} + 183131036395 T^{9} + 2004145990212 T^{10} + 20569760330974 T^{11} + 2004145990212 p T^{12} + 183131036395 p^{2} T^{13} + 15427260042 p^{3} T^{14} + 1201983042 p^{4} T^{15} + 86844721 p^{5} T^{16} + 5591948 p^{6} T^{17} + 339933 p^{7} T^{18} + 16799 p^{8} T^{19} + 835 p^{9} T^{20} + 25 p^{10} T^{21} + p^{11} T^{22} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{22} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.61098232213773353702685824437, −2.57432810518823866123754484372, −2.55448433494666597135971806753, −2.02042739493307917433567643895, −1.98642950358766981309839414651, −1.96404337405570060441582811549, −1.94248094561944986697166995438, −1.91846382821899570382782137768, −1.83034231659917583407250872683, −1.70041281478148869697539614907, −1.69453497619281752773748076084, −1.58366217607258385530556200553, −1.38848755002095931361260894362, −1.22833109058845380354468229434, −1.22503403586371407082801988598, −1.00685727346929222851253481678, −0.877757750366246441533822502724, −0.822440200553613168532414137751, −0.67926266834859741784148343409, −0.66334243853835471514983069187, −0.62569250806196730673913586196, −0.53260817183614215126087350810, −0.50868390065528439472583494855, −0.47239665641727916691482202650, −0.41261040992178619457746279508, 0.41261040992178619457746279508, 0.47239665641727916691482202650, 0.50868390065528439472583494855, 0.53260817183614215126087350810, 0.62569250806196730673913586196, 0.66334243853835471514983069187, 0.67926266834859741784148343409, 0.822440200553613168532414137751, 0.877757750366246441533822502724, 1.00685727346929222851253481678, 1.22503403586371407082801988598, 1.22833109058845380354468229434, 1.38848755002095931361260894362, 1.58366217607258385530556200553, 1.69453497619281752773748076084, 1.70041281478148869697539614907, 1.83034231659917583407250872683, 1.91846382821899570382782137768, 1.94248094561944986697166995438, 1.96404337405570060441582811549, 1.98642950358766981309839414651, 2.02042739493307917433567643895, 2.55448433494666597135971806753, 2.57432810518823866123754484372, 2.61098232213773353702685824437
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.