Dirichlet series
| L(s) = 1 | + 5·2-s − 3-s + 9·4-s − 5·6-s + 8·7-s + 4·8-s − 7·9-s − 9·12-s + 9·13-s + 40·14-s − 7·16-s + 19·17-s − 35·18-s − 19-s − 8·21-s − 2·23-s − 4·24-s + 45·26-s + 9·27-s + 72·28-s − 7·29-s − 5·31-s − 3·32-s + 95·34-s − 63·36-s + 8·37-s − 5·38-s + ⋯ |
| L(s) = 1 | + 3.53·2-s − 0.577·3-s + 9/2·4-s − 2.04·6-s + 3.02·7-s + 1.41·8-s − 7/3·9-s − 2.59·12-s + 2.49·13-s + 10.6·14-s − 7/4·16-s + 4.60·17-s − 8.24·18-s − 0.229·19-s − 1.74·21-s − 0.417·23-s − 0.816·24-s + 8.82·26-s + 1.73·27-s + 13.6·28-s − 1.29·29-s − 0.898·31-s − 0.530·32-s + 16.2·34-s − 10.5·36-s + 1.31·37-s − 0.811·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(5^{16} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.15883\times 10^{11}\) |
| Root analytic conductor: | \(4.91474\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 5^{16} \cdot 11^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(140.4432080\) |
| \(L(\frac12)\) | \(\approx\) | \(140.4432080\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) | |
| good | 2 | \( 1 - 5 T + p^{4} T^{2} - 39 T^{3} + 39 p T^{4} - 135 T^{5} + 105 p T^{6} - 307 T^{7} + 439 T^{8} - 307 p T^{9} + 105 p^{3} T^{10} - 135 p^{3} T^{11} + 39 p^{5} T^{12} - 39 p^{5} T^{13} + p^{10} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 3 | \( 1 + T + 8 T^{2} + 2 p T^{3} + 38 T^{4} + 31 T^{5} + 5 p^{3} T^{6} + 4 p^{3} T^{7} + 400 T^{8} + 4 p^{4} T^{9} + 5 p^{5} T^{10} + 31 p^{3} T^{11} + 38 p^{4} T^{12} + 2 p^{6} T^{13} + 8 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 - 8 T + 66 T^{2} - 342 T^{3} + 1706 T^{4} - 6586 T^{5} + 24272 T^{6} - 73662 T^{7} + 213063 T^{8} - 73662 p T^{9} + 24272 p^{2} T^{10} - 6586 p^{3} T^{11} + 1706 p^{4} T^{12} - 342 p^{5} T^{13} + 66 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 9 T + 100 T^{2} - 578 T^{3} + 3816 T^{4} - 17111 T^{5} + 86824 T^{6} - 324855 T^{7} + 1353369 T^{8} - 324855 p T^{9} + 86824 p^{2} T^{10} - 17111 p^{3} T^{11} + 3816 p^{4} T^{12} - 578 p^{5} T^{13} + 100 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 19 T + 243 T^{2} - 2255 T^{3} + 17192 T^{4} - 110091 T^{5} + 614358 T^{6} - 3013746 T^{7} + 13172311 T^{8} - 3013746 p T^{9} + 614358 p^{2} T^{10} - 110091 p^{3} T^{11} + 17192 p^{4} T^{12} - 2255 p^{5} T^{13} + 243 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + T + 88 T^{2} + 14 T^{3} + 3828 T^{4} - 1989 T^{5} + 109205 T^{6} - 96150 T^{7} + 2342996 T^{8} - 96150 p T^{9} + 109205 p^{2} T^{10} - 1989 p^{3} T^{11} + 3828 p^{4} T^{12} + 14 p^{5} T^{13} + 88 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 2 T + 122 T^{2} + 70 T^{3} + 6387 T^{4} - 5332 T^{5} + 202212 T^{6} - 383798 T^{7} + 4953251 T^{8} - 383798 p T^{9} + 202212 p^{2} T^{10} - 5332 p^{3} T^{11} + 6387 p^{4} T^{12} + 70 p^{5} T^{13} + 122 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 7 T + 170 T^{2} + 918 T^{3} + 13326 T^{4} + 59257 T^{5} + 654940 T^{6} + 2455665 T^{7} + 22414391 T^{8} + 2455665 p T^{9} + 654940 p^{2} T^{10} + 59257 p^{3} T^{11} + 13326 p^{4} T^{12} + 918 p^{5} T^{13} + 170 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 5 T + 148 T^{2} + 796 T^{3} + 11764 T^{4} + 59575 T^{5} + 614734 T^{6} + 2772563 T^{7} + 22507763 T^{8} + 2772563 p T^{9} + 614734 p^{2} T^{10} + 59575 p^{3} T^{11} + 11764 p^{4} T^{12} + 796 p^{5} T^{13} + 148 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 8 T + 242 T^{2} - 1432 T^{3} + 25323 T^{4} - 116468 T^{5} + 1591380 T^{6} - 5970396 T^{7} + 69191025 T^{8} - 5970396 p T^{9} + 1591380 p^{2} T^{10} - 116468 p^{3} T^{11} + 25323 p^{4} T^{12} - 1432 p^{5} T^{13} + 242 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 131 T^{2} + 396 T^{3} + 8670 T^{4} + 55224 T^{5} + 386253 T^{6} + 3832556 T^{7} + 15372562 T^{8} + 3832556 p T^{9} + 386253 p^{2} T^{10} + 55224 p^{3} T^{11} + 8670 p^{4} T^{12} + 396 p^{5} T^{13} + 131 p^{6} T^{14} + p^{8} T^{16} \) | |
| 43 | \( 1 - 14 T + 316 T^{2} - 3482 T^{3} + 45224 T^{4} - 397176 T^{5} + 3771144 T^{6} - 26845676 T^{7} + 200490753 T^{8} - 26845676 p T^{9} + 3771144 p^{2} T^{10} - 397176 p^{3} T^{11} + 45224 p^{4} T^{12} - 3482 p^{5} T^{13} + 316 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 11 T + 268 T^{2} + 2202 T^{3} + 33246 T^{4} + 226791 T^{5} + 2636010 T^{6} + 15279331 T^{7} + 146334409 T^{8} + 15279331 p T^{9} + 2636010 p^{2} T^{10} + 226791 p^{3} T^{11} + 33246 p^{4} T^{12} + 2202 p^{5} T^{13} + 268 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 11 T + 208 T^{2} + 2336 T^{3} + 26908 T^{4} + 248741 T^{5} + 2323595 T^{6} + 18511828 T^{7} + 141234900 T^{8} + 18511828 p T^{9} + 2323595 p^{2} T^{10} + 248741 p^{3} T^{11} + 26908 p^{4} T^{12} + 2336 p^{5} T^{13} + 208 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 17 T + 332 T^{2} - 3180 T^{3} + 34422 T^{4} - 180587 T^{5} + 1293838 T^{6} - 715695 T^{7} + 25085249 T^{8} - 715695 p T^{9} + 1293838 p^{2} T^{10} - 180587 p^{3} T^{11} + 34422 p^{4} T^{12} - 3180 p^{5} T^{13} + 332 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 2 T + 206 T^{2} - 900 T^{3} + 24732 T^{4} - 147702 T^{5} + 2088260 T^{6} - 14233262 T^{7} + 139856405 T^{8} - 14233262 p T^{9} + 2088260 p^{2} T^{10} - 147702 p^{3} T^{11} + 24732 p^{4} T^{12} - 900 p^{5} T^{13} + 206 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 7 T + 213 T^{2} + 1343 T^{3} + 30322 T^{4} + 160543 T^{5} + 2816234 T^{6} + 13348426 T^{7} + 216962197 T^{8} + 13348426 p T^{9} + 2816234 p^{2} T^{10} + 160543 p^{3} T^{11} + 30322 p^{4} T^{12} + 1343 p^{5} T^{13} + 213 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 17 T + 469 T^{2} - 5559 T^{3} + 92058 T^{4} - 845601 T^{5} + 10705312 T^{6} - 81702172 T^{7} + 877883563 T^{8} - 81702172 p T^{9} + 10705312 p^{2} T^{10} - 845601 p^{3} T^{11} + 92058 p^{4} T^{12} - 5559 p^{5} T^{13} + 469 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 34 T + 874 T^{2} - 16024 T^{3} + 251241 T^{4} - 3277098 T^{5} + 37994238 T^{6} - 383504944 T^{7} + 3492170683 T^{8} - 383504944 p T^{9} + 37994238 p^{2} T^{10} - 3277098 p^{3} T^{11} + 251241 p^{4} T^{12} - 16024 p^{5} T^{13} + 874 p^{6} T^{14} - 34 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 23 T + 590 T^{2} + 9202 T^{3} + 145476 T^{4} + 1756103 T^{5} + 21095090 T^{6} + 207222795 T^{7} + 2020553951 T^{8} + 207222795 p T^{9} + 21095090 p^{2} T^{10} + 1756103 p^{3} T^{11} + 145476 p^{4} T^{12} + 9202 p^{5} T^{13} + 590 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 41 T + 1232 T^{2} - 26306 T^{3} + 468782 T^{4} - 6890529 T^{5} + 88264466 T^{6} - 973658437 T^{7} + 9506981157 T^{8} - 973658437 p T^{9} + 88264466 p^{2} T^{10} - 6890529 p^{3} T^{11} + 468782 p^{4} T^{12} - 26306 p^{5} T^{13} + 1232 p^{6} T^{14} - 41 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 11 T + 454 T^{2} + 3752 T^{3} + 95432 T^{4} + 645131 T^{5} + 13099840 T^{6} + 76357755 T^{7} + 14991989 p T^{8} + 76357755 p T^{9} + 13099840 p^{2} T^{10} + 645131 p^{3} T^{11} + 95432 p^{4} T^{12} + 3752 p^{5} T^{13} + 454 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 2 T + 519 T^{2} + 2588 T^{3} + 125014 T^{4} + 877988 T^{5} + 19506009 T^{6} + 146142078 T^{7} + 2202033922 T^{8} + 146142078 p T^{9} + 19506009 p^{2} T^{10} + 877988 p^{3} T^{11} + 125014 p^{4} T^{12} + 2588 p^{5} T^{13} + 519 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.76725069921921034594062073017, −3.60104385590852007104501371890, −3.42092707161964320477100172096, −3.27529128293486376786057378951, −3.24564600522360181624965570232, −3.21712183205321903662459202847, −3.21460863386365846848000245770, −2.98853270785849711968273767631, −2.74898186897139858070205940487, −2.65150432684785943648411891789, −2.57846199263661353338620223198, −2.36897748001345719800628685572, −2.11303746156629329761543081863, −1.96639562323323367362520276253, −1.86437400090255359755478188132, −1.74647388374511815406485215768, −1.71945125673432038578140940728, −1.58946241737947470219751595996, −1.19438808095927837214794558007, −1.16549500974790204026667288882, −1.09195322118842036378564564823, −0.797875594893437197619029378141, −0.70570294791171902296597013003, −0.44959022994132176032553300907, −0.37835219024869659023878354447, 0.37835219024869659023878354447, 0.44959022994132176032553300907, 0.70570294791171902296597013003, 0.797875594893437197619029378141, 1.09195322118842036378564564823, 1.16549500974790204026667288882, 1.19438808095927837214794558007, 1.58946241737947470219751595996, 1.71945125673432038578140940728, 1.74647388374511815406485215768, 1.86437400090255359755478188132, 1.96639562323323367362520276253, 2.11303746156629329761543081863, 2.36897748001345719800628685572, 2.57846199263661353338620223198, 2.65150432684785943648411891789, 2.74898186897139858070205940487, 2.98853270785849711968273767631, 3.21460863386365846848000245770, 3.21712183205321903662459202847, 3.24564600522360181624965570232, 3.27529128293486376786057378951, 3.42092707161964320477100172096, 3.60104385590852007104501371890, 3.76725069921921034594062073017
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.