Dirichlet series
| L(s) = 1 | − 3·2-s + 4-s + 8·5-s − 11·7-s + 10·8-s − 24·10-s − 6·11-s − 8·13-s + 33·14-s − 20·16-s + 2·17-s − 10·19-s + 8·20-s + 18·22-s − 6·23-s + 36·25-s + 24·26-s − 11·28-s − 14·29-s − 31·31-s + 2·32-s − 6·34-s − 88·35-s + 37-s + 30·38-s + 80·40-s + 12·41-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1/2·4-s + 3.57·5-s − 4.15·7-s + 3.53·8-s − 7.58·10-s − 1.80·11-s − 2.21·13-s + 8.81·14-s − 5·16-s + 0.485·17-s − 2.29·19-s + 1.78·20-s + 3.83·22-s − 1.25·23-s + 36/5·25-s + 4.70·26-s − 2.07·28-s − 2.59·29-s − 5.56·31-s + 0.353·32-s − 1.02·34-s − 14.8·35-s + 0.164·37-s + 4.86·38-s + 12.6·40-s + 1.87·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{32} \cdot 5^{8} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.75896\times 10^{12}\) |
| Root analytic conductor: | \(6.48392\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 3^{32} \cdot 5^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 - T )^{8} \) | |
| 13 | \( ( 1 + T )^{8} \) | |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} + 11 T^{3} + 15 T^{4} + 3 p^{2} T^{5} + 13 p T^{6} + 39 T^{7} + 81 T^{8} + 39 p T^{9} + 13 p^{3} T^{10} + 3 p^{5} T^{11} + 15 p^{4} T^{12} + 11 p^{5} T^{13} + p^{9} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 + 11 T + 80 T^{2} + 433 T^{3} + 285 p T^{4} + 7841 T^{5} + 27428 T^{6} + 84957 T^{7} + 237931 T^{8} + 84957 p T^{9} + 27428 p^{2} T^{10} + 7841 p^{3} T^{11} + 285 p^{5} T^{12} + 433 p^{5} T^{13} + 80 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 6 T + 50 T^{2} + 212 T^{3} + 1293 T^{4} + 4716 T^{5} + 22736 T^{6} + 71271 T^{7} + 291738 T^{8} + 71271 p T^{9} + 22736 p^{2} T^{10} + 4716 p^{3} T^{11} + 1293 p^{4} T^{12} + 212 p^{5} T^{13} + 50 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 2 T + 58 T^{2} - 165 T^{3} + 1668 T^{4} - 393 p T^{5} + 35766 T^{6} - 10139 p T^{7} + 655698 T^{8} - 10139 p^{2} T^{9} + 35766 p^{2} T^{10} - 393 p^{4} T^{11} + 1668 p^{4} T^{12} - 165 p^{5} T^{13} + 58 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 10 T + 115 T^{2} + 789 T^{3} + 5477 T^{4} + 29161 T^{5} + 155913 T^{6} + 705427 T^{7} + 3285644 T^{8} + 705427 p T^{9} + 155913 p^{2} T^{10} + 29161 p^{3} T^{11} + 5477 p^{4} T^{12} + 789 p^{5} T^{13} + 115 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 6 T + 105 T^{2} + 264 T^{3} + 3713 T^{4} - 1053 T^{5} + 80616 T^{6} - 210189 T^{7} + 1787385 T^{8} - 210189 p T^{9} + 80616 p^{2} T^{10} - 1053 p^{3} T^{11} + 3713 p^{4} T^{12} + 264 p^{5} T^{13} + 105 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 14 T + 237 T^{2} + 2416 T^{3} + 24507 T^{4} + 190305 T^{5} + 1429000 T^{6} + 8793883 T^{7} + 51805935 T^{8} + 8793883 p T^{9} + 1429000 p^{2} T^{10} + 190305 p^{3} T^{11} + 24507 p^{4} T^{12} + 2416 p^{5} T^{13} + 237 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + p T + 524 T^{2} + 5780 T^{3} + 45302 T^{4} + 248738 T^{5} + 874155 T^{6} + 1080594 T^{7} - 3448436 T^{8} + 1080594 p T^{9} + 874155 p^{2} T^{10} + 248738 p^{3} T^{11} + 45302 p^{4} T^{12} + 5780 p^{5} T^{13} + 524 p^{6} T^{14} + p^{8} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - T + 176 T^{2} - 470 T^{3} + 14414 T^{4} - 60011 T^{5} + 765222 T^{6} - 3845703 T^{7} + 31208770 T^{8} - 3845703 p T^{9} + 765222 p^{2} T^{10} - 60011 p^{3} T^{11} + 14414 p^{4} T^{12} - 470 p^{5} T^{13} + 176 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 12 T + 6 p T^{2} - 2004 T^{3} + 25289 T^{4} - 168366 T^{5} + 1701162 T^{6} - 9850491 T^{7} + 82712001 T^{8} - 9850491 p T^{9} + 1701162 p^{2} T^{10} - 168366 p^{3} T^{11} + 25289 p^{4} T^{12} - 2004 p^{5} T^{13} + 6 p^{7} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 15 T + 177 T^{2} - 1342 T^{3} + 12228 T^{4} - 97414 T^{5} + 821407 T^{6} - 5535612 T^{7} + 38829358 T^{8} - 5535612 p T^{9} + 821407 p^{2} T^{10} - 97414 p^{3} T^{11} + 12228 p^{4} T^{12} - 1342 p^{5} T^{13} + 177 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 18 T + 290 T^{2} - 2732 T^{3} + 26796 T^{4} - 205566 T^{5} + 1868756 T^{6} - 13855674 T^{7} + 108387837 T^{8} - 13855674 p T^{9} + 1868756 p^{2} T^{10} - 205566 p^{3} T^{11} + 26796 p^{4} T^{12} - 2732 p^{5} T^{13} + 290 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 2 T + 142 T^{2} + 708 T^{3} + 12986 T^{4} + 68564 T^{5} + 1017309 T^{6} + 4483381 T^{7} + 61490950 T^{8} + 4483381 p T^{9} + 1017309 p^{2} T^{10} + 68564 p^{3} T^{11} + 12986 p^{4} T^{12} + 708 p^{5} T^{13} + 142 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 24 T + 532 T^{2} + 7266 T^{3} + 94861 T^{4} + 946458 T^{5} + 9417766 T^{6} + 77238627 T^{7} + 647040058 T^{8} + 77238627 p T^{9} + 9417766 p^{2} T^{10} + 946458 p^{3} T^{11} + 94861 p^{4} T^{12} + 7266 p^{5} T^{13} + 532 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 9 T + 112 T^{2} + 597 T^{3} + 7864 T^{4} + 47187 T^{5} + 646369 T^{6} + 79281 p T^{7} + 53798023 T^{8} + 79281 p^{2} T^{9} + 646369 p^{2} T^{10} + 47187 p^{3} T^{11} + 7864 p^{4} T^{12} + 597 p^{5} T^{13} + 112 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 18 T + 190 T^{2} + 1659 T^{3} + 16234 T^{4} + 151020 T^{5} + 1366477 T^{6} + 10497138 T^{7} + 80129401 T^{8} + 10497138 p T^{9} + 1366477 p^{2} T^{10} + 151020 p^{3} T^{11} + 16234 p^{4} T^{12} + 1659 p^{5} T^{13} + 190 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 10 T + 357 T^{2} + 3055 T^{3} + 59273 T^{4} + 450587 T^{5} + 6317853 T^{6} + 43591239 T^{7} + 504141882 T^{8} + 43591239 p T^{9} + 6317853 p^{2} T^{10} + 450587 p^{3} T^{11} + 59273 p^{4} T^{12} + 3055 p^{5} T^{13} + 357 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 6 T + 284 T^{2} - 2319 T^{3} + 42202 T^{4} - 421371 T^{5} + 4380068 T^{6} - 46287561 T^{7} + 355062718 T^{8} - 46287561 p T^{9} + 4380068 p^{2} T^{10} - 421371 p^{3} T^{11} + 42202 p^{4} T^{12} - 2319 p^{5} T^{13} + 284 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 3 T + 134 T^{2} + 714 T^{3} + 23764 T^{4} + 92835 T^{5} + 1999280 T^{6} + 9627495 T^{7} + 204722980 T^{8} + 9627495 p T^{9} + 1999280 p^{2} T^{10} + 92835 p^{3} T^{11} + 23764 p^{4} T^{12} + 714 p^{5} T^{13} + 134 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 10 T + 549 T^{2} - 4721 T^{3} + 135483 T^{4} - 999924 T^{5} + 20087908 T^{6} - 126389984 T^{7} + 2000814123 T^{8} - 126389984 p T^{9} + 20087908 p^{2} T^{10} - 999924 p^{3} T^{11} + 135483 p^{4} T^{12} - 4721 p^{5} T^{13} + 549 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 13 T + 484 T^{2} - 5856 T^{3} + 116018 T^{4} - 1250692 T^{5} + 17794923 T^{6} - 165272834 T^{7} + 1887496867 T^{8} - 165272834 p T^{9} + 17794923 p^{2} T^{10} - 1250692 p^{3} T^{11} + 116018 p^{4} T^{12} - 5856 p^{5} T^{13} + 484 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 34 T + 938 T^{2} + 18959 T^{3} + 329889 T^{4} + 4890448 T^{5} + 64222982 T^{6} + 745880430 T^{7} + 7775456074 T^{8} + 745880430 p T^{9} + 64222982 p^{2} T^{10} + 4890448 p^{3} T^{11} + 329889 p^{4} T^{12} + 18959 p^{5} T^{13} + 938 p^{6} T^{14} + 34 p^{7} T^{15} + p^{8} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.75484933032842467641076180488, −3.58758661896784282361284287918, −3.50013205053082113496509137713, −3.49211576577292451680866562934, −3.34547425583030082332063418201, −3.10148321570925046583942350715, −2.99847867665541398806009586615, −2.87045625270784925757504346296, −2.80102490458673287187750956131, −2.68156309043291675796941682119, −2.66215417185522318205825271255, −2.50684981939273258538497717354, −2.28180131471235831962652773856, −2.27096504300616008440831624927, −2.12928168543405627247897213639, −2.12667362889965779960754371801, −2.05810384488673534611488940155, −1.89361027630178017955377991202, −1.73540558200051382415866947191, −1.37360681828738837835402271084, −1.33019418498566178546759213681, −1.26721484264216815446746383889, −1.23717982262840381434251560862, −1.01268224970995548806560796035, −0.994974127505992111266578984161, 0, 0, 0, 0, 0, 0, 0, 0, 0.994974127505992111266578984161, 1.01268224970995548806560796035, 1.23717982262840381434251560862, 1.26721484264216815446746383889, 1.33019418498566178546759213681, 1.37360681828738837835402271084, 1.73540558200051382415866947191, 1.89361027630178017955377991202, 2.05810384488673534611488940155, 2.12667362889965779960754371801, 2.12928168543405627247897213639, 2.27096504300616008440831624927, 2.28180131471235831962652773856, 2.50684981939273258538497717354, 2.66215417185522318205825271255, 2.68156309043291675796941682119, 2.80102490458673287187750956131, 2.87045625270784925757504346296, 2.99847867665541398806009586615, 3.10148321570925046583942350715, 3.34547425583030082332063418201, 3.49211576577292451680866562934, 3.50013205053082113496509137713, 3.58758661896784282361284287918, 3.75484933032842467641076180488
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.