Dirichlet series
| L(s) = 1 | − 8·2-s − 4·3-s + 36·4-s + 2·5-s + 32·6-s + 2·7-s − 120·8-s − 4·9-s − 16·10-s + 10·11-s − 144·12-s − 16·14-s − 8·15-s + 330·16-s + 2·17-s + 32·18-s + 8·19-s + 72·20-s − 8·21-s − 80·22-s − 8·23-s + 480·24-s − 25·25-s + 34·27-s + 72·28-s + 8·29-s + 64·30-s + ⋯ |
| L(s) = 1 | − 5.65·2-s − 2.30·3-s + 18·4-s + 0.894·5-s + 13.0·6-s + 0.755·7-s − 42.4·8-s − 4/3·9-s − 5.05·10-s + 3.01·11-s − 41.5·12-s − 4.27·14-s − 2.06·15-s + 82.5·16-s + 0.485·17-s + 7.54·18-s + 1.83·19-s + 16.0·20-s − 1.74·21-s − 17.0·22-s − 1.66·23-s + 97.9·24-s − 5·25-s + 6.54·27-s + 13.6·28-s + 1.48·29-s + 11.6·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 13^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 13^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 13^{16} \cdot 19^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.78166\times 10^{13}\) |
| Root analytic conductor: | \(7.16100\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{8} \cdot 13^{16} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
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 | 2 | \( ( 1 + T )^{8} \) |
| 13 | \( 1 \) | |
| 19 | \( ( 1 - T )^{8} \) | |
| good | 3 | \( 1 + 4 T + 20 T^{2} + 62 T^{3} + 61 p T^{4} + 454 T^{5} + 1015 T^{6} + 682 p T^{7} + 3712 T^{8} + 682 p^{2} T^{9} + 1015 p^{2} T^{10} + 454 p^{3} T^{11} + 61 p^{5} T^{12} + 62 p^{5} T^{13} + 20 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 - 2 T + 29 T^{2} - 2 p^{2} T^{3} + 79 p T^{4} - 586 T^{5} + 3373 T^{6} - 4278 T^{7} + 19977 T^{8} - 4278 p T^{9} + 3373 p^{2} T^{10} - 586 p^{3} T^{11} + 79 p^{5} T^{12} - 2 p^{7} T^{13} + 29 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 - 2 T + 37 T^{2} - 60 T^{3} + 660 T^{4} - 948 T^{5} + 7656 T^{6} - 9770 T^{7} + 62956 T^{8} - 9770 p T^{9} + 7656 p^{2} T^{10} - 948 p^{3} T^{11} + 660 p^{4} T^{12} - 60 p^{5} T^{13} + 37 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 10 T + 106 T^{2} - 700 T^{3} + 4325 T^{4} - 21458 T^{5} + 96617 T^{6} - 378386 T^{7} + 1332028 T^{8} - 378386 p T^{9} + 96617 p^{2} T^{10} - 21458 p^{3} T^{11} + 4325 p^{4} T^{12} - 700 p^{5} T^{13} + 106 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 2 T + 77 T^{2} - 146 T^{3} + 2994 T^{4} - 5438 T^{5} + 79107 T^{6} - 132590 T^{7} + 1548154 T^{8} - 132590 p T^{9} + 79107 p^{2} T^{10} - 5438 p^{3} T^{11} + 2994 p^{4} T^{12} - 146 p^{5} T^{13} + 77 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 8 T + 162 T^{2} + 1160 T^{3} + 11989 T^{4} + 74760 T^{5} + 525575 T^{6} + 2770680 T^{7} + 14875490 T^{8} + 2770680 p T^{9} + 525575 p^{2} T^{10} + 74760 p^{3} T^{11} + 11989 p^{4} T^{12} + 1160 p^{5} T^{13} + 162 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 8 T + 157 T^{2} - 842 T^{3} + 9893 T^{4} - 35698 T^{5} + 358487 T^{6} - 907684 T^{7} + 10385737 T^{8} - 907684 p T^{9} + 358487 p^{2} T^{10} - 35698 p^{3} T^{11} + 9893 p^{4} T^{12} - 842 p^{5} T^{13} + 157 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 12 T + 219 T^{2} - 1832 T^{3} + 19780 T^{4} - 129422 T^{5} + 1055926 T^{6} - 5721514 T^{7} + 38605652 T^{8} - 5721514 p T^{9} + 1055926 p^{2} T^{10} - 129422 p^{3} T^{11} + 19780 p^{4} T^{12} - 1832 p^{5} T^{13} + 219 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 141 T^{2} + 196 T^{3} + 9902 T^{4} + 28372 T^{5} + 470299 T^{6} + 1865848 T^{7} + 497786 p T^{8} + 1865848 p T^{9} + 470299 p^{2} T^{10} + 28372 p^{3} T^{11} + 9902 p^{4} T^{12} + 196 p^{5} T^{13} + 141 p^{6} T^{14} + p^{8} T^{16} \) | |
| 41 | \( 1 + 2 T + 140 T^{2} + 270 T^{3} + 12623 T^{4} + 20902 T^{5} + 755977 T^{6} + 1115562 T^{7} + 35795383 T^{8} + 1115562 p T^{9} + 755977 p^{2} T^{10} + 20902 p^{3} T^{11} + 12623 p^{4} T^{12} + 270 p^{5} T^{13} + 140 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 16 T + 355 T^{2} + 3936 T^{3} + 51013 T^{4} + 437444 T^{5} + 4184730 T^{6} + 28999360 T^{7} + 220840874 T^{8} + 28999360 p T^{9} + 4184730 p^{2} T^{10} + 437444 p^{3} T^{11} + 51013 p^{4} T^{12} + 3936 p^{5} T^{13} + 355 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 12 T + 335 T^{2} + 3080 T^{3} + 48254 T^{4} + 356304 T^{5} + 4084834 T^{6} + 24948302 T^{7} + 230589774 T^{8} + 24948302 p T^{9} + 4084834 p^{2} T^{10} + 356304 p^{3} T^{11} + 48254 p^{4} T^{12} + 3080 p^{5} T^{13} + 335 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 24 T + 316 T^{2} + 2864 T^{3} + 23551 T^{4} + 183314 T^{5} + 1279061 T^{6} + 7262870 T^{7} + 45384983 T^{8} + 7262870 p T^{9} + 1279061 p^{2} T^{10} + 183314 p^{3} T^{11} + 23551 p^{4} T^{12} + 2864 p^{5} T^{13} + 316 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 4 T + 178 T^{2} - 152 T^{3} + 13695 T^{4} + 29440 T^{5} + 791841 T^{6} + 3067418 T^{7} + 47962712 T^{8} + 3067418 p T^{9} + 791841 p^{2} T^{10} + 29440 p^{3} T^{11} + 13695 p^{4} T^{12} - 152 p^{5} T^{13} + 178 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 26 T + 535 T^{2} + 7784 T^{3} + 92491 T^{4} + 924640 T^{5} + 8144949 T^{6} + 65618002 T^{7} + 520528261 T^{8} + 65618002 p T^{9} + 8144949 p^{2} T^{10} + 924640 p^{3} T^{11} + 92491 p^{4} T^{12} + 7784 p^{5} T^{13} + 535 p^{6} T^{14} + 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 10 T + 334 T^{2} + 2730 T^{3} + 53880 T^{4} + 369354 T^{5} + 5659554 T^{6} + 33594778 T^{7} + 435057454 T^{8} + 33594778 p T^{9} + 5659554 p^{2} T^{10} + 369354 p^{3} T^{11} + 53880 p^{4} T^{12} + 2730 p^{5} T^{13} + 334 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 36 T + 925 T^{2} - 17158 T^{3} + 263911 T^{4} - 3419374 T^{5} + 38711114 T^{6} - 386904034 T^{7} + 3448285040 T^{8} - 386904034 p T^{9} + 38711114 p^{2} T^{10} - 3419374 p^{3} T^{11} + 263911 p^{4} T^{12} - 17158 p^{5} T^{13} + 925 p^{6} T^{14} - 36 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 20 T + 424 T^{2} + 5400 T^{3} + 75558 T^{4} + 780156 T^{5} + 8948664 T^{6} + 1096748 p T^{7} + 774069247 T^{8} + 1096748 p^{2} T^{9} + 8948664 p^{2} T^{10} + 780156 p^{3} T^{11} + 75558 p^{4} T^{12} + 5400 p^{5} T^{13} + 424 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 22 T + 750 T^{2} + 11526 T^{3} + 221681 T^{4} + 2598778 T^{5} + 35787217 T^{6} + 332225248 T^{7} + 3549079388 T^{8} + 332225248 p T^{9} + 35787217 p^{2} T^{10} + 2598778 p^{3} T^{11} + 221681 p^{4} T^{12} + 11526 p^{5} T^{13} + 750 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 18 T + 523 T^{2} + 7294 T^{3} + 127848 T^{4} + 1454578 T^{5} + 19159590 T^{6} + 181024400 T^{7} + 1919997810 T^{8} + 181024400 p T^{9} + 19159590 p^{2} T^{10} + 1454578 p^{3} T^{11} + 127848 p^{4} T^{12} + 7294 p^{5} T^{13} + 523 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 18 T + 539 T^{2} - 7280 T^{3} + 121252 T^{4} - 1273794 T^{5} + 16066674 T^{6} - 140588680 T^{7} + 1573448600 T^{8} - 140588680 p T^{9} + 16066674 p^{2} T^{10} - 1273794 p^{3} T^{11} + 121252 p^{4} T^{12} - 7280 p^{5} T^{13} + 539 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 8 T + 468 T^{2} - 3272 T^{3} + 96708 T^{4} - 632696 T^{5} + 12469996 T^{6} - 80561080 T^{7} + 1276414198 T^{8} - 80561080 p T^{9} + 12469996 p^{2} T^{10} - 632696 p^{3} T^{11} + 96708 p^{4} T^{12} - 3272 p^{5} T^{13} + 468 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.67727271525461319723870025113, −3.27585520582534809588333660917, −3.24744585114637088031006385312, −3.22603230519642183445576226131, −3.17392888523664531214971396180, −3.02164441052423059993825556752, −2.96486873984333480067799076797, −2.95764599182345669338336407596, −2.75529549254233850840584992600, −2.40131959779618747696096449891, −2.30740487923882923907172766190, −2.26559845841867326813456084918, −2.24539422758976895782411909884, −2.20354174915350718424224165098, −2.04233893832482255958666034402, −1.81169864350088040938001411904, −1.68639566886076598261463424776, −1.53596162877104204813177364949, −1.36215000169410020924461658594, −1.35445357624630814529047050194, −1.31201034744111752246026967202, −1.17885855013737209228523979543, −1.09547527050503367007135879697, −1.00817903991399240560729964430, −0.991973080361210316215755650144, 0, 0, 0, 0, 0, 0, 0, 0, 0.991973080361210316215755650144, 1.00817903991399240560729964430, 1.09547527050503367007135879697, 1.17885855013737209228523979543, 1.31201034744111752246026967202, 1.35445357624630814529047050194, 1.36215000169410020924461658594, 1.53596162877104204813177364949, 1.68639566886076598261463424776, 1.81169864350088040938001411904, 2.04233893832482255958666034402, 2.20354174915350718424224165098, 2.24539422758976895782411909884, 2.26559845841867326813456084918, 2.30740487923882923907172766190, 2.40131959779618747696096449891, 2.75529549254233850840584992600, 2.95764599182345669338336407596, 2.96486873984333480067799076797, 3.02164441052423059993825556752, 3.17392888523664531214971396180, 3.22603230519642183445576226131, 3.24744585114637088031006385312, 3.27585520582534809588333660917, 3.67727271525461319723870025113
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.