Dirichlet series
| L(s) = 1 | + 5·3-s + 2·5-s + 7-s + 13·9-s − 3·11-s − 2·13-s + 10·15-s − 13·17-s − 15·19-s + 5·21-s − 10·23-s + 25-s + 25·27-s − 9·29-s + 10·31-s − 15·33-s + 2·35-s + 24·37-s − 10·39-s + 8·41-s + 38·43-s + 26·45-s + 8·49-s − 65·51-s + 13·53-s − 6·55-s − 75·57-s + ⋯ |
| L(s) = 1 | + 2.88·3-s + 0.894·5-s + 0.377·7-s + 13/3·9-s − 0.904·11-s − 0.554·13-s + 2.58·15-s − 3.15·17-s − 3.44·19-s + 1.09·21-s − 2.08·23-s + 1/5·25-s + 4.81·27-s − 1.67·29-s + 1.79·31-s − 2.61·33-s + 0.338·35-s + 3.94·37-s − 1.60·39-s + 1.24·41-s + 5.79·43-s + 3.87·45-s + 8/7·49-s − 9.10·51-s + 1.78·53-s − 0.809·55-s − 9.93·57-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 5^{8} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.94399\times 10^{6}\) |
| Root analytic conductor: | \(2.65081\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 5^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(28.11899713\) |
| \(L(\frac12)\) | \(\approx\) | \(28.11899713\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) | |
| 11 | \( 1 + 3 T + 18 T^{2} - 9 T^{3} + 75 T^{4} - 9 p T^{5} + 18 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 3 | \( 1 - 5 T + 4 p T^{2} - 20 T^{3} + 10 p T^{4} - 5 p T^{5} - 26 p T^{6} + 250 T^{7} - 461 T^{8} + 250 p T^{9} - 26 p^{3} T^{10} - 5 p^{4} T^{11} + 10 p^{5} T^{12} - 20 p^{5} T^{13} + 4 p^{7} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - T - p T^{2} + 3 T^{3} - 19 T^{4} + 244 T^{5} + 60 p T^{6} - 1206 T^{7} - 1671 T^{8} - 1206 p T^{9} + 60 p^{3} T^{10} + 244 p^{3} T^{11} - 19 p^{4} T^{12} + 3 p^{5} T^{13} - p^{7} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 2 T - 5 T^{2} - 17 T^{3} - T^{4} - 379 T^{5} + 659 T^{6} + 226 T^{7} - 4079 T^{8} + 226 p T^{9} + 659 p^{2} T^{10} - 379 p^{3} T^{11} - p^{4} T^{12} - 17 p^{5} T^{13} - 5 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 13 T + 62 T^{2} + 211 T^{3} + 1328 T^{4} + 5444 T^{5} + 6114 T^{6} + 32568 T^{7} + 312583 T^{8} + 32568 p T^{9} + 6114 p^{2} T^{10} + 5444 p^{3} T^{11} + 1328 p^{4} T^{12} + 211 p^{5} T^{13} + 62 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 15 T + 97 T^{2} + 360 T^{3} + 673 T^{4} - 610 T^{5} - 686 T^{6} + 56575 T^{7} + 375305 T^{8} + 56575 p T^{9} - 686 p^{2} T^{10} - 610 p^{3} T^{11} + 673 p^{4} T^{12} + 360 p^{5} T^{13} + 97 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 + 5 T + 96 T^{2} + 335 T^{3} + 3347 T^{4} + 335 p T^{5} + 96 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 + 9 T + 41 T^{2} + 450 T^{3} + 4023 T^{4} + 21060 T^{5} + 127504 T^{6} + 821169 T^{7} + 4563203 T^{8} + 821169 p T^{9} + 127504 p^{2} T^{10} + 21060 p^{3} T^{11} + 4023 p^{4} T^{12} + 450 p^{5} T^{13} + 41 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 10 T + 63 T^{2} - 95 T^{3} - 1617 T^{4} + 16015 T^{5} - 48789 T^{6} - 115000 T^{7} + 1966355 T^{8} - 115000 p T^{9} - 48789 p^{2} T^{10} + 16015 p^{3} T^{11} - 1617 p^{4} T^{12} - 95 p^{5} T^{13} + 63 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 24 T + 285 T^{2} - 2504 T^{3} + 20454 T^{4} - 167192 T^{5} + 1266601 T^{6} - 8436018 T^{7} + 52041811 T^{8} - 8436018 p T^{9} + 1266601 p^{2} T^{10} - 167192 p^{3} T^{11} + 20454 p^{4} T^{12} - 2504 p^{5} T^{13} + 285 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 8 T + 11 T^{2} - 52 T^{3} + 2396 T^{4} - 11404 T^{5} - 17571 T^{6} - 488156 T^{7} + 6637407 T^{8} - 488156 p T^{9} - 17571 p^{2} T^{10} - 11404 p^{3} T^{11} + 2396 p^{4} T^{12} - 52 p^{5} T^{13} + 11 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 - 19 T + 293 T^{2} - 2740 T^{3} + 21711 T^{4} - 2740 p T^{5} + 293 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 - 71 T^{2} + 145 T^{3} + 3737 T^{4} - 7995 T^{5} - 136173 T^{6} + 2730 p T^{7} + 3965375 T^{8} + 2730 p^{2} T^{9} - 136173 p^{2} T^{10} - 7995 p^{3} T^{11} + 3737 p^{4} T^{12} + 145 p^{5} T^{13} - 71 p^{6} T^{14} + p^{8} T^{16} \) | |
| 53 | \( 1 - 13 T - 21 T^{2} + 1139 T^{3} - 4299 T^{4} - 68344 T^{5} + 622530 T^{6} + 1982270 T^{7} - 45891153 T^{8} + 1982270 p T^{9} + 622530 p^{2} T^{10} - 68344 p^{3} T^{11} - 4299 p^{4} T^{12} + 1139 p^{5} T^{13} - 21 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 27 T + 267 T^{2} - 1053 T^{3} + 1707 T^{4} - 10170 T^{5} - 256612 T^{6} + 7292916 T^{7} - 76795335 T^{8} + 7292916 p T^{9} - 256612 p^{2} T^{10} - 10170 p^{3} T^{11} + 1707 p^{4} T^{12} - 1053 p^{5} T^{13} + 267 p^{6} T^{14} - 27 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 6 T - 112 T^{2} + 1194 T^{3} + 2297 T^{4} - 63510 T^{5} + 270712 T^{6} + 1244412 T^{7} - 19660775 T^{8} + 1244412 p T^{9} + 270712 p^{2} T^{10} - 63510 p^{3} T^{11} + 2297 p^{4} T^{12} + 1194 p^{5} T^{13} - 112 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 - 19 T + 290 T^{2} - 2805 T^{3} + 25803 T^{4} - 2805 p T^{5} + 290 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 - 20 T + p T^{2} + 450 T^{3} + 5350 T^{4} - 64050 T^{5} - 426451 T^{6} + 1694530 T^{7} + 35725759 T^{8} + 1694530 p T^{9} - 426451 p^{2} T^{10} - 64050 p^{3} T^{11} + 5350 p^{4} T^{12} + 450 p^{5} T^{13} + p^{7} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 13 T - 90 T^{2} + 1953 T^{3} - 1346 T^{4} - 97114 T^{5} + 6288 p T^{6} + 1063856 T^{7} - 22325469 T^{8} + 1063856 p T^{9} + 6288 p^{3} T^{10} - 97114 p^{3} T^{11} - 1346 p^{4} T^{12} + 1953 p^{5} T^{13} - 90 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 37 T + 540 T^{2} + 3230 T^{3} - 14490 T^{4} - 396919 T^{5} - 1141518 T^{6} + 39718360 T^{7} + 582114095 T^{8} + 39718360 p T^{9} - 1141518 p^{2} T^{10} - 396919 p^{3} T^{11} - 14490 p^{4} T^{12} + 3230 p^{5} T^{13} + 540 p^{6} T^{14} + 37 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 27 T + 327 T^{2} + 4051 T^{3} + 60921 T^{4} + 736952 T^{5} + 7664840 T^{6} + 76449288 T^{7} + 718143749 T^{8} + 76449288 p T^{9} + 7664840 p^{2} T^{10} + 736952 p^{3} T^{11} + 60921 p^{4} T^{12} + 4051 p^{5} T^{13} + 327 p^{6} T^{14} + 27 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 8 T + 254 T^{2} + 1664 T^{3} + 31231 T^{4} + 1664 p T^{5} + 254 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 24 T + 555 T^{2} - 8744 T^{3} + 140724 T^{4} - 1774272 T^{5} + 22382321 T^{6} - 237375808 T^{7} + 2542677991 T^{8} - 237375808 p T^{9} + 22382321 p^{2} T^{10} - 1774272 p^{3} T^{11} + 140724 p^{4} T^{12} - 8744 p^{5} T^{13} + 555 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.29902455455232689039548042433, −4.12604934046040278654595024825, −4.11172224995221627346444682822, −4.10300153941495192293538363988, −4.03131581987928409548210357523, −3.76598887600749613638144355664, −3.72292530106244254073304403450, −3.27869797593987076232954429755, −3.25853473679431278766848164245, −3.03462462009339411885213971764, −2.74678421671360840190480098910, −2.68833115532689616752372706878, −2.50455307991654834827182182696, −2.47079342859900000874052449017, −2.27273727117260405468084850338, −2.26810184678963567269061836016, −2.18177667691331622909892891515, −2.17257794492365013591415070593, −2.10379146800534435647610295489, −1.65548405652745611923177857021, −1.57615467347272262414299914567, −0.930607650645638655614440456585, −0.78502487521828555497362423461, −0.63046740337158463986626242252, −0.52944999379856866112900339289, 0.52944999379856866112900339289, 0.63046740337158463986626242252, 0.78502487521828555497362423461, 0.930607650645638655614440456585, 1.57615467347272262414299914567, 1.65548405652745611923177857021, 2.10379146800534435647610295489, 2.17257794492365013591415070593, 2.18177667691331622909892891515, 2.26810184678963567269061836016, 2.27273727117260405468084850338, 2.47079342859900000874052449017, 2.50455307991654834827182182696, 2.68833115532689616752372706878, 2.74678421671360840190480098910, 3.03462462009339411885213971764, 3.25853473679431278766848164245, 3.27869797593987076232954429755, 3.72292530106244254073304403450, 3.76598887600749613638144355664, 4.03131581987928409548210357523, 4.10300153941495192293538363988, 4.11172224995221627346444682822, 4.12604934046040278654595024825, 4.29902455455232689039548042433
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.