Dirichlet series
| L(s) = 1 | + 5·2-s + 5·3-s + 12·4-s + 2·5-s + 25·6-s − 3·7-s + 20·8-s + 11·9-s + 10·10-s − 11-s + 60·12-s + 10·13-s − 15·14-s + 10·15-s + 30·16-s − 5·17-s + 55·18-s − 11·19-s + 24·20-s − 15·21-s − 5·22-s + 100·24-s + 25-s + 50·26-s + 5·27-s − 36·28-s − 5·29-s + ⋯ |
| L(s) = 1 | + 3.53·2-s + 2.88·3-s + 6·4-s + 0.894·5-s + 10.2·6-s − 1.13·7-s + 7.07·8-s + 11/3·9-s + 3.16·10-s − 0.301·11-s + 17.3·12-s + 2.77·13-s − 4.00·14-s + 2.58·15-s + 15/2·16-s − 1.21·17-s + 12.9·18-s − 2.52·19-s + 5.36·20-s − 3.27·21-s − 1.06·22-s + 20.4·24-s + 1/5·25-s + 9.80·26-s + 0.962·27-s − 6.80·28-s − 0.928·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 5^{8} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(90.6981\) |
| Root analytic conductor: | \(1.32540\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{16} \cdot 5^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(48.27286280\) |
| \(L(\frac12)\) | \(\approx\) | \(48.27286280\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - 5 T + 13 T^{2} - 25 T^{3} + 39 T^{4} - 25 p T^{5} + 13 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) | |
| 11 | \( 1 + T + 10 T^{2} - 31 T^{3} - 51 T^{4} - 31 p T^{5} + 10 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 3 | \( 1 - 5 T + 14 T^{2} - 20 T^{3} + 4 p T^{4} + 5 p T^{5} - 2 p^{2} T^{6} - 20 T^{7} + 115 T^{8} - 20 p T^{9} - 2 p^{4} T^{10} + 5 p^{4} T^{11} + 4 p^{5} T^{12} - 20 p^{5} T^{13} + 14 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 + 3 T + 9 T^{2} + 23 T^{3} + 33 T^{4} - 16 p T^{5} - 344 T^{6} - 1866 T^{7} - 6767 T^{8} - 1866 p T^{9} - 344 p^{2} T^{10} - 16 p^{4} T^{11} + 33 p^{4} T^{12} + 23 p^{5} T^{13} + 9 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 10 T + 63 T^{2} - 255 T^{3} + 985 T^{4} - 3725 T^{5} + 17943 T^{6} - 5880 p T^{7} + 309609 T^{8} - 5880 p^{2} T^{9} + 17943 p^{2} T^{10} - 3725 p^{3} T^{11} + 985 p^{4} T^{12} - 255 p^{5} T^{13} + 63 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 5 T + 22 T^{2} + 95 T^{3} + 630 T^{4} + 2570 T^{5} + 11552 T^{6} + 38480 T^{7} + 157739 T^{8} + 38480 p T^{9} + 11552 p^{2} T^{10} + 2570 p^{3} T^{11} + 630 p^{4} T^{12} + 95 p^{5} T^{13} + 22 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 11 T + p T^{2} - 104 T^{3} + 281 T^{4} + 4988 T^{5} + 16586 T^{6} + 34243 T^{7} + 100777 T^{8} + 34243 p T^{9} + 16586 p^{2} T^{10} + 4988 p^{3} T^{11} + 281 p^{4} T^{12} - 104 p^{5} T^{13} + p^{7} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 3 p T^{2} + 2652 T^{4} - 73647 T^{6} + 1794335 T^{8} - 73647 p^{2} T^{10} + 2652 p^{4} T^{12} - 3 p^{7} T^{14} + p^{8} T^{16} \) | |
| 29 | \( 1 + 5 T + 65 T^{2} + 390 T^{3} + 101 p T^{4} + 16620 T^{5} + 100910 T^{6} + 18895 p T^{7} + 94619 p T^{8} + 18895 p^{2} T^{9} + 100910 p^{2} T^{10} + 16620 p^{3} T^{11} + 101 p^{5} T^{12} + 390 p^{5} T^{13} + 65 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 10 T + 125 T^{2} - 1025 T^{3} + 7269 T^{4} - 47065 T^{5} + 8105 p T^{6} - 1412070 T^{7} + 7458211 T^{8} - 1412070 p T^{9} + 8105 p^{3} T^{10} - 47065 p^{3} T^{11} + 7269 p^{4} T^{12} - 1025 p^{5} T^{13} + 125 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 20 T + 121 T^{2} - 100 T^{3} + 422 T^{4} - 24080 T^{5} + 162413 T^{6} - 352450 T^{7} - 62645 T^{8} - 352450 p T^{9} + 162413 p^{2} T^{10} - 24080 p^{3} T^{11} + 422 p^{4} T^{12} - 100 p^{5} T^{13} + 121 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 10 T + 139 T^{2} - 790 T^{3} + 4350 T^{4} - 100 T^{5} - 182029 T^{6} + 1962350 T^{7} - 16190221 T^{8} + 1962350 p T^{9} - 182029 p^{2} T^{10} - 100 p^{3} T^{11} + 4350 p^{4} T^{12} - 790 p^{5} T^{13} + 139 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 + 17 T + 261 T^{2} + 2356 T^{3} + 18809 T^{4} + 2356 p T^{5} + 261 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 + 79 T^{2} - 615 T^{3} + 4647 T^{4} - 48585 T^{5} + 415847 T^{6} - 2255820 T^{7} + 23961215 T^{8} - 2255820 p T^{9} + 415847 p^{2} T^{10} - 48585 p^{3} T^{11} + 4647 p^{4} T^{12} - 615 p^{5} T^{13} + 79 p^{6} T^{14} + p^{8} T^{16} \) | |
| 53 | \( 1 + 7 T - 63 T^{2} - 699 T^{3} - 1039 T^{4} + 60732 T^{5} + 451460 T^{6} - 1646222 T^{7} - 29558181 T^{8} - 1646222 p T^{9} + 451460 p^{2} T^{10} + 60732 p^{3} T^{11} - 1039 p^{4} T^{12} - 699 p^{5} T^{13} - 63 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 5 T + 55 T^{2} + 1055 T^{3} + 7269 T^{4} + 59120 T^{5} + 675980 T^{6} + 4611020 T^{7} + 35189921 T^{8} + 4611020 p T^{9} + 675980 p^{2} T^{10} + 59120 p^{3} T^{11} + 7269 p^{4} T^{12} + 1055 p^{5} T^{13} + 55 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 10 T + 40 T^{2} - 970 T^{3} - 11951 T^{4} - 57310 T^{5} + 116720 T^{6} + 4738300 T^{7} + 39395361 T^{8} + 4738300 p T^{9} + 116720 p^{2} T^{10} - 57310 p^{3} T^{11} - 11951 p^{4} T^{12} - 970 p^{5} T^{13} + 40 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 349 T^{2} + 59052 T^{4} - 6537947 T^{6} + 515546375 T^{8} - 6537947 p^{2} T^{10} + 59052 p^{4} T^{12} - 349 p^{6} T^{14} + p^{8} T^{16} \) | |
| 71 | \( 1 - 90 T + 3925 T^{2} - 110820 T^{3} + 2283354 T^{4} - 36667110 T^{5} + 477857375 T^{6} - 5182638450 T^{7} + 47432387951 T^{8} - 5182638450 p T^{9} + 477857375 p^{2} T^{10} - 36667110 p^{3} T^{11} + 2283354 p^{4} T^{12} - 110820 p^{5} T^{13} + 3925 p^{6} T^{14} - 90 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 5 T + 248 T^{2} - 805 T^{3} + 20520 T^{4} - 4970 T^{5} + 425438 T^{6} + 6710670 T^{7} - 20435801 T^{8} + 6710670 p T^{9} + 425438 p^{2} T^{10} - 4970 p^{3} T^{11} + 20520 p^{4} T^{12} - 805 p^{5} T^{13} + 248 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 21 T + 184 T^{2} + 1056 T^{3} + 9486 T^{4} + 95943 T^{5} + 278336 T^{6} - 4558662 T^{7} - 59027833 T^{8} - 4558662 p T^{9} + 278336 p^{2} T^{10} + 95943 p^{3} T^{11} + 9486 p^{4} T^{12} + 1056 p^{5} T^{13} + 184 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 19 T - 49 T^{2} + 4489 T^{3} - 30697 T^{4} - 331754 T^{5} + 5033994 T^{6} + 7836182 T^{7} - 444554067 T^{8} + 7836182 p T^{9} + 5033994 p^{2} T^{10} - 331754 p^{3} T^{11} - 30697 p^{4} T^{12} + 4489 p^{5} T^{13} - 49 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 - 10 T + 236 T^{2} - 1540 T^{3} + 26461 T^{4} - 1540 p T^{5} + 236 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 + 44 T + 713 T^{2} + 4508 T^{3} - 6314 T^{4} - 300796 T^{5} - 1224575 T^{6} + 43495854 T^{7} + 786406539 T^{8} + 43495854 p T^{9} - 1224575 p^{2} T^{10} - 300796 p^{3} T^{11} - 6314 p^{4} T^{12} + 4508 p^{5} T^{13} + 713 p^{6} T^{14} + 44 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−5.61612435607151835289827886636, −5.47427156556099582481359172384, −5.22573931640927602183731235124, −4.96130055507239827965535109454, −4.84581392550468243132209204006, −4.66955891635857849735655446844, −4.61954457434242279951035309727, −4.48924300063501911734850840226, −4.19752633136315987590021497591, −3.96100825597302374629343378617, −3.83880409490315311350009556010, −3.73459298300824883293314798711, −3.69470324510167727779705716799, −3.65812977026697535832247971015, −3.37106599445265061565678770946, −3.24057716808216265642205423763, −2.85893416084571325850330162329, −2.74984404934146479115254744435, −2.53953284929822799054687769771, −2.47123309731214470634783270092, −2.26720114673310848632125787792, −1.94965721694096998715290383455, −1.68325905157608750826784759236, −1.62907234149835594352884690199, −0.964619208924802567703869705844, 0.964619208924802567703869705844, 1.62907234149835594352884690199, 1.68325905157608750826784759236, 1.94965721694096998715290383455, 2.26720114673310848632125787792, 2.47123309731214470634783270092, 2.53953284929822799054687769771, 2.74984404934146479115254744435, 2.85893416084571325850330162329, 3.24057716808216265642205423763, 3.37106599445265061565678770946, 3.65812977026697535832247971015, 3.69470324510167727779705716799, 3.73459298300824883293314798711, 3.83880409490315311350009556010, 3.96100825597302374629343378617, 4.19752633136315987590021497591, 4.48924300063501911734850840226, 4.61954457434242279951035309727, 4.66955891635857849735655446844, 4.84581392550468243132209204006, 4.96130055507239827965535109454, 5.22573931640927602183731235124, 5.47427156556099582481359172384, 5.61612435607151835289827886636
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.