Dirichlet series
| L(s) = 1 | + 2·2-s − 3·3-s + 4-s + 5·5-s − 6·6-s + 5·7-s + 8·9-s + 10·10-s − 3·12-s + 10·13-s + 10·14-s − 15·15-s + 15·17-s + 16·18-s + 10·19-s + 5·20-s − 15·21-s + 7·25-s + 20·26-s − 20·27-s + 5·28-s − 22·29-s − 30·30-s − 2·31-s − 2·32-s + 30·34-s + 25·35-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.73·3-s + 1/2·4-s + 2.23·5-s − 2.44·6-s + 1.88·7-s + 8/3·9-s + 3.16·10-s − 0.866·12-s + 2.77·13-s + 2.67·14-s − 3.87·15-s + 3.63·17-s + 3.77·18-s + 2.29·19-s + 1.11·20-s − 3.27·21-s + 7/5·25-s + 3.92·26-s − 3.84·27-s + 0.944·28-s − 4.08·29-s − 5.47·30-s − 0.359·31-s − 0.353·32-s + 5.14·34-s + 4.22·35-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.27558\times 10^{6}\) |
| Root analytic conductor: | \(2.40772\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 11^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(27.28423370\) |
| \(L(\frac12)\) | \(\approx\) | \(27.28423370\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 3 | \( 1 + p T + T^{2} - T^{3} + 4 T^{4} - p T^{5} + p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - p T + 18 T^{2} - 2 p^{2} T^{3} + 124 T^{4} - 11 p^{2} T^{5} + 527 T^{6} - 182 p T^{7} + 1596 T^{8} - 182 p^{2} T^{9} + 527 p^{2} T^{10} - 11 p^{5} T^{11} + 124 p^{4} T^{12} - 2 p^{7} T^{13} + 18 p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 5 T + 16 T^{2} - 50 T^{3} + 192 T^{4} - 605 T^{5} + 1783 T^{6} - 4440 T^{7} + 11440 T^{8} - 4440 p T^{9} + 1783 p^{2} T^{10} - 605 p^{3} T^{11} + 192 p^{4} T^{12} - 50 p^{5} T^{13} + 16 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 10 T + 58 T^{2} - 20 p T^{3} + 955 T^{4} - 230 p T^{5} + 6988 T^{6} - 12040 T^{7} + 25949 T^{8} - 12040 p T^{9} + 6988 p^{2} T^{10} - 230 p^{4} T^{11} + 955 p^{4} T^{12} - 20 p^{6} T^{13} + 58 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 15 T + 116 T^{2} - 610 T^{3} + 2342 T^{4} - 4085 T^{5} - 21877 T^{6} + 230250 T^{7} - 1158820 T^{8} + 230250 p T^{9} - 21877 p^{2} T^{10} - 4085 p^{3} T^{11} + 2342 p^{4} T^{12} - 610 p^{5} T^{13} + 116 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 10 T + 58 T^{2} - 200 T^{3} + 253 T^{4} + 1510 T^{5} - 15914 T^{6} + 110600 T^{7} - 526945 T^{8} + 110600 p T^{9} - 15914 p^{2} T^{10} + 1510 p^{3} T^{11} + 253 p^{4} T^{12} - 200 p^{5} T^{13} + 58 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 96 T^{2} + 5212 T^{4} - 191648 T^{6} + 5108870 T^{8} - 191648 p^{2} T^{10} + 5212 p^{4} T^{12} - 96 p^{6} T^{14} + p^{8} T^{16} \) | |
| 29 | \( 1 + 22 T + 200 T^{2} + 1130 T^{3} + 7565 T^{4} + 69406 T^{5} + 495452 T^{6} + 2393620 T^{7} + 10842845 T^{8} + 2393620 p T^{9} + 495452 p^{2} T^{10} + 69406 p^{3} T^{11} + 7565 p^{4} T^{12} + 1130 p^{5} T^{13} + 200 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 2 T - 64 T^{2} + 8 p T^{3} + 1511 T^{4} - 19634 T^{5} + 61614 T^{6} + 314644 T^{7} - 4714293 T^{8} + 314644 p T^{9} + 61614 p^{2} T^{10} - 19634 p^{3} T^{11} + 1511 p^{4} T^{12} + 8 p^{6} T^{13} - 64 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 6 T - 2 T^{2} + 18 T^{3} + 651 T^{4} - 438 p T^{5} + 81680 T^{6} - 129816 T^{7} + 1029569 T^{8} - 129816 p T^{9} + 81680 p^{2} T^{10} - 438 p^{4} T^{11} + 651 p^{4} T^{12} + 18 p^{5} T^{13} - 2 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 3 T + 16 T^{2} + 62 T^{3} + 3606 T^{4} + 3169 T^{5} - 52901 T^{6} - 124934 T^{7} + 4934612 T^{8} - 124934 p T^{9} - 52901 p^{2} T^{10} + 3169 p^{3} T^{11} + 3606 p^{4} T^{12} + 62 p^{5} T^{13} + 16 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 231 T^{2} + 27167 T^{4} - 2024653 T^{6} + 103981560 T^{8} - 2024653 p^{2} T^{10} + 27167 p^{4} T^{12} - 231 p^{6} T^{14} + p^{8} T^{16} \) | |
| 47 | \( 1 - 40 T + 18 p T^{2} - 12670 T^{3} + 150647 T^{4} - 1508060 T^{5} + 280584 p T^{6} - 103324220 T^{7} + 738745865 T^{8} - 103324220 p T^{9} + 280584 p^{3} T^{10} - 1508060 p^{3} T^{11} + 150647 p^{4} T^{12} - 12670 p^{5} T^{13} + 18 p^{7} T^{14} - 40 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 30 T + 476 T^{2} + 4830 T^{3} + 33057 T^{4} + 138510 T^{5} + 54028 T^{6} - 4846860 T^{7} - 48839635 T^{8} - 4846860 p T^{9} + 54028 p^{2} T^{10} + 138510 p^{3} T^{11} + 33057 p^{4} T^{12} + 4830 p^{5} T^{13} + 476 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 35 T + 760 T^{2} + 12690 T^{3} + 173064 T^{4} + 2021015 T^{5} + 20625295 T^{6} + 186841480 T^{7} + 1516880536 T^{8} + 186841480 p T^{9} + 20625295 p^{2} T^{10} + 2021015 p^{3} T^{11} + 173064 p^{4} T^{12} + 12690 p^{5} T^{13} + 760 p^{6} T^{14} + 35 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 20 T + 154 T^{2} - 830 T^{3} + 12335 T^{4} - 160200 T^{5} + 1250776 T^{6} - 8865500 T^{7} + 70089009 T^{8} - 8865500 p T^{9} + 1250776 p^{2} T^{10} - 160200 p^{3} T^{11} + 12335 p^{4} T^{12} - 830 p^{5} T^{13} + 154 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 + T + 119 T^{2} - 83 T^{3} + 10044 T^{4} - 83 p T^{5} + 119 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( ( 1 - 10 T + 151 T^{2} - 1970 T^{3} + 13641 T^{4} - 1970 p T^{5} + 151 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 73 | \( 1 - 5 T + 176 T^{2} - 980 T^{3} + 13952 T^{4} - 84235 T^{5} + 312183 T^{6} - 4512640 T^{7} - 10927800 T^{8} - 4512640 p T^{9} + 312183 p^{2} T^{10} - 84235 p^{3} T^{11} + 13952 p^{4} T^{12} - 980 p^{5} T^{13} + 176 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 25 T + 490 T^{2} - 6950 T^{3} + 82574 T^{4} - 826725 T^{5} + 7165235 T^{6} - 60780120 T^{7} + 511302616 T^{8} - 60780120 p T^{9} + 7165235 p^{2} T^{10} - 826725 p^{3} T^{11} + 82574 p^{4} T^{12} - 6950 p^{5} T^{13} + 490 p^{6} T^{14} - 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 9 T - 54 T^{2} - 306 T^{3} + 11238 T^{4} - 7689 T^{5} + 330539 T^{6} - 6229728 T^{7} + 24790368 T^{8} - 6229728 p T^{9} + 330539 p^{2} T^{10} - 7689 p^{3} T^{11} + 11238 p^{4} T^{12} - 306 p^{5} T^{13} - 54 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 199 T^{2} + 19855 T^{4} - 1662361 T^{6} + 149152624 T^{8} - 1662361 p^{2} T^{10} + 19855 p^{4} T^{12} - 199 p^{6} T^{14} + p^{8} T^{16} \) | |
| 97 | \( 1 + T - 272 T^{2} + 732 T^{3} + 19856 T^{4} - 189929 T^{5} + 2645495 T^{6} + 11592976 T^{7} - 547077416 T^{8} + 11592976 p T^{9} + 2645495 p^{2} T^{10} - 189929 p^{3} T^{11} + 19856 p^{4} T^{12} + 732 p^{5} T^{13} - 272 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.51564537682364533010728828021, −4.37322821490804266708286504982, −4.35013855626226470950799874414, −4.05716113868085360607153576728, −3.95188472549062780035469331228, −3.91697111390558801517815821000, −3.63643171769511166093709601609, −3.58303183405946853052658227067, −3.57489308240367398309874547402, −3.35789471127563020755786700708, −3.34250869786431232203447622968, −3.10508125555208738160361304535, −2.86700099680959891593024283901, −2.54408815967774206762180287657, −2.38080360920870854744590942729, −2.14334577360541694650910586395, −2.02073259052050004722895290592, −1.85517066879107661472738042212, −1.84456768872076685191609717538, −1.36899351103411708508267698025, −1.26228144921106264279953861637, −1.25869264197700598748715674409, −1.16559497313187596998660163683, −0.919058834027984726741559190089, −0.44797973157732636538599264887, 0.44797973157732636538599264887, 0.919058834027984726741559190089, 1.16559497313187596998660163683, 1.25869264197700598748715674409, 1.26228144921106264279953861637, 1.36899351103411708508267698025, 1.84456768872076685191609717538, 1.85517066879107661472738042212, 2.02073259052050004722895290592, 2.14334577360541694650910586395, 2.38080360920870854744590942729, 2.54408815967774206762180287657, 2.86700099680959891593024283901, 3.10508125555208738160361304535, 3.34250869786431232203447622968, 3.35789471127563020755786700708, 3.57489308240367398309874547402, 3.58303183405946853052658227067, 3.63643171769511166093709601609, 3.91697111390558801517815821000, 3.95188472549062780035469331228, 4.05716113868085360607153576728, 4.35013855626226470950799874414, 4.37322821490804266708286504982, 4.51564537682364533010728828021
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.