Dirichlet series
| L(s) = 1 | − 2·2-s + 2·3-s + 4-s − 5·5-s − 4·6-s + 5·7-s + 3·9-s + 10·10-s + 2·12-s + 10·13-s − 10·14-s − 10·15-s − 15·17-s − 6·18-s + 10·19-s − 5·20-s + 10·21-s + 7·25-s − 20·26-s + 10·27-s + 5·28-s + 22·29-s + 20·30-s − 2·31-s + 2·32-s + 30·34-s − 25·35-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s − 2.23·5-s − 1.63·6-s + 1.88·7-s + 9-s + 3.16·10-s + 0.577·12-s + 2.77·13-s − 2.67·14-s − 2.58·15-s − 3.63·17-s − 1.41·18-s + 2.29·19-s − 1.11·20-s + 2.18·21-s + 7/5·25-s − 3.92·26-s + 1.92·27-s + 0.944·28-s + 4.08·29-s + 3.65·30-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
| \(L(1)\) | \(\approx\) | \(3.131255766\) |
| \(L(\frac12)\) | \(\approx\) | \(3.131255766\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 - 2 T + T^{2} - 2 p T^{3} + 19 T^{4} - 2 p^{2} T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | |
| 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
−4.45437580760826660987482589481, −4.22435352401668084952294391724, −4.10992529895201145675049859818, −3.94840697321950893892982248319, −3.94816596040186669011376492426, −3.90616767686351147954249689700, −3.85338130965640959213445772727, −3.76776124127801019377587167437, −3.32684294363012644326333309437, −3.24778067543725685231298806686, −3.02470156702244872795848902261, −2.92872910000468661286311452396, −2.76234005076677528716652771761, −2.58649163417704968815280346072, −2.52449447742441296182318311011, −2.25928546005967512443504872301, −2.21298597688393118052777787431, −1.93598495650111921370844833406, −1.46825837769477478050911618811, −1.41948392840268191502165653367, −1.33169728091595032092236977261, −1.15671363148599607377134854653, −0.890166059390733697026589040515, −0.53642421743452471320128486534, −0.40784543359820088599865920917, 0.40784543359820088599865920917, 0.53642421743452471320128486534, 0.890166059390733697026589040515, 1.15671363148599607377134854653, 1.33169728091595032092236977261, 1.41948392840268191502165653367, 1.46825837769477478050911618811, 1.93598495650111921370844833406, 2.21298597688393118052777787431, 2.25928546005967512443504872301, 2.52449447742441296182318311011, 2.58649163417704968815280346072, 2.76234005076677528716652771761, 2.92872910000468661286311452396, 3.02470156702244872795848902261, 3.24778067543725685231298806686, 3.32684294363012644326333309437, 3.76776124127801019377587167437, 3.85338130965640959213445772727, 3.90616767686351147954249689700, 3.94816596040186669011376492426, 3.94840697321950893892982248319, 4.10992529895201145675049859818, 4.22435352401668084952294391724, 4.45437580760826660987482589481
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.