Dirichlet series
| L(s) = 1 | + 3-s − 7-s + 3·9-s + 3·11-s + 2·13-s + 18·17-s − 8·19-s − 21-s + 3·23-s + 8·27-s − 9·29-s − 2·31-s + 3·33-s + 2·37-s + 2·39-s + 9·41-s + 8·43-s − 12·47-s + 10·49-s + 18·51-s + 24·53-s − 8·57-s − 15·59-s + 61-s − 3·63-s + 11·67-s + 3·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 9-s + 0.904·11-s + 0.554·13-s + 4.36·17-s − 1.83·19-s − 0.218·21-s + 0.625·23-s + 1.53·27-s − 1.67·29-s − 0.359·31-s + 0.522·33-s + 0.328·37-s + 0.320·39-s + 1.40·41-s + 1.21·43-s − 1.75·47-s + 10/7·49-s + 2.52·51-s + 3.29·53-s − 1.05·57-s − 1.95·59-s + 0.128·61-s − 0.377·63-s + 1.34·67-s + 0.361·69-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.11470\times 10^{6}\) |
| Root analytic conductor: | \(2.68077\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{16} \cdot 3^{16} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(17.77184462\) |
| \(L(\frac12)\) | \(\approx\) | \(17.77184462\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T - 2 T^{2} - p T^{3} + 5 p T^{4} - p^{2} T^{5} - 2 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + T - 9 T^{2} - 52 T^{3} + 26 T^{4} + 9 p^{2} T^{5} + 1294 T^{6} - 2174 T^{7} - 11493 T^{8} - 2174 p T^{9} + 1294 p^{2} T^{10} + 9 p^{5} T^{11} + 26 p^{4} T^{12} - 52 p^{5} T^{13} - 9 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 - 3 T - 20 T^{2} + 21 T^{3} + 310 T^{4} + 54 T^{5} - 2549 T^{6} + 6 T^{7} + 11641 T^{8} + 6 p T^{9} - 2549 p^{2} T^{10} + 54 p^{3} T^{11} + 310 p^{4} T^{12} + 21 p^{5} T^{13} - 20 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 2 T - 30 T^{2} + 86 T^{3} + 413 T^{4} - 1263 T^{5} - 4607 T^{6} + 6829 T^{7} + 63495 T^{8} + 6829 p T^{9} - 4607 p^{2} T^{10} - 1263 p^{3} T^{11} + 413 p^{4} T^{12} + 86 p^{5} T^{13} - 30 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( ( 1 - 9 T + 80 T^{2} - 441 T^{3} + 2115 T^{4} - 441 p T^{5} + 80 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 19 | \( ( 1 + 4 T + 49 T^{2} + 148 T^{3} + 1117 T^{4} + 148 p T^{5} + 49 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 23 | \( 1 - 3 T - 14 T^{2} - 141 T^{3} + 610 T^{4} + 1818 T^{5} + 27493 T^{6} - 110118 T^{7} - 369755 T^{8} - 110118 p T^{9} + 27493 p^{2} T^{10} + 1818 p^{3} T^{11} + 610 p^{4} T^{12} - 141 p^{5} T^{13} - 14 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 9 T + 25 T^{2} + 270 T^{3} + 1678 T^{4} + 2655 T^{5} + 54664 T^{6} + 310734 T^{7} + 373405 T^{8} + 310734 p T^{9} + 54664 p^{2} T^{10} + 2655 p^{3} T^{11} + 1678 p^{4} T^{12} + 270 p^{5} T^{13} + 25 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 2 T - 102 T^{2} - 86 T^{3} + 6263 T^{4} + 1713 T^{5} - 278855 T^{6} - 19339 T^{7} + 9684405 T^{8} - 19339 p T^{9} - 278855 p^{2} T^{10} + 1713 p^{3} T^{11} + 6263 p^{4} T^{12} - 86 p^{5} T^{13} - 102 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( ( 1 - T + 109 T^{2} - 88 T^{3} + 5425 T^{4} - 88 p T^{5} + 109 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 41 | \( 1 - 9 T - 38 T^{2} + 333 T^{3} + 1366 T^{4} + 4266 T^{5} - 136919 T^{6} - 117378 T^{7} + 6281365 T^{8} - 117378 p T^{9} - 136919 p^{2} T^{10} + 4266 p^{3} T^{11} + 1366 p^{4} T^{12} + 333 p^{5} T^{13} - 38 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 8 T - 54 T^{2} + 272 T^{3} + 2573 T^{4} + 5904 T^{5} - 168590 T^{6} - 155624 T^{7} + 7198236 T^{8} - 155624 p T^{9} - 168590 p^{2} T^{10} + 5904 p^{3} T^{11} + 2573 p^{4} T^{12} + 272 p^{5} T^{13} - 54 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 12 T + 7 T^{2} - 588 T^{3} - 2084 T^{4} + 26280 T^{5} + 199141 T^{6} - 1167900 T^{7} - 19309289 T^{8} - 1167900 p T^{9} + 199141 p^{2} T^{10} + 26280 p^{3} T^{11} - 2084 p^{4} T^{12} - 588 p^{5} T^{13} + 7 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( ( 1 - 12 T + 134 T^{2} - 1287 T^{3} + 11367 T^{4} - 1287 p T^{5} + 134 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 59 | \( 1 + 15 T + 103 T^{2} + 1776 T^{3} + 17116 T^{4} + 83997 T^{5} + 1147912 T^{6} + 8991024 T^{7} + 35402743 T^{8} + 8991024 p T^{9} + 1147912 p^{2} T^{10} + 83997 p^{3} T^{11} + 17116 p^{4} T^{12} + 1776 p^{5} T^{13} + 103 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - T - 168 T^{2} - 425 T^{3} + 15830 T^{4} + 55158 T^{5} - 891281 T^{6} - 1848982 T^{7} + 44687097 T^{8} - 1848982 p T^{9} - 891281 p^{2} T^{10} + 55158 p^{3} T^{11} + 15830 p^{4} T^{12} - 425 p^{5} T^{13} - 168 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 11 T - 63 T^{2} + 1202 T^{3} - 2422 T^{4} + 2907 T^{5} - 345500 T^{6} - 2520956 T^{7} + 69881643 T^{8} - 2520956 p T^{9} - 345500 p^{2} T^{10} + 2907 p^{3} T^{11} - 2422 p^{4} T^{12} + 1202 p^{5} T^{13} - 63 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( ( 1 + 12 T + 212 T^{2} + 1665 T^{3} + 19293 T^{4} + 1665 p T^{5} + 212 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 73 | \( ( 1 - 10 T + 280 T^{2} - 1897 T^{3} + 29707 T^{4} - 1897 p T^{5} + 280 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 79 | \( 1 - 7 T - 30 T^{2} + 13 p T^{3} - 14782 T^{4} + 92082 T^{5} - 641 p T^{6} - 117484 p T^{7} + 161146809 T^{8} - 117484 p^{2} T^{9} - 641 p^{3} T^{10} + 92082 p^{3} T^{11} - 14782 p^{4} T^{12} + 13 p^{6} T^{13} - 30 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 12 T - 92 T^{2} - 1758 T^{3} - 941 T^{4} + 33777 T^{5} - 634079 T^{6} + 3239787 T^{7} + 142998925 T^{8} + 3239787 p T^{9} - 634079 p^{2} T^{10} + 33777 p^{3} T^{11} - 941 p^{4} T^{12} - 1758 p^{5} T^{13} - 92 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 - 3 T + 113 T^{2} - 1314 T^{3} + 10185 T^{4} - 1314 p T^{5} + 113 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 5 T - 228 T^{2} + 311 T^{3} + 27794 T^{4} + 46950 T^{5} - 3165857 T^{6} - 3221834 T^{7} + 336287745 T^{8} - 3221834 p T^{9} - 3165857 p^{2} T^{10} + 46950 p^{3} T^{11} + 27794 p^{4} T^{12} + 311 p^{5} T^{13} - 228 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.26351004061246734984516989670, −4.17927527662628418212093490259, −4.12500489802437272922322327416, −4.02024352939577709614007182210, −3.77081684847438041048593449940, −3.53319262511845990786755417986, −3.50245550944784135073818288144, −3.44360227988192995022981670120, −3.33047409404700272048681138131, −3.23797259287458233184581961090, −3.08140494910483944347787991956, −2.83967304169391600802534426431, −2.74649974093181215221891046517, −2.46699257661948749758894440323, −2.37925802480055375802088135283, −2.16058408346592217329412982557, −1.98040683677355152970339884005, −1.92070818782146938998829385616, −1.67840495869674384509511319092, −1.45372608014731236649718346941, −1.21619461015966694845110778101, −1.09526816492604661025867385443, −0.808549741574295044698852557591, −0.69208547089590597002966602674, −0.49180132686209533017322178753, 0.49180132686209533017322178753, 0.69208547089590597002966602674, 0.808549741574295044698852557591, 1.09526816492604661025867385443, 1.21619461015966694845110778101, 1.45372608014731236649718346941, 1.67840495869674384509511319092, 1.92070818782146938998829385616, 1.98040683677355152970339884005, 2.16058408346592217329412982557, 2.37925802480055375802088135283, 2.46699257661948749758894440323, 2.74649974093181215221891046517, 2.83967304169391600802534426431, 3.08140494910483944347787991956, 3.23797259287458233184581961090, 3.33047409404700272048681138131, 3.44360227988192995022981670120, 3.50245550944784135073818288144, 3.53319262511845990786755417986, 3.77081684847438041048593449940, 4.02024352939577709614007182210, 4.12500489802437272922322327416, 4.17927527662628418212093490259, 4.26351004061246734984516989670
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.