Dirichlet series
| L(s) = 1 | − 2·3-s + 5·5-s − 8·7-s + 9-s − 2·11-s − 10·15-s + 7·17-s + 5·19-s + 16·21-s + 7·23-s + 15·25-s + 27·29-s − 3·31-s + 4·33-s − 40·35-s − 9·37-s + 20·41-s − 68·43-s + 5·45-s − 7·47-s − 14·51-s − 11·53-s − 10·55-s − 10·57-s + 2·59-s − 14·61-s − 8·63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 2.23·5-s − 3.02·7-s + 1/3·9-s − 0.603·11-s − 2.58·15-s + 1.69·17-s + 1.14·19-s + 3.49·21-s + 1.45·23-s + 3·25-s + 5.01·29-s − 0.538·31-s + 0.696·33-s − 6.76·35-s − 1.47·37-s + 3.12·41-s − 10.3·43-s + 0.745·45-s − 1.02·47-s − 1.96·51-s − 1.51·53-s − 1.34·55-s − 1.32·57-s + 0.260·59-s − 1.79·61-s − 1.00·63-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 3^{8} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1084.39\) |
| Root analytic conductor: | \(1.54774\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{16} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.166521062\) |
| \(L(\frac12)\) | \(\approx\) | \(1.166521062\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) | |
| 5 | \( 1 - p T + 2 p T^{2} - p^{2} T^{3} + 3 p^{2} T^{4} - p^{3} T^{5} + 2 p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) | |
| good | 7 | \( ( 1 + 4 T + 24 T^{2} + 83 T^{3} + 239 T^{4} + 83 p T^{5} + 24 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 + 2 T - 19 T^{2} - 12 T^{3} + 26 p T^{4} + 186 T^{5} - 2221 T^{6} + 74 T^{7} + 23167 T^{8} + 74 p T^{9} - 2221 p^{2} T^{10} + 186 p^{3} T^{11} + 26 p^{5} T^{12} - 12 p^{5} T^{13} - 19 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 9 T^{2} + 50 T^{3} + 24 p T^{4} + 950 T^{5} + 2137 T^{6} + 21300 T^{7} + 49555 T^{8} + 21300 p T^{9} + 2137 p^{2} T^{10} + 950 p^{3} T^{11} + 24 p^{5} T^{12} + 50 p^{5} T^{13} + 9 p^{6} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 - 7 T - T^{2} + 138 T^{3} - 347 T^{4} - 2532 T^{5} + 13916 T^{6} + 24899 T^{7} - 325277 T^{8} + 24899 p T^{9} + 13916 p^{2} T^{10} - 2532 p^{3} T^{11} - 347 p^{4} T^{12} + 138 p^{5} T^{13} - p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 5 T + 7 T^{2} - 25 T^{3} + 17 p T^{4} - 2930 T^{5} + 9374 T^{6} - 12250 T^{7} + 79165 T^{8} - 12250 p T^{9} + 9374 p^{2} T^{10} - 2930 p^{3} T^{11} + 17 p^{5} T^{12} - 25 p^{5} T^{13} + 7 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 7 T + 12 T^{2} - 21 T^{3} + 416 T^{4} - 5712 T^{5} + 25820 T^{6} - 56278 T^{7} + 245589 T^{8} - 56278 p T^{9} + 25820 p^{2} T^{10} - 5712 p^{3} T^{11} + 416 p^{4} T^{12} - 21 p^{5} T^{13} + 12 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 27 T + 370 T^{2} - 3330 T^{3} + 20130 T^{4} - 65421 T^{5} - 141208 T^{6} + 3507120 T^{7} - 25208005 T^{8} + 3507120 p T^{9} - 141208 p^{2} T^{10} - 65421 p^{3} T^{11} + 20130 p^{4} T^{12} - 3330 p^{5} T^{13} + 370 p^{6} T^{14} - 27 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 3 T - 44 T^{2} + 177 T^{3} + 1806 T^{4} - 3786 T^{5} + 33794 T^{6} + 7146 p T^{7} - 23953 p T^{8} + 7146 p^{2} T^{9} + 33794 p^{2} T^{10} - 3786 p^{3} T^{11} + 1806 p^{4} T^{12} + 177 p^{5} T^{13} - 44 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 9 T + 3 T^{2} - 62 T^{3} + 1311 T^{4} + 1954 T^{5} - 42200 T^{6} - 255951 T^{7} - 1117421 T^{8} - 255951 p T^{9} - 42200 p^{2} T^{10} + 1954 p^{3} T^{11} + 1311 p^{4} T^{12} - 62 p^{5} T^{13} + 3 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 20 T + 3 p T^{2} - 300 T^{3} + 4268 T^{4} - 69120 T^{5} + 547721 T^{6} - 2200400 T^{7} + 6713175 T^{8} - 2200400 p T^{9} + 547721 p^{2} T^{10} - 69120 p^{3} T^{11} + 4268 p^{4} T^{12} - 300 p^{5} T^{13} + 3 p^{7} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 + 34 T + 573 T^{2} + 6230 T^{3} + 47951 T^{4} + 6230 p T^{5} + 573 p^{2} T^{6} + 34 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 + 7 T - 6 T^{2} - 198 T^{3} - 2362 T^{4} - 26403 T^{5} + 34526 T^{6} + 818356 T^{7} + 3049983 T^{8} + 818356 p T^{9} + 34526 p^{2} T^{10} - 26403 p^{3} T^{11} - 2362 p^{4} T^{12} - 198 p^{5} T^{13} - 6 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 11 T + 11 T^{2} - 66 T^{3} + 4903 T^{4} + 8286 T^{5} - 453856 T^{6} - 2044573 T^{7} + 4229323 T^{8} - 2044573 p T^{9} - 453856 p^{2} T^{10} + 8286 p^{3} T^{11} + 4903 p^{4} T^{12} - 66 p^{5} T^{13} + 11 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 2 T + 65 T^{2} + 15 T^{3} + 6025 T^{4} + 7959 T^{5} + 334337 T^{6} + 389020 T^{7} + 32161615 T^{8} + 389020 p T^{9} + 334337 p^{2} T^{10} + 7959 p^{3} T^{11} + 6025 p^{4} T^{12} + 15 p^{5} T^{13} + 65 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 14 T + 115 T^{2} + 670 T^{3} + 8030 T^{4} - 1228 T^{5} - 627577 T^{6} - 7156070 T^{7} - 38667965 T^{8} - 7156070 p T^{9} - 627577 p^{2} T^{10} - 1228 p^{3} T^{11} + 8030 p^{4} T^{12} + 670 p^{5} T^{13} + 115 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 28 T + 289 T^{2} - 1808 T^{3} + 21683 T^{4} - 358408 T^{5} + 3866051 T^{6} - 24777764 T^{7} + 142623568 T^{8} - 24777764 p T^{9} + 3866051 p^{2} T^{10} - 358408 p^{3} T^{11} + 21683 p^{4} T^{12} - 1808 p^{5} T^{13} + 289 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 15 T + 133 T^{2} + 1980 T^{3} + 24873 T^{4} + 209490 T^{5} + 2093006 T^{6} + 19987875 T^{7} + 165378005 T^{8} + 19987875 p T^{9} + 2093006 p^{2} T^{10} + 209490 p^{3} T^{11} + 24873 p^{4} T^{12} + 1980 p^{5} T^{13} + 133 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 6 T - 239 T^{2} + 2646 T^{3} + 18873 T^{4} - 347586 T^{5} + 527099 T^{6} + 14424498 T^{7} - 134795212 T^{8} + 14424498 p T^{9} + 527099 p^{2} T^{10} - 347586 p^{3} T^{11} + 18873 p^{4} T^{12} + 2646 p^{5} T^{13} - 239 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 24 T + 319 T^{2} - 2754 T^{3} + 18126 T^{4} - 402 T^{5} - 1818079 T^{6} + 27900558 T^{7} - 270706753 T^{8} + 27900558 p T^{9} - 1818079 p^{2} T^{10} - 402 p^{3} T^{11} + 18126 p^{4} T^{12} - 2754 p^{5} T^{13} + 319 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 2 T - 133 T^{2} + 1134 T^{3} + 12286 T^{4} - 24972 T^{5} - 900625 T^{6} + 1006102 T^{7} + 140255179 T^{8} + 1006102 p T^{9} - 900625 p^{2} T^{10} - 24972 p^{3} T^{11} + 12286 p^{4} T^{12} + 1134 p^{5} T^{13} - 133 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 73 T^{2} - 480 T^{3} + 11868 T^{4} + 18000 T^{5} - 927491 T^{6} + 418500 T^{7} + 125679815 T^{8} + 418500 p T^{9} - 927491 p^{2} T^{10} + 18000 p^{3} T^{11} + 11868 p^{4} T^{12} - 480 p^{5} T^{13} - 73 p^{6} T^{14} + p^{8} T^{16} \) | |
| 97 | \( 1 - 34 T + 373 T^{2} + 322 T^{3} - 45904 T^{4} + 447956 T^{5} - 882625 T^{6} - 28402484 T^{7} + 432272239 T^{8} - 28402484 p T^{9} - 882625 p^{2} T^{10} + 447956 p^{3} T^{11} - 45904 p^{4} T^{12} + 322 p^{5} T^{13} + 373 p^{6} T^{14} - 34 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−5.32080992557715849671829713460, −5.05498044443543253267349679617, −4.92604053570603960784421293138, −4.87526299176482912353337160312, −4.83003744634610638714514718882, −4.80952024617629072322721104249, −4.55211796492519213775239744477, −4.51601674762235185969225233901, −3.87718474866200714560064479084, −3.53064220823209389421307818106, −3.48931671844367102238583722718, −3.48591427016371792279880665852, −3.40496730084144707608362494979, −3.12263568842217983236342604030, −3.01848874051742557511064495197, −2.87989761038925967401245941719, −2.83650825450134602968918587389, −2.67694257915881551962844570464, −1.95811561132110517878834212113, −1.94752981332501780926111968456, −1.79397027610451142920268268071, −1.46395762947731839080065056928, −1.23198642295838561295038337509, −0.78480386283946699059256817189, −0.35959520115248816312758783364, 0.35959520115248816312758783364, 0.78480386283946699059256817189, 1.23198642295838561295038337509, 1.46395762947731839080065056928, 1.79397027610451142920268268071, 1.94752981332501780926111968456, 1.95811561132110517878834212113, 2.67694257915881551962844570464, 2.83650825450134602968918587389, 2.87989761038925967401245941719, 3.01848874051742557511064495197, 3.12263568842217983236342604030, 3.40496730084144707608362494979, 3.48591427016371792279880665852, 3.48931671844367102238583722718, 3.53064220823209389421307818106, 3.87718474866200714560064479084, 4.51601674762235185969225233901, 4.55211796492519213775239744477, 4.80952024617629072322721104249, 4.83003744634610638714514718882, 4.87526299176482912353337160312, 4.92604053570603960784421293138, 5.05498044443543253267349679617, 5.32080992557715849671829713460
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.