Dirichlet series
| L(s) = 1 | − 106·3-s − 2.56e3·4-s + 1.13e3·5-s − 2.50e5·9-s + 9.54e4·11-s + 2.71e5·12-s − 1.20e5·15-s + 3.93e6·16-s − 2.91e6·20-s + 1.74e7·23-s − 4.89e7·25-s + 4.72e7·27-s − 9.10e7·31-s − 1.01e7·33-s + 6.41e8·36-s − 8.26e7·37-s − 2.44e8·44-s − 2.85e8·45-s + 3.52e8·47-s − 4.16e8·48-s + 1.22e9·49-s + 5.71e8·53-s + 1.08e8·55-s − 1.50e9·59-s + 3.08e8·60-s − 4.69e9·64-s + 3.14e9·67-s + ⋯ |
| L(s) = 1 | − 0.436·3-s − 5/2·4-s + 0.364·5-s − 4.24·9-s + 0.592·11-s + 1.09·12-s − 0.158·15-s + 15/4·16-s − 0.910·20-s + 2.71·23-s − 5.01·25-s + 3.29·27-s − 3.18·31-s − 0.258·33-s + 10.6·36-s − 1.19·37-s − 1.48·44-s − 1.54·45-s + 1.53·47-s − 1.63·48-s + 4.33·49-s + 1.36·53-s + 0.215·55-s − 2.11·59-s + 0.397·60-s − 4.37·64-s + 2.33·67-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s+5)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 11^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.84712\times 10^{11}\) |
| Root analytic conductor: | \(3.73869\) |
| Motivic weight: | \(10\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 11^{10} ,\ ( \ : [5]^{10} ),\ 1 )\) |
Particular Values
| \(L(\frac{11}{2})\) | \(\approx\) | \(0.1384021251\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1384021251\) |
| \(L(6)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{9} T^{2} )^{5} \) |
| 11 | \( 1 - 8674 p T - 33874521 p^{3} T^{2} + 14903567848 p^{5} T^{3} + 26218660180486 p^{7} T^{4} + 279912357933468 p^{10} T^{5} + 26218660180486 p^{17} T^{6} + 14903567848 p^{25} T^{7} - 33874521 p^{33} T^{8} - 8674 p^{41} T^{9} + p^{50} T^{10} \) | |
| good | 3 | \( ( 1 + 53 T + 43172 p T^{2} - 370885 p^{2} T^{3} + 130175599 p^{4} T^{4} - 1388710760 p^{5} T^{5} + 130175599 p^{14} T^{6} - 370885 p^{22} T^{7} + 43172 p^{31} T^{8} + 53 p^{40} T^{9} + p^{50} T^{10} )^{2} \) |
| 5 | \( ( 1 - 569 T + 4989912 p T^{2} + 3785354017 p T^{3} + 12085775211011 p^{2} T^{4} + 15564004550500176 p^{2} T^{5} + 12085775211011 p^{12} T^{6} + 3785354017 p^{21} T^{7} + 4989912 p^{31} T^{8} - 569 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 7 | \( 1 - 1225073506 T^{2} + 891927891503680845 T^{4} - \)\(93\!\cdots\!96\)\( p^{2} T^{6} + \)\(76\!\cdots\!30\)\( p^{4} T^{8} - \)\(49\!\cdots\!04\)\( p^{6} T^{10} + \)\(76\!\cdots\!30\)\( p^{24} T^{12} - \)\(93\!\cdots\!96\)\( p^{42} T^{14} + 891927891503680845 p^{60} T^{16} - 1225073506 p^{80} T^{18} + p^{100} T^{20} \) | |
| 13 | \( 1 - 60524553754 p T^{2} + \)\(30\!\cdots\!57\)\( T^{4} - \)\(80\!\cdots\!00\)\( T^{6} + \)\(15\!\cdots\!38\)\( T^{8} - \)\(23\!\cdots\!52\)\( T^{10} + \)\(15\!\cdots\!38\)\( p^{20} T^{12} - \)\(80\!\cdots\!00\)\( p^{40} T^{14} + \)\(30\!\cdots\!57\)\( p^{60} T^{16} - 60524553754 p^{81} T^{18} + p^{100} T^{20} \) | |
| 17 | \( 1 - 11023653357826 T^{2} + \)\(54\!\cdots\!37\)\( T^{4} - \)\(16\!\cdots\!08\)\( T^{6} + \)\(39\!\cdots\!62\)\( T^{8} - \)\(81\!\cdots\!04\)\( T^{10} + \)\(39\!\cdots\!62\)\( p^{20} T^{12} - \)\(16\!\cdots\!08\)\( p^{40} T^{14} + \)\(54\!\cdots\!37\)\( p^{60} T^{16} - 11023653357826 p^{80} T^{18} + p^{100} T^{20} \) | |
| 19 | \( 1 - 23293362935698 T^{2} + \)\(25\!\cdots\!17\)\( T^{4} - \)\(10\!\cdots\!60\)\( p T^{6} + \)\(16\!\cdots\!18\)\( T^{8} - \)\(11\!\cdots\!48\)\( T^{10} + \)\(16\!\cdots\!18\)\( p^{20} T^{12} - \)\(10\!\cdots\!60\)\( p^{41} T^{14} + \)\(25\!\cdots\!17\)\( p^{60} T^{16} - 23293362935698 p^{80} T^{18} + p^{100} T^{20} \) | |
| 23 | \( ( 1 - 8748419 T + 107415541769892 T^{2} - \)\(19\!\cdots\!37\)\( T^{3} + \)\(71\!\cdots\!19\)\( T^{4} + \)\(17\!\cdots\!80\)\( T^{5} + \)\(71\!\cdots\!19\)\( p^{10} T^{6} - \)\(19\!\cdots\!37\)\( p^{20} T^{7} + 107415541769892 p^{30} T^{8} - 8748419 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 29 | \( 1 - 2096773873053466 T^{2} + \)\(22\!\cdots\!45\)\( T^{4} - \)\(17\!\cdots\!32\)\( T^{6} + \)\(98\!\cdots\!34\)\( T^{8} - \)\(45\!\cdots\!12\)\( T^{10} + \)\(98\!\cdots\!34\)\( p^{20} T^{12} - \)\(17\!\cdots\!32\)\( p^{40} T^{14} + \)\(22\!\cdots\!45\)\( p^{60} T^{16} - 2096773873053466 p^{80} T^{18} + p^{100} T^{20} \) | |
| 31 | \( ( 1 + 45525485 T + 2551402080637044 T^{2} + \)\(54\!\cdots\!11\)\( T^{3} + \)\(20\!\cdots\!99\)\( T^{4} + \)\(33\!\cdots\!80\)\( T^{5} + \)\(20\!\cdots\!99\)\( p^{10} T^{6} + \)\(54\!\cdots\!11\)\( p^{20} T^{7} + 2551402080637044 p^{30} T^{8} + 45525485 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 37 | \( ( 1 + 41338487 T + 13547525047120152 T^{2} + \)\(13\!\cdots\!57\)\( p T^{3} + \)\(10\!\cdots\!59\)\( T^{4} + \)\(31\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!59\)\( p^{10} T^{6} + \)\(13\!\cdots\!57\)\( p^{21} T^{7} + 13547525047120152 p^{30} T^{8} + 41338487 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 41 | \( 1 - 41482220254917922 T^{2} + \)\(12\!\cdots\!85\)\( T^{4} - \)\(27\!\cdots\!32\)\( T^{6} + \)\(48\!\cdots\!94\)\( T^{8} - \)\(71\!\cdots\!40\)\( T^{10} + \)\(48\!\cdots\!94\)\( p^{20} T^{12} - \)\(27\!\cdots\!32\)\( p^{40} T^{14} + \)\(12\!\cdots\!85\)\( p^{60} T^{16} - 41482220254917922 p^{80} T^{18} + p^{100} T^{20} \) | |
| 43 | \( 1 - 13532254637523034 T^{2} - \)\(46\!\cdots\!55\)\( T^{4} - \)\(14\!\cdots\!08\)\( T^{6} + \)\(32\!\cdots\!94\)\( T^{8} - \)\(55\!\cdots\!88\)\( T^{10} + \)\(32\!\cdots\!94\)\( p^{20} T^{12} - \)\(14\!\cdots\!08\)\( p^{40} T^{14} - \)\(46\!\cdots\!55\)\( p^{60} T^{16} - 13532254637523034 p^{80} T^{18} + p^{100} T^{20} \) | |
| 47 | \( ( 1 - 176253998 T + 190238609571899757 T^{2} - \)\(35\!\cdots\!36\)\( T^{3} + \)\(17\!\cdots\!14\)\( T^{4} - \)\(27\!\cdots\!00\)\( T^{5} + \)\(17\!\cdots\!14\)\( p^{10} T^{6} - \)\(35\!\cdots\!36\)\( p^{20} T^{7} + 190238609571899757 p^{30} T^{8} - 176253998 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 53 | \( ( 1 - 285564938 T + 671961218036288613 T^{2} - \)\(15\!\cdots\!76\)\( T^{3} + \)\(20\!\cdots\!14\)\( T^{4} - \)\(36\!\cdots\!04\)\( T^{5} + \)\(20\!\cdots\!14\)\( p^{10} T^{6} - \)\(15\!\cdots\!76\)\( p^{20} T^{7} + 671961218036288613 p^{30} T^{8} - 285564938 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 59 | \( ( 1 + 754323805 T + 1753906201459505004 T^{2} + \)\(12\!\cdots\!95\)\( T^{3} + \)\(16\!\cdots\!91\)\( T^{4} + \)\(83\!\cdots\!00\)\( T^{5} + \)\(16\!\cdots\!91\)\( p^{10} T^{6} + \)\(12\!\cdots\!95\)\( p^{20} T^{7} + 1753906201459505004 p^{30} T^{8} + 754323805 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 61 | \( 1 - 881414775641099002 T^{2} + \)\(10\!\cdots\!89\)\( T^{4} - \)\(53\!\cdots\!36\)\( T^{6} + \)\(29\!\cdots\!26\)\( T^{8} - \)\(52\!\cdots\!36\)\( p^{2} T^{10} + \)\(29\!\cdots\!26\)\( p^{20} T^{12} - \)\(53\!\cdots\!36\)\( p^{40} T^{14} + \)\(10\!\cdots\!89\)\( p^{60} T^{16} - 881414775641099002 p^{80} T^{18} + p^{100} T^{20} \) | |
| 67 | \( ( 1 - 1573405891 T + 4880877240531504732 T^{2} - \)\(92\!\cdots\!65\)\( T^{3} + \)\(16\!\cdots\!47\)\( T^{4} - \)\(21\!\cdots\!92\)\( T^{5} + \)\(16\!\cdots\!47\)\( p^{10} T^{6} - \)\(92\!\cdots\!65\)\( p^{20} T^{7} + 4880877240531504732 p^{30} T^{8} - 1573405891 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 71 | \( ( 1 + 164288725 T + 12473504962270182804 T^{2} - \)\(25\!\cdots\!21\)\( T^{3} + \)\(69\!\cdots\!19\)\( T^{4} - \)\(45\!\cdots\!60\)\( T^{5} + \)\(69\!\cdots\!19\)\( p^{10} T^{6} - \)\(25\!\cdots\!21\)\( p^{20} T^{7} + 12473504962270182804 p^{30} T^{8} + 164288725 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 73 | \( 1 - 19624813630376072866 T^{2} + \)\(23\!\cdots\!13\)\( T^{4} - \)\(18\!\cdots\!68\)\( T^{6} + \)\(11\!\cdots\!86\)\( T^{8} - \)\(56\!\cdots\!32\)\( T^{10} + \)\(11\!\cdots\!86\)\( p^{20} T^{12} - \)\(18\!\cdots\!68\)\( p^{40} T^{14} + \)\(23\!\cdots\!13\)\( p^{60} T^{16} - 19624813630376072866 p^{80} T^{18} + p^{100} T^{20} \) | |
| 79 | \( 1 - 22063400817744304714 T^{2} + \)\(45\!\cdots\!77\)\( T^{4} - \)\(62\!\cdots\!72\)\( T^{6} + \)\(77\!\cdots\!42\)\( T^{8} - \)\(77\!\cdots\!36\)\( T^{10} + \)\(77\!\cdots\!42\)\( p^{20} T^{12} - \)\(62\!\cdots\!72\)\( p^{40} T^{14} + \)\(45\!\cdots\!77\)\( p^{60} T^{16} - 22063400817744304714 p^{80} T^{18} + p^{100} T^{20} \) | |
| 83 | \( 1 - 56392752955470960730 T^{2} + \)\(23\!\cdots\!89\)\( T^{4} - \)\(65\!\cdots\!92\)\( T^{6} + \)\(14\!\cdots\!98\)\( T^{8} - \)\(25\!\cdots\!48\)\( T^{10} + \)\(14\!\cdots\!98\)\( p^{20} T^{12} - \)\(65\!\cdots\!92\)\( p^{40} T^{14} + \)\(23\!\cdots\!89\)\( p^{60} T^{16} - 56392752955470960730 p^{80} T^{18} + p^{100} T^{20} \) | |
| 89 | \( ( 1 - 8895713489 T + \)\(10\!\cdots\!08\)\( T^{2} - \)\(76\!\cdots\!59\)\( T^{3} + \)\(56\!\cdots\!87\)\( T^{4} - \)\(32\!\cdots\!72\)\( T^{5} + \)\(56\!\cdots\!87\)\( p^{10} T^{6} - \)\(76\!\cdots\!59\)\( p^{20} T^{7} + \)\(10\!\cdots\!08\)\( p^{30} T^{8} - 8895713489 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 97 | \( ( 1 + 31292594807 T + \)\(69\!\cdots\!68\)\( T^{2} + \)\(10\!\cdots\!93\)\( T^{3} + \)\(12\!\cdots\!91\)\( T^{4} + \)\(11\!\cdots\!56\)\( T^{5} + \)\(12\!\cdots\!91\)\( p^{10} T^{6} + \)\(10\!\cdots\!93\)\( p^{20} T^{7} + \)\(69\!\cdots\!68\)\( p^{30} T^{8} + 31292594807 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.38802565148475463958502108482, −5.22114775167156014022976652109, −5.01036446915337421119697874038, −4.84801873287245121481283690498, −4.42316101614948242678299983303, −4.25674848715465604938625416160, −4.02013755611193576139783852299, −4.00897497038895651444915411344, −3.70237908704551677622059674113, −3.54773883347161527699557307170, −3.41688463706375517013764863627, −3.13315121202650833558531904784, −2.87487275791592997475970885051, −2.65988135307503269661226487668, −2.59785374822927017213180202839, −2.14806899682348860208750136842, −1.96150068556283932864200040003, −1.93596469503109269348027762775, −1.34496887384156749649676299234, −1.23726965852637209199040209409, −0.807209915395196950482653525426, −0.70087097956598029777254302448, −0.36920525618093745636560400249, −0.36092918562689343749649907528, −0.07866926166237412298020858725, 0.07866926166237412298020858725, 0.36092918562689343749649907528, 0.36920525618093745636560400249, 0.70087097956598029777254302448, 0.807209915395196950482653525426, 1.23726965852637209199040209409, 1.34496887384156749649676299234, 1.93596469503109269348027762775, 1.96150068556283932864200040003, 2.14806899682348860208750136842, 2.59785374822927017213180202839, 2.65988135307503269661226487668, 2.87487275791592997475970885051, 3.13315121202650833558531904784, 3.41688463706375517013764863627, 3.54773883347161527699557307170, 3.70237908704551677622059674113, 4.00897497038895651444915411344, 4.02013755611193576139783852299, 4.25674848715465604938625416160, 4.42316101614948242678299983303, 4.84801873287245121481283690498, 5.01036446915337421119697874038, 5.22114775167156014022976652109, 5.38802565148475463958502108482
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.