Dirichlet series
| L(s) = 1 | + 2·3-s − 5·5-s + 8·7-s + 9-s + 8·11-s − 10·15-s + 3·17-s + 5·19-s + 16·21-s − 7·23-s + 15·25-s − 3·29-s − 3·31-s + 16·33-s − 40·35-s − 37-s + 10·41-s − 12·43-s − 5·45-s − 33·47-s + 6·51-s − 19·53-s − 40·55-s + 10·57-s − 38·59-s + 46·61-s + 8·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2.23·5-s + 3.02·7-s + 1/3·9-s + 2.41·11-s − 2.58·15-s + 0.727·17-s + 1.14·19-s + 3.49·21-s − 1.45·23-s + 3·25-s − 0.557·29-s − 0.538·31-s + 2.78·33-s − 6.76·35-s − 0.164·37-s + 1.56·41-s − 1.82·43-s − 0.745·45-s − 4.81·47-s + 0.840·51-s − 2.60·53-s − 5.39·55-s + 1.32·57-s − 4.94·59-s + 5.88·61-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
| \(L(1)\) | \(\approx\) | \(1.483579401\) |
| \(L(\frac12)\) | \(\approx\) | \(1.483579401\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 + p T + 2 p T^{2} - p T^{3} - 9 p T^{4} - p^{2} T^{5} + 2 p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) | |
| good | 7 | \( ( 1 - 4 T + 24 T^{2} - 73 T^{3} + 239 T^{4} - 73 p T^{5} + 24 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 - 8 T + T^{2} + 138 T^{3} - 164 T^{4} - 2194 T^{5} + 8689 T^{6} + 6024 T^{7} - 103413 T^{8} + 6024 p T^{9} + 8689 p^{2} T^{10} - 2194 p^{3} T^{11} - 164 p^{4} T^{12} + 138 p^{5} T^{13} + p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 31 T^{2} - 70 T^{3} + 232 T^{4} + 150 p T^{5} + 509 p T^{6} - 14700 T^{7} - 162685 T^{8} - 14700 p T^{9} + 509 p^{3} T^{10} + 150 p^{4} T^{11} + 232 p^{4} T^{12} - 70 p^{5} T^{13} - 31 p^{6} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 - 3 T - 21 T^{2} - 18 T^{3} + 643 T^{4} - 378 T^{5} - 4104 T^{6} - 10869 T^{7} + 122923 T^{8} - 10869 p T^{9} - 4104 p^{2} T^{10} - 378 p^{3} T^{11} + 643 p^{4} T^{12} - 18 p^{5} T^{13} - 21 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 5 T - 33 T^{2} + 235 T^{3} + 103 T^{4} - 4430 T^{5} + 16014 T^{6} + 32750 T^{7} - 459395 T^{8} + 32750 p T^{9} + 16014 p^{2} T^{10} - 4430 p^{3} T^{11} + 103 p^{4} T^{12} + 235 p^{5} T^{13} - 33 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 7 T - 28 T^{2} - 429 T^{3} - 664 T^{4} + 13402 T^{5} + 69240 T^{6} - 148932 T^{7} - 2177471 T^{8} - 148932 p T^{9} + 69240 p^{2} T^{10} + 13402 p^{3} T^{11} - 664 p^{4} T^{12} - 429 p^{5} T^{13} - 28 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 3 T - 50 T^{2} - 240 T^{3} + 1410 T^{4} + 3699 T^{5} - 1502 p T^{6} - 22530 T^{7} + 1657235 T^{8} - 22530 p T^{9} - 1502 p^{3} T^{10} + 3699 p^{3} T^{11} + 1410 p^{4} T^{12} - 240 p^{5} T^{13} - 50 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 3 T - 4 T^{2} + 77 T^{3} - 374 T^{4} + 13734 T^{5} + 62294 T^{6} - 84654 T^{7} + 867377 T^{8} - 84654 p T^{9} + 62294 p^{2} T^{10} + 13734 p^{3} T^{11} - 374 p^{4} T^{12} + 77 p^{5} T^{13} - 4 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + T - 17 T^{2} + 2 T^{3} + 551 T^{4} + 4726 T^{5} + 19280 T^{6} - 23839 T^{7} + 189739 T^{8} - 23839 p T^{9} + 19280 p^{2} T^{10} + 4726 p^{3} T^{11} + 551 p^{4} T^{12} + 2 p^{5} T^{13} - 17 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 10 T + 73 T^{2} - 110 T^{3} - 1532 T^{4} + 16340 T^{5} - 35729 T^{6} - 499600 T^{7} + 4584375 T^{8} - 499600 p T^{9} - 35729 p^{2} T^{10} + 16340 p^{3} T^{11} - 1532 p^{4} T^{12} - 110 p^{5} T^{13} + 73 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 + 6 T + 153 T^{2} + 710 T^{3} + 9591 T^{4} + 710 p T^{5} + 153 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 + 33 T + 474 T^{2} + 3848 T^{3} + 16738 T^{4} - 19027 T^{5} - 1009424 T^{6} - 9858306 T^{7} - 71145257 T^{8} - 9858306 p T^{9} - 1009424 p^{2} T^{10} - 19027 p^{3} T^{11} + 16738 p^{4} T^{12} + 3848 p^{5} T^{13} + 474 p^{6} T^{14} + 33 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 19 T + 111 T^{2} + 446 T^{3} + 7253 T^{4} + 73084 T^{5} + 427084 T^{6} + 2113623 T^{7} + 11974763 T^{8} + 2113623 p T^{9} + 427084 p^{2} T^{10} + 73084 p^{3} T^{11} + 7253 p^{4} T^{12} + 446 p^{5} T^{13} + 111 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 38 T + 655 T^{2} + 7725 T^{3} + 84055 T^{4} + 903379 T^{5} + 8675467 T^{6} + 73018500 T^{7} + 571352775 T^{8} + 73018500 p T^{9} + 8675467 p^{2} T^{10} + 903379 p^{3} T^{11} + 84055 p^{4} T^{12} + 7725 p^{5} T^{13} + 655 p^{6} T^{14} + 38 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 46 T + 15 p T^{2} - 9670 T^{3} + 46450 T^{4} + 140252 T^{5} - 4106397 T^{6} + 37868110 T^{7} - 287746685 T^{8} + 37868110 p T^{9} - 4106397 p^{2} T^{10} + 140252 p^{3} T^{11} + 46450 p^{4} T^{12} - 9670 p^{5} T^{13} + 15 p^{7} T^{14} - 46 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 8 T + 49 T^{2} - 532 T^{3} - 1617 T^{4} + 50348 T^{5} + 422051 T^{6} + 783864 T^{7} - 37943752 T^{8} + 783864 p T^{9} + 422051 p^{2} T^{10} + 50348 p^{3} T^{11} - 1617 p^{4} T^{12} - 532 p^{5} T^{13} + 49 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 25 T + 133 T^{2} - 1800 T^{3} - 27677 T^{4} - 176200 T^{5} - 447794 T^{6} + 13292325 T^{7} + 208696005 T^{8} + 13292325 p T^{9} - 447794 p^{2} T^{10} - 176200 p^{3} T^{11} - 27677 p^{4} T^{12} - 1800 p^{5} T^{13} + 133 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 26 T + 121 T^{2} - 2546 T^{3} - 24607 T^{4} + 189686 T^{5} + 3345779 T^{6} - 2617118 T^{7} - 244133852 T^{8} - 2617118 p T^{9} + 3345779 p^{2} T^{10} + 189686 p^{3} T^{11} - 24607 p^{4} T^{12} - 2546 p^{5} T^{13} + 121 p^{6} T^{14} + 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 16 T - 41 T^{2} - 1274 T^{3} + 1086 T^{4} + 107038 T^{5} + 1563401 T^{6} + 712638 T^{7} - 136069393 T^{8} + 712638 p T^{9} + 1563401 p^{2} T^{10} + 107038 p^{3} T^{11} + 1086 p^{4} T^{12} - 1274 p^{5} T^{13} - 41 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 8 T - 183 T^{2} + 556 T^{3} + 20366 T^{4} + 43312 T^{5} - 1683575 T^{6} - 3727302 T^{7} + 129580619 T^{8} - 3727302 p T^{9} - 1683575 p^{2} T^{10} + 43312 p^{3} T^{11} + 20366 p^{4} T^{12} + 556 p^{5} T^{13} - 183 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 30 T + 447 T^{2} + 5790 T^{3} + 74838 T^{4} + 911880 T^{5} + 10426699 T^{6} + 104334750 T^{7} + 967648155 T^{8} + 104334750 p T^{9} + 10426699 p^{2} T^{10} + 911880 p^{3} T^{11} + 74838 p^{4} T^{12} + 5790 p^{5} T^{13} + 447 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 14 T - 107 T^{2} - 1102 T^{3} + 7296 T^{4} - 20716 T^{5} + 708335 T^{6} + 6619644 T^{7} - 118197841 T^{8} + 6619644 p T^{9} + 708335 p^{2} T^{10} - 20716 p^{3} T^{11} + 7296 p^{4} T^{12} - 1102 p^{5} T^{13} - 107 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−5.21430579520330894216592877292, −5.06965294413434562002964270325, −4.68992610641936173285053949771, −4.62882013492822122160086438804, −4.56682582474784768637955087125, −4.54513802939548934376194359505, −4.46789721949115305135216121691, −4.42214207991710728987637383563, −4.02781204414478963335257559368, −3.87789613103136790405302001379, −3.62112097583148634819482119417, −3.41697897100048037267751350460, −3.30966819492146361550953339878, −3.30127038855072042878500478438, −3.26521246908136153526298154657, −2.92355213689828465024673889288, −2.79268241098197412373312381121, −2.49924201694392635208321723206, −1.99150017254603040029405385034, −1.69663220115879902833391500232, −1.60832468618487252793367087971, −1.53963131806076570823056915314, −1.52068892563759896701500530207, −1.23843541769502200255114355943, −0.25077322556532707292336558227, 0.25077322556532707292336558227, 1.23843541769502200255114355943, 1.52068892563759896701500530207, 1.53963131806076570823056915314, 1.60832468618487252793367087971, 1.69663220115879902833391500232, 1.99150017254603040029405385034, 2.49924201694392635208321723206, 2.79268241098197412373312381121, 2.92355213689828465024673889288, 3.26521246908136153526298154657, 3.30127038855072042878500478438, 3.30966819492146361550953339878, 3.41697897100048037267751350460, 3.62112097583148634819482119417, 3.87789613103136790405302001379, 4.02781204414478963335257559368, 4.42214207991710728987637383563, 4.46789721949115305135216121691, 4.54513802939548934376194359505, 4.56682582474784768637955087125, 4.62882013492822122160086438804, 4.68992610641936173285053949771, 5.06965294413434562002964270325, 5.21430579520330894216592877292
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.