Dirichlet series
| L(s) = 1 | + 8·2-s − 5·3-s + 36·4-s + 5-s − 40·6-s − 4·7-s + 120·8-s + 8·9-s + 8·10-s − 8·11-s − 180·12-s − 3·13-s − 32·14-s − 5·15-s + 330·16-s − 6·17-s + 64·18-s + 36·20-s + 20·21-s − 64·22-s + 9·23-s − 600·24-s − 19·25-s − 24·26-s + 2·27-s − 144·28-s − 4·29-s + ⋯ |
| L(s) = 1 | + 5.65·2-s − 2.88·3-s + 18·4-s + 0.447·5-s − 16.3·6-s − 1.51·7-s + 42.4·8-s + 8/3·9-s + 2.52·10-s − 2.41·11-s − 51.9·12-s − 0.832·13-s − 8.55·14-s − 1.29·15-s + 82.5·16-s − 1.45·17-s + 15.0·18-s + 8.04·20-s + 4.36·21-s − 13.6·22-s + 1.87·23-s − 122.·24-s − 3.79·25-s − 4.70·26-s + 0.384·27-s − 27.2·28-s − 0.742·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{8} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{8} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 11^{8} \cdot 19^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.61610\times 10^{14}\) |
| Root analytic conductor: | \(7.96349\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{8} \cdot 11^{8} \cdot 19^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{8} \) |
| 11 | \( ( 1 + T )^{8} \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + 5 T + 17 T^{2} + 43 T^{3} + 10 p^{2} T^{4} + 182 T^{5} + 14 p^{3} T^{6} + 760 T^{7} + 1420 T^{8} + 760 p T^{9} + 14 p^{5} T^{10} + 182 p^{3} T^{11} + 10 p^{6} T^{12} + 43 p^{5} T^{13} + 17 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 - T + 4 p T^{2} - 12 T^{3} + 174 T^{4} - 56 T^{5} + 953 T^{6} - 141 T^{7} + 4561 T^{8} - 141 p T^{9} + 953 p^{2} T^{10} - 56 p^{3} T^{11} + 174 p^{4} T^{12} - 12 p^{5} T^{13} + 4 p^{7} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 + 4 T + 40 T^{2} + 141 T^{3} + 746 T^{4} + 2383 T^{5} + 1263 p T^{6} + 25082 T^{7} + 73288 T^{8} + 25082 p T^{9} + 1263 p^{3} T^{10} + 2383 p^{3} T^{11} + 746 p^{4} T^{12} + 141 p^{5} T^{13} + 40 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 3 T + 32 T^{2} + 32 T^{3} + 458 T^{4} - 644 T^{5} + 2607 T^{6} - 29023 T^{7} - 625 T^{8} - 29023 p T^{9} + 2607 p^{2} T^{10} - 644 p^{3} T^{11} + 458 p^{4} T^{12} + 32 p^{5} T^{13} + 32 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 6 T + 116 T^{2} + 587 T^{3} + 6153 T^{4} + 26305 T^{5} + 195117 T^{6} + 699036 T^{7} + 4050559 T^{8} + 699036 p T^{9} + 195117 p^{2} T^{10} + 26305 p^{3} T^{11} + 6153 p^{4} T^{12} + 587 p^{5} T^{13} + 116 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 9 T + 122 T^{2} - 32 p T^{3} + 5373 T^{4} - 26844 T^{5} + 144595 T^{6} - 725377 T^{7} + 3403722 T^{8} - 725377 p T^{9} + 144595 p^{2} T^{10} - 26844 p^{3} T^{11} + 5373 p^{4} T^{12} - 32 p^{6} T^{13} + 122 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 4 T + 132 T^{2} + 149 T^{3} + 7210 T^{4} - 8455 T^{5} + 261495 T^{6} - 702233 T^{7} + 7976455 T^{8} - 702233 p T^{9} + 261495 p^{2} T^{10} - 8455 p^{3} T^{11} + 7210 p^{4} T^{12} + 149 p^{5} T^{13} + 132 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 11 T + 222 T^{2} + 1765 T^{3} + 20656 T^{4} + 130021 T^{5} + 1137593 T^{6} + 5928155 T^{7} + 42218816 T^{8} + 5928155 p T^{9} + 1137593 p^{2} T^{10} + 130021 p^{3} T^{11} + 20656 p^{4} T^{12} + 1765 p^{5} T^{13} + 222 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 10 T + 221 T^{2} - 1934 T^{3} + 23831 T^{4} - 175415 T^{5} + 42997 p T^{6} - 9764443 T^{7} + 71129081 T^{8} - 9764443 p T^{9} + 42997 p^{3} T^{10} - 175415 p^{3} T^{11} + 23831 p^{4} T^{12} - 1934 p^{5} T^{13} + 221 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 29 T + 591 T^{2} + 8870 T^{3} + 109589 T^{4} + 1132509 T^{5} + 10137134 T^{6} + 78739477 T^{7} + 538618569 T^{8} + 78739477 p T^{9} + 10137134 p^{2} T^{10} + 1132509 p^{3} T^{11} + 109589 p^{4} T^{12} + 8870 p^{5} T^{13} + 591 p^{6} T^{14} + 29 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 2 T + 236 T^{2} - 465 T^{3} + 27460 T^{4} - 51575 T^{5} + 2034075 T^{6} - 3419244 T^{7} + 104089696 T^{8} - 3419244 p T^{9} + 2034075 p^{2} T^{10} - 51575 p^{3} T^{11} + 27460 p^{4} T^{12} - 465 p^{5} T^{13} + 236 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 117 T^{2} + 136 T^{3} + 154 p T^{4} + 18338 T^{5} + 426984 T^{6} + 1406698 T^{7} + 22678952 T^{8} + 1406698 p T^{9} + 426984 p^{2} T^{10} + 18338 p^{3} T^{11} + 154 p^{5} T^{12} + 136 p^{5} T^{13} + 117 p^{6} T^{14} + p^{8} T^{16} \) | |
| 53 | \( 1 + 59 T + 1773 T^{2} + 35994 T^{3} + 555641 T^{4} + 6950535 T^{5} + 73081626 T^{6} + 659060411 T^{7} + 5148041929 T^{8} + 659060411 p T^{9} + 73081626 p^{2} T^{10} + 6950535 p^{3} T^{11} + 555641 p^{4} T^{12} + 35994 p^{5} T^{13} + 1773 p^{6} T^{14} + 59 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 23 T + 391 T^{2} + 5027 T^{3} + 56886 T^{4} + 553890 T^{5} + 5066388 T^{6} + 41897990 T^{7} + 333030964 T^{8} + 41897990 p T^{9} + 5066388 p^{2} T^{10} + 553890 p^{3} T^{11} + 56886 p^{4} T^{12} + 5027 p^{5} T^{13} + 391 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 2 T + 59 T^{2} + 36 T^{3} + 8639 T^{4} + 3895 T^{5} + 529997 T^{6} - 186683 T^{7} + 37362403 T^{8} - 186683 p T^{9} + 529997 p^{2} T^{10} + 3895 p^{3} T^{11} + 8639 p^{4} T^{12} + 36 p^{5} T^{13} + 59 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 3 T + 339 T^{2} - 841 T^{3} + 56502 T^{4} - 122764 T^{5} + 6176868 T^{6} - 11961514 T^{7} + 484321692 T^{8} - 11961514 p T^{9} + 6176868 p^{2} T^{10} - 122764 p^{3} T^{11} + 56502 p^{4} T^{12} - 841 p^{5} T^{13} + 339 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + T + 200 T^{2} + 642 T^{3} + 22644 T^{4} + 133805 T^{5} + 1838211 T^{6} + 14593132 T^{7} + 133386112 T^{8} + 14593132 p T^{9} + 1838211 p^{2} T^{10} + 133805 p^{3} T^{11} + 22644 p^{4} T^{12} + 642 p^{5} T^{13} + 200 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 6 T + 155 T^{2} + 1840 T^{3} + 23849 T^{4} + 267699 T^{5} + 2403405 T^{6} + 25891991 T^{7} + 218901495 T^{8} + 25891991 p T^{9} + 2403405 p^{2} T^{10} + 267699 p^{3} T^{11} + 23849 p^{4} T^{12} + 1840 p^{5} T^{13} + 155 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 222 T^{2} - 1008 T^{3} + 23865 T^{4} - 199880 T^{5} + 2046977 T^{6} - 22369928 T^{7} + 157302126 T^{8} - 22369928 p T^{9} + 2046977 p^{2} T^{10} - 199880 p^{3} T^{11} + 23865 p^{4} T^{12} - 1008 p^{5} T^{13} + 222 p^{6} T^{14} + p^{8} T^{16} \) | |
| 83 | \( 1 - 15 T + 425 T^{2} - 5251 T^{3} + 90501 T^{4} - 941833 T^{5} + 12398464 T^{6} - 111235053 T^{7} + 1207619210 T^{8} - 111235053 p T^{9} + 12398464 p^{2} T^{10} - 941833 p^{3} T^{11} + 90501 p^{4} T^{12} - 5251 p^{5} T^{13} + 425 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 4 T + 615 T^{2} + 2228 T^{3} + 173044 T^{4} + 549813 T^{5} + 29004737 T^{6} + 78354964 T^{7} + 3163082895 T^{8} + 78354964 p T^{9} + 29004737 p^{2} T^{10} + 549813 p^{3} T^{11} + 173044 p^{4} T^{12} + 2228 p^{5} T^{13} + 615 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 62 T + 2150 T^{2} + 52739 T^{3} + 1017422 T^{4} + 16273323 T^{5} + 222841507 T^{6} + 2658362297 T^{7} + 27886183171 T^{8} + 2658362297 p T^{9} + 222841507 p^{2} T^{10} + 16273323 p^{3} T^{11} + 1017422 p^{4} T^{12} + 52739 p^{5} T^{13} + 2150 p^{6} T^{14} + 62 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.64908404039973833116864310985, −3.57717846435173523668317316653, −3.30637861791742570564466997033, −3.21785953698397577767453393550, −3.18271994820844461338509969160, −3.14921108605357625637670752412, −3.13014410442255156776337085301, −3.07181088960356476542134712123, −2.97354129892694325771752549499, −2.68272697089408182477276556649, −2.50723317184675712955960883712, −2.45511936349161204277270975532, −2.44490057383513629653025831134, −2.27101028384090827778624545519, −2.25152758590992857517398808092, −2.18854117088884262207869100009, −2.16216206319439227930244226121, −1.74892997219920836412079456263, −1.58796233473676478163332758978, −1.51786059131022507392827202116, −1.37130446486533707872307212185, −1.31429040404889462024778991618, −1.30711384512383907495092731927, −1.30165720220255559326268160222, −1.10876524187700719941787450513, 0, 0, 0, 0, 0, 0, 0, 0, 1.10876524187700719941787450513, 1.30165720220255559326268160222, 1.30711384512383907495092731927, 1.31429040404889462024778991618, 1.37130446486533707872307212185, 1.51786059131022507392827202116, 1.58796233473676478163332758978, 1.74892997219920836412079456263, 2.16216206319439227930244226121, 2.18854117088884262207869100009, 2.25152758590992857517398808092, 2.27101028384090827778624545519, 2.44490057383513629653025831134, 2.45511936349161204277270975532, 2.50723317184675712955960883712, 2.68272697089408182477276556649, 2.97354129892694325771752549499, 3.07181088960356476542134712123, 3.13014410442255156776337085301, 3.14921108605357625637670752412, 3.18271994820844461338509969160, 3.21785953698397577767453393550, 3.30637861791742570564466997033, 3.57717846435173523668317316653, 3.64908404039973833116864310985
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.