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: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 11^{8} \cdot 19^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.324131944\) |
| \(L(\frac12)\) | \(\approx\) | \(2.324131944\) |
| \(L(\frac{3}{2})\) | 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
−3.08351047984136269126662976171, −2.74497306445632990296385862578, −2.74183010101730594632717439814, −2.57737502694857664743361357650, −2.57709177083257225066204230589, −2.53069452239049757476847540186, −2.52536120076818578992151134687, −2.43073277151161422841424371625, −2.35483076326657674772814912955, −2.04887990584138918442774850554, −2.03265134136195557396116568077, −2.01117215929650084401577053621, −1.85875542195603187290474445371, −1.77567473624622087683125183683, −1.75460470617714496374176581214, −1.24153175884468027961507289511, −1.21241357458060606731142985918, −1.21032489753899351961314220122, −0.957272294718045335459950003103, −0.844550390328305882231882763188, −0.71927805363189081686457828772, −0.44172840048349894198269642245, −0.38874091002499552831149092781, −0.35387914043484898965873833828, −0.34910515548840773384424325709, 0.34910515548840773384424325709, 0.35387914043484898965873833828, 0.38874091002499552831149092781, 0.44172840048349894198269642245, 0.71927805363189081686457828772, 0.844550390328305882231882763188, 0.957272294718045335459950003103, 1.21032489753899351961314220122, 1.21241357458060606731142985918, 1.24153175884468027961507289511, 1.75460470617714496374176581214, 1.77567473624622087683125183683, 1.85875542195603187290474445371, 2.01117215929650084401577053621, 2.03265134136195557396116568077, 2.04887990584138918442774850554, 2.35483076326657674772814912955, 2.43073277151161422841424371625, 2.52536120076818578992151134687, 2.53069452239049757476847540186, 2.57709177083257225066204230589, 2.57737502694857664743361357650, 2.74183010101730594632717439814, 2.74497306445632990296385862578, 3.08351047984136269126662976171
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.