Dirichlet series
| L(s) = 1 | − 6·5-s − 6·7-s − 36·11-s − 14·13-s + 4·19-s − 102·23-s − 27·25-s − 114·29-s + 50·31-s + 36·35-s − 120·37-s − 264·41-s − 28·43-s + 150·47-s + 163·49-s + 216·55-s + 108·59-s − 14·61-s + 84·65-s − 20·67-s − 76·73-s + 216·77-s − 26·79-s − 246·83-s + 84·91-s − 24·95-s − 236·97-s + ⋯ |
| L(s) = 1 | − 6/5·5-s − 6/7·7-s − 3.27·11-s − 1.07·13-s + 4/19·19-s − 4.43·23-s − 1.07·25-s − 3.93·29-s + 1.61·31-s + 1.02·35-s − 3.24·37-s − 6.43·41-s − 0.651·43-s + 3.19·47-s + 3.32·49-s + 3.92·55-s + 1.83·59-s − 0.229·61-s + 1.29·65-s − 0.298·67-s − 1.04·73-s + 2.80·77-s − 0.329·79-s − 2.96·83-s + 0.923·91-s − 0.252·95-s − 2.43·97-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{48} \cdot 3^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.41562\times 10^{13}\) |
| Root analytic conductor: | \(6.86182\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.002332049091\) |
| \(L(\frac12)\) | \(\approx\) | \(0.002332049091\) |
| \(L(2)\) | 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 | \( 1 \) | |
| good | 5 | \( 1 + 6 T + 63 T^{2} + 306 T^{3} + 2461 T^{4} + 15132 T^{5} + 86562 T^{6} + 473928 T^{7} + 2131866 T^{8} + 473928 p^{2} T^{9} + 86562 p^{4} T^{10} + 15132 p^{6} T^{11} + 2461 p^{8} T^{12} + 306 p^{10} T^{13} + 63 p^{12} T^{14} + 6 p^{14} T^{15} + p^{16} T^{16} \) |
| 7 | \( 1 + 6 T - 127 T^{2} - 102 p T^{3} + 1459 p T^{4} + 43992 T^{5} - 588922 T^{6} - 917052 T^{7} + 31082074 T^{8} - 917052 p^{2} T^{9} - 588922 p^{4} T^{10} + 43992 p^{6} T^{11} + 1459 p^{9} T^{12} - 102 p^{11} T^{13} - 127 p^{12} T^{14} + 6 p^{14} T^{15} + p^{16} T^{16} \) | |
| 11 | \( 1 + 36 T + 828 T^{2} + 1296 p T^{3} + 189103 T^{4} + 184500 p T^{5} + 18127476 T^{6} + 142796448 T^{7} + 1322450688 T^{8} + 142796448 p^{2} T^{9} + 18127476 p^{4} T^{10} + 184500 p^{7} T^{11} + 189103 p^{8} T^{12} + 1296 p^{11} T^{13} + 828 p^{12} T^{14} + 36 p^{14} T^{15} + p^{16} T^{16} \) | |
| 13 | \( 1 + 14 T - 129 T^{2} - 1886 T^{3} + 9461 T^{4} + 158556 T^{5} + 4694986 T^{6} + 8508992 T^{7} - 1078024230 T^{8} + 8508992 p^{2} T^{9} + 4694986 p^{4} T^{10} + 158556 p^{6} T^{11} + 9461 p^{8} T^{12} - 1886 p^{10} T^{13} - 129 p^{12} T^{14} + 14 p^{14} T^{15} + p^{16} T^{16} \) | |
| 17 | \( 1 - 858 T^{2} + 505105 T^{4} - 204328674 T^{6} + 68217206628 T^{8} - 204328674 p^{4} T^{10} + 505105 p^{8} T^{12} - 858 p^{12} T^{14} + p^{16} T^{16} \) | |
| 19 | \( ( 1 - 2 T + 265 T^{2} - 6170 T^{3} + 157036 T^{4} - 6170 p^{2} T^{5} + 265 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 23 | \( 1 + 102 T + 6033 T^{2} + 261630 T^{3} + 9057109 T^{4} + 271036392 T^{5} + 7331036166 T^{6} + 184354538700 T^{7} + 4374045926298 T^{8} + 184354538700 p^{2} T^{9} + 7331036166 p^{4} T^{10} + 271036392 p^{6} T^{11} + 9057109 p^{8} T^{12} + 261630 p^{10} T^{13} + 6033 p^{12} T^{14} + 102 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( 1 + 114 T + 8599 T^{2} + 486438 T^{3} + 22645933 T^{4} + 890300916 T^{5} + 31365917458 T^{6} + 1000841966952 T^{7} + 29952505780714 T^{8} + 1000841966952 p^{2} T^{9} + 31365917458 p^{4} T^{10} + 890300916 p^{6} T^{11} + 22645933 p^{8} T^{12} + 486438 p^{10} T^{13} + 8599 p^{12} T^{14} + 114 p^{14} T^{15} + p^{16} T^{16} \) | |
| 31 | \( 1 - 50 T - 21 p T^{2} + 68990 T^{3} - 334663 T^{4} - 5769840 T^{5} - 1286987822 T^{6} - 24925573940 T^{7} + 3245120663634 T^{8} - 24925573940 p^{2} T^{9} - 1286987822 p^{4} T^{10} - 5769840 p^{6} T^{11} - 334663 p^{8} T^{12} + 68990 p^{10} T^{13} - 21 p^{13} T^{14} - 50 p^{14} T^{15} + p^{16} T^{16} \) | |
| 37 | \( ( 1 + 60 T + 5200 T^{2} + 219060 T^{3} + 10695774 T^{4} + 219060 p^{2} T^{5} + 5200 p^{4} T^{6} + 60 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 41 | \( 1 + 264 T + 38158 T^{2} + 3940464 T^{3} + 321724129 T^{4} + 21802117680 T^{5} + 1260648967822 T^{6} + 63190156979928 T^{7} + 2768191141578052 T^{8} + 63190156979928 p^{2} T^{9} + 1260648967822 p^{4} T^{10} + 21802117680 p^{6} T^{11} + 321724129 p^{8} T^{12} + 3940464 p^{10} T^{13} + 38158 p^{12} T^{14} + 264 p^{14} T^{15} + p^{16} T^{16} \) | |
| 43 | \( 1 + 28 T - 2364 T^{2} + 141848 T^{3} + 7021127 T^{4} - 338054892 T^{5} + 12175270060 T^{6} + 693178978408 T^{7} - 23735268373680 T^{8} + 693178978408 p^{2} T^{9} + 12175270060 p^{4} T^{10} - 338054892 p^{6} T^{11} + 7021127 p^{8} T^{12} + 141848 p^{10} T^{13} - 2364 p^{12} T^{14} + 28 p^{14} T^{15} + p^{16} T^{16} \) | |
| 47 | \( 1 - 150 T + 16249 T^{2} - 1312350 T^{3} + 88432117 T^{4} - 5053131000 T^{5} + 262267521478 T^{6} - 12644089287900 T^{7} + 595883042253514 T^{8} - 12644089287900 p^{2} T^{9} + 262267521478 p^{4} T^{10} - 5053131000 p^{6} T^{11} + 88432117 p^{8} T^{12} - 1312350 p^{10} T^{13} + 16249 p^{12} T^{14} - 150 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( 1 - 15456 T^{2} + 115082620 T^{4} - 547509829536 T^{6} + 1820499555493830 T^{8} - 547509829536 p^{4} T^{10} + 115082620 p^{8} T^{12} - 15456 p^{12} T^{14} + p^{16} T^{16} \) | |
| 59 | \( 1 - 108 T + 14820 T^{2} - 1180656 T^{3} + 104177575 T^{4} - 7489417356 T^{5} + 509063599692 T^{6} - 32920352322696 T^{7} + 1903340903262096 T^{8} - 32920352322696 p^{2} T^{9} + 509063599692 p^{4} T^{10} - 7489417356 p^{6} T^{11} + 104177575 p^{8} T^{12} - 1180656 p^{10} T^{13} + 14820 p^{12} T^{14} - 108 p^{14} T^{15} + p^{16} T^{16} \) | |
| 61 | \( 1 + 14 T - 6633 T^{2} - 472862 T^{3} + 19891733 T^{4} + 2361270204 T^{5} + 99631636810 T^{6} - 5929775489824 T^{7} - 599620944879702 T^{8} - 5929775489824 p^{2} T^{9} + 99631636810 p^{4} T^{10} + 2361270204 p^{6} T^{11} + 19891733 p^{8} T^{12} - 472862 p^{10} T^{13} - 6633 p^{12} T^{14} + 14 p^{14} T^{15} + p^{16} T^{16} \) | |
| 67 | \( 1 + 20 T - 15756 T^{2} - 141560 T^{3} + 151491335 T^{4} + 601531260 T^{5} - 1010166803204 T^{6} - 1012785207880 T^{7} + 5163639663893904 T^{8} - 1012785207880 p^{2} T^{9} - 1010166803204 p^{4} T^{10} + 601531260 p^{6} T^{11} + 151491335 p^{8} T^{12} - 141560 p^{10} T^{13} - 15756 p^{12} T^{14} + 20 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( 1 - 13464 T^{2} + 105773404 T^{4} - 696180849192 T^{6} + 3915811319568198 T^{8} - 696180849192 p^{4} T^{10} + 105773404 p^{8} T^{12} - 13464 p^{12} T^{14} + p^{16} T^{16} \) | |
| 73 | \( ( 1 + 38 T + 11485 T^{2} + 120314 T^{3} + 68572624 T^{4} + 120314 p^{2} T^{5} + 11485 p^{4} T^{6} + 38 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 79 | \( 1 + 26 T - 12687 T^{2} - 773558 T^{3} + 64925093 T^{4} + 5836272936 T^{5} - 59707396922 T^{6} - 18106848362500 T^{7} - 289214375790438 T^{8} - 18106848362500 p^{2} T^{9} - 59707396922 p^{4} T^{10} + 5836272936 p^{6} T^{11} + 64925093 p^{8} T^{12} - 773558 p^{10} T^{13} - 12687 p^{12} T^{14} + 26 p^{14} T^{15} + p^{16} T^{16} \) | |
| 83 | \( 1 + 246 T + 54825 T^{2} + 8524638 T^{3} + 1254860197 T^{4} + 148166306376 T^{5} + 16481417300694 T^{6} + 1562002778781228 T^{7} + 140096195439334362 T^{8} + 1562002778781228 p^{2} T^{9} + 16481417300694 p^{4} T^{10} + 148166306376 p^{6} T^{11} + 1254860197 p^{8} T^{12} + 8524638 p^{10} T^{13} + 54825 p^{12} T^{14} + 246 p^{14} T^{15} + p^{16} T^{16} \) | |
| 89 | \( 1 - 28448 T^{2} + 504887356 T^{4} - 6266477720288 T^{6} + 56631012864364294 T^{8} - 6266477720288 p^{4} T^{10} + 504887356 p^{8} T^{12} - 28448 p^{12} T^{14} + p^{16} T^{16} \) | |
| 97 | \( 1 + 236 T + 438 T^{2} - 1185512 T^{3} + 510091505 T^{4} + 51879290952 T^{5} - 3191784386426 T^{6} + 104812309462772 T^{7} + 86158688483213604 T^{8} + 104812309462772 p^{2} T^{9} - 3191784386426 p^{4} T^{10} + 51879290952 p^{6} T^{11} + 510091505 p^{8} T^{12} - 1185512 p^{10} T^{13} + 438 p^{12} T^{14} + 236 p^{14} T^{15} + p^{16} T^{16} \) | |
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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
−3.74273968079015240718389142484, −3.74257723936243326658040572850, −3.46656430786351724543778848938, −3.39323568092044107735177015431, −3.26966803884910400060778369421, −3.11749171821479223255776252988, −2.94535125272394237060220547327, −2.67305193790346076097972474633, −2.65749646502157407507744756790, −2.55149372343305084463616915866, −2.49389780739279722177409394365, −2.38646097009609461877663783173, −2.13604492219071260463169556821, −2.05304782125940523301585241491, −1.80445464429976423030723350698, −1.74983691617457404528609120518, −1.64055709549905143855350142156, −1.45245661409859772613231178396, −1.28353801180217620337091925667, −1.21398193733308094846140476665, −0.51173296511356399201294720288, −0.37176663470732499498831274930, −0.36762567240788976085118012160, −0.06757730456858675625325337391, −0.04882074530511757047563925387, 0.04882074530511757047563925387, 0.06757730456858675625325337391, 0.36762567240788976085118012160, 0.37176663470732499498831274930, 0.51173296511356399201294720288, 1.21398193733308094846140476665, 1.28353801180217620337091925667, 1.45245661409859772613231178396, 1.64055709549905143855350142156, 1.74983691617457404528609120518, 1.80445464429976423030723350698, 2.05304782125940523301585241491, 2.13604492219071260463169556821, 2.38646097009609461877663783173, 2.49389780739279722177409394365, 2.55149372343305084463616915866, 2.65749646502157407507744756790, 2.67305193790346076097972474633, 2.94535125272394237060220547327, 3.11749171821479223255776252988, 3.26966803884910400060778369421, 3.39323568092044107735177015431, 3.46656430786351724543778848938, 3.74257723936243326658040572850, 3.74273968079015240718389142484
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.