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
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{24} \cdot 3^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.43982\times 10^{6}\) |
| Root analytic conductor: | \(2.42602\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{24} \cdot 3^{24} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.3493846037\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3493846037\) |
| \(L(2)\) | 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 | \( 1 + 236 T + 438 T^{2} - 1185512 T^{3} + 510091505 T^{4} + 51879290952 T^{5} - 3191784386426 T^{6} + 104812309462772 T^{7} + 86158688483213604 T^{8} + 104812309462772 p^{2} T^{9} - 3191784386426 p^{4} T^{10} + 51879290952 p^{6} T^{11} + 510091505 p^{8} T^{12} - 1185512 p^{10} T^{13} + 438 p^{12} T^{14} + 236 p^{14} T^{15} + p^{16} T^{16} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.34076172212214225075655792566, −5.14000222707174394574673533527, −5.02703927177714132515297135633, −4.96975423254316763118185470601, −4.82258269911389802090620900267, −4.63843602866098716391908460851, −4.49341275933664568170475580854, −4.28124105892902277117401275342, −4.16103808126711847490594391816, −3.78287520451304394684977145696, −3.61770792692957709534675385333, −3.33960968152408349756510724051, −3.08080513970757130848849454877, −3.04117251112805491993060898691, −3.02473180882254094199143850690, −2.75672404048624938244993687737, −2.55650683163885217225789943043, −2.39292226703255033597477585824, −1.96278079796447593016728979215, −1.76429894830359508876465030460, −1.73986901727507146758740957247, −1.21597481944238800614590382699, −0.989399795376746464926906543620, −0.858767024485041830863964339343, −0.07022404959542682344813204512, 0.07022404959542682344813204512, 0.858767024485041830863964339343, 0.989399795376746464926906543620, 1.21597481944238800614590382699, 1.73986901727507146758740957247, 1.76429894830359508876465030460, 1.96278079796447593016728979215, 2.39292226703255033597477585824, 2.55650683163885217225789943043, 2.75672404048624938244993687737, 3.02473180882254094199143850690, 3.04117251112805491993060898691, 3.08080513970757130848849454877, 3.33960968152408349756510724051, 3.61770792692957709534675385333, 3.78287520451304394684977145696, 4.16103808126711847490594391816, 4.28124105892902277117401275342, 4.49341275933664568170475580854, 4.63843602866098716391908460851, 4.82258269911389802090620900267, 4.96975423254316763118185470601, 5.02703927177714132515297135633, 5.14000222707174394574673533527, 5.34076172212214225075655792566
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.