Dirichlet series
| L(s) = 1 | + 1.45e3·4-s − 690·5-s − 2.62e4·9-s − 7.19e4·11-s + 5.23e5·16-s + 8.51e5·19-s − 1.00e6·20-s + 1.08e6·25-s − 7.35e4·29-s + 4.74e5·31-s − 3.80e7·36-s + 9.33e7·41-s − 1.04e8·44-s + 1.81e7·45-s + 1.87e8·49-s + 4.96e7·55-s + 2.36e8·59-s − 3.57e8·61-s − 3.89e8·64-s − 1.56e8·71-s + 1.23e9·76-s + 8.63e8·79-s − 3.61e8·80-s + 4.30e8·81-s + 3.57e8·89-s − 5.87e8·95-s + 1.88e9·99-s + ⋯ |
| L(s) = 1 | + 2.83·4-s − 0.493·5-s − 4/3·9-s − 1.48·11-s + 1.99·16-s + 1.49·19-s − 1.39·20-s + 0.555·25-s − 0.0193·29-s + 0.0922·31-s − 3.77·36-s + 5.15·41-s − 4.20·44-s + 0.658·45-s + 4.63·49-s + 0.731·55-s + 2.54·59-s − 3.30·61-s − 2.90·64-s − 0.732·71-s + 4.24·76-s + 2.49·79-s − 0.985·80-s + 10/9·81-s + 0.603·89-s − 0.740·95-s + 1.97·99-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 5^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.26890\times 10^{7}\) |
| Root analytic conductor: | \(2.77948\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 5^{8} ,\ ( \ : [9/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(8.344074414\) |
| \(L(\frac12)\) | \(\approx\) | \(8.344074414\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p^{8} T^{2} )^{4} \) |
| 5 | \( 1 + 138 p T - 24388 p^{2} T^{2} - 543978 p^{5} T^{3} - 1299858 p^{9} T^{4} - 543978 p^{14} T^{5} - 24388 p^{20} T^{6} + 138 p^{28} T^{7} + p^{36} T^{8} \) | |
| good | 2 | \( 1 - 1451 T^{2} + 395485 p^{2} T^{4} - 17915303 p^{6} T^{6} + 41866597 p^{14} T^{8} - 17915303 p^{24} T^{10} + 395485 p^{38} T^{12} - 1451 p^{54} T^{14} + p^{72} T^{16} \) |
| 7 | \( 1 - 26725604 p T^{2} + 15253712953314292 T^{4} - \)\(15\!\cdots\!84\)\( p^{2} T^{6} + \)\(12\!\cdots\!14\)\( p^{4} T^{8} - \)\(15\!\cdots\!84\)\( p^{20} T^{10} + 15253712953314292 p^{36} T^{12} - 26725604 p^{55} T^{14} + p^{72} T^{16} \) | |
| 11 | \( ( 1 + 35994 T + 5523704672 T^{2} + 186668393765490 T^{3} + 15626722625814770286 T^{4} + 186668393765490 p^{9} T^{5} + 5523704672 p^{18} T^{6} + 35994 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 13 | \( 1 - 57204710684 T^{2} + \)\(16\!\cdots\!00\)\( T^{4} - \)\(29\!\cdots\!48\)\( T^{6} + \)\(37\!\cdots\!18\)\( T^{8} - \)\(29\!\cdots\!48\)\( p^{18} T^{10} + \)\(16\!\cdots\!00\)\( p^{36} T^{12} - 57204710684 p^{54} T^{14} + p^{72} T^{16} \) | |
| 17 | \( 1 - 118674378380 T^{2} + \)\(17\!\cdots\!36\)\( T^{4} - \)\(19\!\cdots\!60\)\( T^{6} + \)\(18\!\cdots\!86\)\( T^{8} - \)\(19\!\cdots\!60\)\( p^{18} T^{10} + \)\(17\!\cdots\!36\)\( p^{36} T^{12} - 118674378380 p^{54} T^{14} + p^{72} T^{16} \) | |
| 19 | \( ( 1 - 425792 T + 640639588492 T^{2} - 117316391220409664 T^{3} + \)\(17\!\cdots\!54\)\( T^{4} - 117316391220409664 p^{9} T^{5} + 640639588492 p^{18} T^{6} - 425792 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 23 | \( 1 - 10272938895992 T^{2} + \)\(48\!\cdots\!32\)\( T^{4} - \)\(14\!\cdots\!84\)\( T^{6} + \)\(30\!\cdots\!94\)\( T^{8} - \)\(14\!\cdots\!84\)\( p^{18} T^{10} + \)\(48\!\cdots\!32\)\( p^{36} T^{12} - 10272938895992 p^{54} T^{14} + p^{72} T^{16} \) | |
| 29 | \( ( 1 + 36786 T + 11698955610980 T^{2} - 38232367405598309058 T^{3} + \)\(21\!\cdots\!18\)\( T^{4} - 38232367405598309058 p^{9} T^{5} + 11698955610980 p^{18} T^{6} + 36786 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 31 | \( ( 1 - 237044 T + 50199397014268 T^{2} - \)\(19\!\cdots\!72\)\( T^{3} + \)\(11\!\cdots\!74\)\( T^{4} - \)\(19\!\cdots\!72\)\( p^{9} T^{5} + 50199397014268 p^{18} T^{6} - 237044 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 37 | \( 1 - 624655700715068 T^{2} + \)\(20\!\cdots\!12\)\( T^{4} - \)\(45\!\cdots\!76\)\( T^{6} + \)\(70\!\cdots\!14\)\( T^{8} - \)\(45\!\cdots\!76\)\( p^{18} T^{10} + \)\(20\!\cdots\!12\)\( p^{36} T^{12} - 624655700715068 p^{54} T^{14} + p^{72} T^{16} \) | |
| 41 | \( ( 1 - 46660044 T + 1629598509234068 T^{2} - \)\(39\!\cdots\!32\)\( T^{3} + \)\(78\!\cdots\!54\)\( T^{4} - \)\(39\!\cdots\!32\)\( p^{9} T^{5} + 1629598509234068 p^{18} T^{6} - 46660044 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 43 | \( 1 - 1288760154444440 T^{2} + \)\(10\!\cdots\!96\)\( T^{4} - \)\(57\!\cdots\!80\)\( T^{6} + \)\(31\!\cdots\!06\)\( T^{8} - \)\(57\!\cdots\!80\)\( p^{18} T^{10} + \)\(10\!\cdots\!96\)\( p^{36} T^{12} - 1288760154444440 p^{54} T^{14} + p^{72} T^{16} \) | |
| 47 | \( 1 - 4836473602067240 T^{2} + \)\(12\!\cdots\!56\)\( T^{4} - \)\(21\!\cdots\!80\)\( T^{6} + \)\(28\!\cdots\!26\)\( T^{8} - \)\(21\!\cdots\!80\)\( p^{18} T^{10} + \)\(12\!\cdots\!56\)\( p^{36} T^{12} - 4836473602067240 p^{54} T^{14} + p^{72} T^{16} \) | |
| 53 | \( 1 - 4341506689340012 T^{2} + \)\(22\!\cdots\!72\)\( T^{4} - \)\(59\!\cdots\!84\)\( T^{6} + \)\(28\!\cdots\!74\)\( T^{8} - \)\(59\!\cdots\!84\)\( p^{18} T^{10} + \)\(22\!\cdots\!72\)\( p^{36} T^{12} - 4341506689340012 p^{54} T^{14} + p^{72} T^{16} \) | |
| 59 | \( ( 1 - 118263018 T + 24386862278408432 T^{2} - \)\(17\!\cdots\!86\)\( T^{3} + \)\(26\!\cdots\!54\)\( T^{4} - \)\(17\!\cdots\!86\)\( p^{9} T^{5} + 24386862278408432 p^{18} T^{6} - 118263018 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 61 | \( ( 1 + 178713880 T + 47272829372304076 T^{2} + \)\(59\!\cdots\!20\)\( T^{3} + \)\(82\!\cdots\!06\)\( T^{4} + \)\(59\!\cdots\!20\)\( p^{9} T^{5} + 47272829372304076 p^{18} T^{6} + 178713880 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 67 | \( 1 - 128915336297443400 T^{2} + \)\(88\!\cdots\!36\)\( T^{4} - \)\(39\!\cdots\!00\)\( T^{6} + \)\(12\!\cdots\!86\)\( T^{8} - \)\(39\!\cdots\!00\)\( p^{18} T^{10} + \)\(88\!\cdots\!36\)\( p^{36} T^{12} - 128915336297443400 p^{54} T^{14} + p^{72} T^{16} \) | |
| 71 | \( ( 1 + 78445332 T + 132742682733898508 T^{2} + \)\(14\!\cdots\!24\)\( T^{3} + \)\(78\!\cdots\!70\)\( T^{4} + \)\(14\!\cdots\!24\)\( p^{9} T^{5} + 132742682733898508 p^{18} T^{6} + 78445332 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 73 | \( 1 - 112790784848235992 T^{2} + \)\(16\!\cdots\!32\)\( T^{4} - \)\(11\!\cdots\!84\)\( T^{6} + \)\(89\!\cdots\!94\)\( T^{8} - \)\(11\!\cdots\!84\)\( p^{18} T^{10} + \)\(16\!\cdots\!32\)\( p^{36} T^{12} - 112790784848235992 p^{54} T^{14} + p^{72} T^{16} \) | |
| 79 | \( ( 1 - 431961140 T + 452072920533003676 T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + \)\(79\!\cdots\!66\)\( T^{4} - \)\(14\!\cdots\!80\)\( p^{9} T^{5} + 452072920533003676 p^{18} T^{6} - 431961140 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 83 | \( 1 - 981210392397464024 T^{2} + \)\(42\!\cdots\!20\)\( T^{4} - \)\(11\!\cdots\!28\)\( T^{6} + \)\(22\!\cdots\!98\)\( T^{8} - \)\(11\!\cdots\!28\)\( p^{18} T^{10} + \)\(42\!\cdots\!20\)\( p^{36} T^{12} - 981210392397464024 p^{54} T^{14} + p^{72} T^{16} \) | |
| 89 | \( ( 1 - 178691112 T + 708605882924008892 T^{2} - \)\(39\!\cdots\!44\)\( T^{3} + \)\(23\!\cdots\!94\)\( T^{4} - \)\(39\!\cdots\!44\)\( p^{9} T^{5} + 708605882924008892 p^{18} T^{6} - 178691112 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 97 | \( 1 - 1313769052010852360 T^{2} + \)\(18\!\cdots\!56\)\( T^{4} - \)\(20\!\cdots\!20\)\( T^{6} + \)\(15\!\cdots\!26\)\( T^{8} - \)\(20\!\cdots\!20\)\( p^{18} T^{10} + \)\(18\!\cdots\!56\)\( p^{36} T^{12} - 1313769052010852360 p^{54} T^{14} + p^{72} T^{16} \) | |
| 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
−7.53064209529914651618619406866, −7.15570224127282995710543137673, −7.07278249468186176882932944629, −6.60750871499045465129459240750, −6.59052075004843036856490050916, −6.07471605351197105544272280434, −5.91832183949341266972285573683, −5.80958119455002037402999776781, −5.67756715236357763390301332935, −5.19400243384815031306730447756, −4.91188250978158227800992380718, −4.66401915695825710434697231049, −4.12379854594605978594935926983, −4.06778738609871268909513987400, −3.54232816679901321022532486156, −3.09078466691506456604579187664, −2.87852774425775009653577283932, −2.60369591536547005127570239773, −2.32569165219823342722075518409, −2.30448770745738778793287466997, −2.11630379973610380778822065997, −1.27286022515273165208726702548, −0.955486554575044734772895162382, −0.66180733059183723265634497397, −0.31333241160882256906953373122, 0.31333241160882256906953373122, 0.66180733059183723265634497397, 0.955486554575044734772895162382, 1.27286022515273165208726702548, 2.11630379973610380778822065997, 2.30448770745738778793287466997, 2.32569165219823342722075518409, 2.60369591536547005127570239773, 2.87852774425775009653577283932, 3.09078466691506456604579187664, 3.54232816679901321022532486156, 4.06778738609871268909513987400, 4.12379854594605978594935926983, 4.66401915695825710434697231049, 4.91188250978158227800992380718, 5.19400243384815031306730447756, 5.67756715236357763390301332935, 5.80958119455002037402999776781, 5.91832183949341266972285573683, 6.07471605351197105544272280434, 6.59052075004843036856490050916, 6.60750871499045465129459240750, 7.07278249468186176882932944629, 7.15570224127282995710543137673, 7.53064209529914651618619406866
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.