Dirichlet series
| L(s) = 1 | + 324·3-s − 256·4-s − 2.22e3·5-s − 140·7-s + 5.68e4·9-s − 7.43e3·11-s − 8.29e4·12-s − 7.21e5·15-s + 1.63e4·16-s + 7.93e3·17-s − 7.73e5·19-s + 5.69e5·20-s − 4.53e4·21-s + 3.91e5·23-s + 1.62e6·25-s + 7.08e6·27-s + 3.58e4·28-s − 2.31e6·29-s − 3.21e6·31-s − 2.40e6·33-s + 3.11e5·35-s − 1.45e7·36-s + 5.44e5·37-s + 5.52e5·43-s + 1.90e6·44-s − 1.26e8·45-s − 1.57e7·47-s + ⋯ |
| L(s) = 1 | + 4·3-s − 4-s − 3.56·5-s − 0.0583·7-s + 26/3·9-s − 0.507·11-s − 4·12-s − 14.2·15-s + 1/4·16-s + 0.0949·17-s − 5.93·19-s + 3.56·20-s − 0.233·21-s + 1.39·23-s + 4.16·25-s + 40/3·27-s + 0.0583·28-s − 3.27·29-s − 3.47·31-s − 2.03·33-s + 0.207·35-s − 8.66·36-s + 0.290·37-s + 0.161·43-s + 0.507·44-s − 30.8·45-s − 3.22·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+4)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.34480\times 10^{9}\) |
| Root analytic conductor: | \(4.13641\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : [4]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(0.1596797225\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1596797225\) |
| \(L(5)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{7} T^{2} + p^{14} T^{4} )^{2} \) |
| 3 | \( ( 1 - p^{4} T + p^{7} T^{2} )^{4} \) | |
| 7 | \( 1 + 20 p T + 886 p^{3} T^{2} + 2000 p^{6} T^{3} - 51053 p^{10} T^{4} + 2000 p^{14} T^{5} + 886 p^{19} T^{6} + 20 p^{25} T^{7} + p^{32} T^{8} \) | |
| good | 5 | \( 1 + 2226 T + 3328231 T^{2} + 3731975814 T^{3} + 3646264212301 T^{4} + 635686311080772 p T^{5} + 99300901350469186 p^{2} T^{6} + 14127933350637058776 p^{3} T^{7} + \)\(18\!\cdots\!94\)\( p^{4} T^{8} + 14127933350637058776 p^{11} T^{9} + 99300901350469186 p^{18} T^{10} + 635686311080772 p^{25} T^{11} + 3646264212301 p^{32} T^{12} + 3731975814 p^{40} T^{13} + 3328231 p^{48} T^{14} + 2226 p^{56} T^{15} + p^{64} T^{16} \) |
| 11 | \( 1 + 7434 T - 324792421 T^{2} + 6971653960038 T^{3} + 111524546009006809 T^{4} - \)\(19\!\cdots\!60\)\( T^{5} + \)\(24\!\cdots\!14\)\( T^{6} + \)\(40\!\cdots\!12\)\( p T^{7} - \)\(62\!\cdots\!98\)\( T^{8} + \)\(40\!\cdots\!12\)\( p^{9} T^{9} + \)\(24\!\cdots\!14\)\( p^{16} T^{10} - \)\(19\!\cdots\!60\)\( p^{24} T^{11} + 111524546009006809 p^{32} T^{12} + 6971653960038 p^{40} T^{13} - 324792421 p^{48} T^{14} + 7434 p^{56} T^{15} + p^{64} T^{16} \) | |
| 13 | \( 1 - 2608551938 T^{2} + 2156805763660455169 T^{4} + \)\(49\!\cdots\!50\)\( T^{6} - \)\(17\!\cdots\!88\)\( T^{8} + \)\(49\!\cdots\!50\)\( p^{16} T^{10} + 2156805763660455169 p^{32} T^{12} - 2608551938 p^{48} T^{14} + p^{64} T^{16} \) | |
| 17 | \( 1 - 7932 T + 19886965360 T^{2} - 157577057681664 T^{3} + \)\(21\!\cdots\!62\)\( T^{4} + \)\(32\!\cdots\!08\)\( T^{5} + \)\(17\!\cdots\!36\)\( T^{6} + \)\(55\!\cdots\!36\)\( T^{7} + \)\(11\!\cdots\!27\)\( T^{8} + \)\(55\!\cdots\!36\)\( p^{8} T^{9} + \)\(17\!\cdots\!36\)\( p^{16} T^{10} + \)\(32\!\cdots\!08\)\( p^{24} T^{11} + \)\(21\!\cdots\!62\)\( p^{32} T^{12} - 157577057681664 p^{40} T^{13} + 19886965360 p^{48} T^{14} - 7932 p^{56} T^{15} + p^{64} T^{16} \) | |
| 19 | \( 1 + 773082 T + 332940457063 T^{2} + 103377966184675710 T^{3} + \)\(25\!\cdots\!05\)\( T^{4} + \)\(53\!\cdots\!60\)\( T^{5} + \)\(97\!\cdots\!30\)\( T^{6} + \)\(15\!\cdots\!68\)\( T^{7} + \)\(21\!\cdots\!78\)\( T^{8} + \)\(15\!\cdots\!68\)\( p^{8} T^{9} + \)\(97\!\cdots\!30\)\( p^{16} T^{10} + \)\(53\!\cdots\!60\)\( p^{24} T^{11} + \)\(25\!\cdots\!05\)\( p^{32} T^{12} + 103377966184675710 p^{40} T^{13} + 332940457063 p^{48} T^{14} + 773082 p^{56} T^{15} + p^{64} T^{16} \) | |
| 23 | \( 1 - 391200 T - 107298377092 T^{2} + 77980181132472384 T^{3} + \)\(39\!\cdots\!50\)\( T^{4} - \)\(80\!\cdots\!16\)\( T^{5} + \)\(10\!\cdots\!00\)\( T^{6} + \)\(28\!\cdots\!80\)\( T^{7} - \)\(12\!\cdots\!13\)\( T^{8} + \)\(28\!\cdots\!80\)\( p^{8} T^{9} + \)\(10\!\cdots\!00\)\( p^{16} T^{10} - \)\(80\!\cdots\!16\)\( p^{24} T^{11} + \)\(39\!\cdots\!50\)\( p^{32} T^{12} + 77980181132472384 p^{40} T^{13} - 107298377092 p^{48} T^{14} - 391200 p^{56} T^{15} + p^{64} T^{16} \) | |
| 29 | \( ( 1 + 1159098 T + 952364940685 T^{2} + 401284295554339278 T^{3} + \)\(12\!\cdots\!52\)\( T^{4} + 401284295554339278 p^{8} T^{5} + 952364940685 p^{16} T^{6} + 1159098 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 31 | \( 1 + 3212544 T + 6034006082110 T^{2} + 8332888624243226112 T^{3} + \)\(89\!\cdots\!45\)\( T^{4} + \)\(76\!\cdots\!88\)\( T^{5} + \)\(50\!\cdots\!82\)\( T^{6} + \)\(27\!\cdots\!48\)\( T^{7} + \)\(18\!\cdots\!24\)\( T^{8} + \)\(27\!\cdots\!48\)\( p^{8} T^{9} + \)\(50\!\cdots\!82\)\( p^{16} T^{10} + \)\(76\!\cdots\!88\)\( p^{24} T^{11} + \)\(89\!\cdots\!45\)\( p^{32} T^{12} + 8332888624243226112 p^{40} T^{13} + 6034006082110 p^{48} T^{14} + 3212544 p^{56} T^{15} + p^{64} T^{16} \) | |
| 37 | \( 1 - 544802 T - 12194313015765 T^{2} + 4256562952939670882 T^{3} + \)\(89\!\cdots\!61\)\( T^{4} - \)\(19\!\cdots\!80\)\( T^{5} - \)\(45\!\cdots\!82\)\( T^{6} + \)\(28\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!06\)\( T^{8} + \)\(28\!\cdots\!00\)\( p^{8} T^{9} - \)\(45\!\cdots\!82\)\( p^{16} T^{10} - \)\(19\!\cdots\!80\)\( p^{24} T^{11} + \)\(89\!\cdots\!61\)\( p^{32} T^{12} + 4256562952939670882 p^{40} T^{13} - 12194313015765 p^{48} T^{14} - 544802 p^{56} T^{15} + p^{64} T^{16} \) | |
| 41 | \( 1 - 53995243882904 T^{2} + \)\(13\!\cdots\!32\)\( T^{4} - \)\(19\!\cdots\!96\)\( T^{6} + \)\(18\!\cdots\!58\)\( T^{8} - \)\(19\!\cdots\!96\)\( p^{16} T^{10} + \)\(13\!\cdots\!32\)\( p^{32} T^{12} - 53995243882904 p^{48} T^{14} + p^{64} T^{16} \) | |
| 43 | \( ( 1 - 276142 T + 10942801421233 T^{2} + 3844406824229610290 T^{3} + \)\(24\!\cdots\!36\)\( T^{4} + 3844406824229610290 p^{8} T^{5} + 10942801421233 p^{16} T^{6} - 276142 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 47 | \( 1 + 15735252 T + 147275905587904 T^{2} + \)\(10\!\cdots\!72\)\( T^{3} + \)\(50\!\cdots\!74\)\( T^{4} + \)\(14\!\cdots\!00\)\( T^{5} - \)\(88\!\cdots\!28\)\( T^{6} - \)\(41\!\cdots\!68\)\( T^{7} - \)\(28\!\cdots\!25\)\( T^{8} - \)\(41\!\cdots\!68\)\( p^{8} T^{9} - \)\(88\!\cdots\!28\)\( p^{16} T^{10} + \)\(14\!\cdots\!00\)\( p^{24} T^{11} + \)\(50\!\cdots\!74\)\( p^{32} T^{12} + \)\(10\!\cdots\!72\)\( p^{40} T^{13} + 147275905587904 p^{48} T^{14} + 15735252 p^{56} T^{15} + p^{64} T^{16} \) | |
| 53 | \( 1 - 10935222 T - 151034141451721 T^{2} + \)\(94\!\cdots\!14\)\( T^{3} + \)\(26\!\cdots\!89\)\( T^{4} - \)\(87\!\cdots\!24\)\( T^{5} - \)\(23\!\cdots\!10\)\( T^{6} + \)\(15\!\cdots\!64\)\( T^{7} + \)\(18\!\cdots\!42\)\( T^{8} + \)\(15\!\cdots\!64\)\( p^{8} T^{9} - \)\(23\!\cdots\!10\)\( p^{16} T^{10} - \)\(87\!\cdots\!24\)\( p^{24} T^{11} + \)\(26\!\cdots\!89\)\( p^{32} T^{12} + \)\(94\!\cdots\!14\)\( p^{40} T^{13} - 151034141451721 p^{48} T^{14} - 10935222 p^{56} T^{15} + p^{64} T^{16} \) | |
| 59 | \( 1 - 63840318 T + 2434828606215835 T^{2} - \)\(68\!\cdots\!86\)\( T^{3} + \)\(15\!\cdots\!17\)\( T^{4} - \)\(30\!\cdots\!88\)\( T^{5} + \)\(51\!\cdots\!06\)\( T^{6} - \)\(75\!\cdots\!76\)\( T^{7} + \)\(96\!\cdots\!82\)\( T^{8} - \)\(75\!\cdots\!76\)\( p^{8} T^{9} + \)\(51\!\cdots\!06\)\( p^{16} T^{10} - \)\(30\!\cdots\!88\)\( p^{24} T^{11} + \)\(15\!\cdots\!17\)\( p^{32} T^{12} - \)\(68\!\cdots\!86\)\( p^{40} T^{13} + 2434828606215835 p^{48} T^{14} - 63840318 p^{56} T^{15} + p^{64} T^{16} \) | |
| 61 | \( 1 - 43108176 T + 1299312631884580 T^{2} - \)\(48\!\cdots\!08\)\( p T^{3} + \)\(58\!\cdots\!38\)\( T^{4} - \)\(16\!\cdots\!12\)\( p T^{5} + \)\(16\!\cdots\!28\)\( T^{6} - \)\(24\!\cdots\!64\)\( T^{7} + \)\(34\!\cdots\!07\)\( T^{8} - \)\(24\!\cdots\!64\)\( p^{8} T^{9} + \)\(16\!\cdots\!28\)\( p^{16} T^{10} - \)\(16\!\cdots\!12\)\( p^{25} T^{11} + \)\(58\!\cdots\!38\)\( p^{32} T^{12} - \)\(48\!\cdots\!08\)\( p^{41} T^{13} + 1299312631884580 p^{48} T^{14} - 43108176 p^{56} T^{15} + p^{64} T^{16} \) | |
| 67 | \( 1 - 48827290 T + 683401072076991 T^{2} - \)\(28\!\cdots\!70\)\( T^{3} + \)\(28\!\cdots\!69\)\( T^{4} + \)\(40\!\cdots\!60\)\( T^{5} - \)\(16\!\cdots\!50\)\( T^{6} + \)\(20\!\cdots\!20\)\( T^{7} - \)\(16\!\cdots\!78\)\( T^{8} + \)\(20\!\cdots\!20\)\( p^{8} T^{9} - \)\(16\!\cdots\!50\)\( p^{16} T^{10} + \)\(40\!\cdots\!60\)\( p^{24} T^{11} + \)\(28\!\cdots\!69\)\( p^{32} T^{12} - \)\(28\!\cdots\!70\)\( p^{40} T^{13} + 683401072076991 p^{48} T^{14} - 48827290 p^{56} T^{15} + p^{64} T^{16} \) | |
| 71 | \( ( 1 + 82263372 T + 5069368028260696 T^{2} + \)\(27\!\cdots\!40\)\( p T^{3} + \)\(58\!\cdots\!54\)\( T^{4} + \)\(27\!\cdots\!40\)\( p^{9} T^{5} + 5069368028260696 p^{16} T^{6} + 82263372 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 73 | \( 1 - 150530022 T + 13065283713690811 T^{2} - \)\(82\!\cdots\!26\)\( T^{3} + \)\(42\!\cdots\!21\)\( T^{4} - \)\(18\!\cdots\!36\)\( T^{5} + \)\(68\!\cdots\!90\)\( T^{6} - \)\(22\!\cdots\!12\)\( T^{7} + \)\(67\!\cdots\!02\)\( T^{8} - \)\(22\!\cdots\!12\)\( p^{8} T^{9} + \)\(68\!\cdots\!90\)\( p^{16} T^{10} - \)\(18\!\cdots\!36\)\( p^{24} T^{11} + \)\(42\!\cdots\!21\)\( p^{32} T^{12} - \)\(82\!\cdots\!26\)\( p^{40} T^{13} + 13065283713690811 p^{48} T^{14} - 150530022 p^{56} T^{15} + p^{64} T^{16} \) | |
| 79 | \( 1 + 18689696 T - 5241619985395602 T^{2} - \)\(39\!\cdots\!84\)\( T^{3} + \)\(17\!\cdots\!21\)\( T^{4} + \)\(52\!\cdots\!40\)\( T^{5} - \)\(39\!\cdots\!30\)\( T^{6} - \)\(31\!\cdots\!00\)\( T^{7} + \)\(68\!\cdots\!80\)\( T^{8} - \)\(31\!\cdots\!00\)\( p^{8} T^{9} - \)\(39\!\cdots\!30\)\( p^{16} T^{10} + \)\(52\!\cdots\!40\)\( p^{24} T^{11} + \)\(17\!\cdots\!21\)\( p^{32} T^{12} - \)\(39\!\cdots\!84\)\( p^{40} T^{13} - 5241619985395602 p^{48} T^{14} + 18689696 p^{56} T^{15} + p^{64} T^{16} \) | |
| 83 | \( 1 - 7978689794105930 T^{2} + \)\(39\!\cdots\!57\)\( T^{4} - \)\(13\!\cdots\!10\)\( T^{6} + \)\(34\!\cdots\!28\)\( T^{8} - \)\(13\!\cdots\!10\)\( p^{16} T^{10} + \)\(39\!\cdots\!57\)\( p^{32} T^{12} - 7978689794105930 p^{48} T^{14} + p^{64} T^{16} \) | |
| 89 | \( 1 + 191605284 T + 24387400236274096 T^{2} + \)\(23\!\cdots\!96\)\( T^{3} + \)\(18\!\cdots\!06\)\( T^{4} + \)\(14\!\cdots\!80\)\( T^{5} + \)\(10\!\cdots\!92\)\( T^{6} + \)\(78\!\cdots\!00\)\( T^{7} + \)\(51\!\cdots\!19\)\( T^{8} + \)\(78\!\cdots\!00\)\( p^{8} T^{9} + \)\(10\!\cdots\!92\)\( p^{16} T^{10} + \)\(14\!\cdots\!80\)\( p^{24} T^{11} + \)\(18\!\cdots\!06\)\( p^{32} T^{12} + \)\(23\!\cdots\!96\)\( p^{40} T^{13} + 24387400236274096 p^{48} T^{14} + 191605284 p^{56} T^{15} + p^{64} T^{16} \) | |
| 97 | \( 1 - 29569205100777170 T^{2} + \)\(47\!\cdots\!73\)\( p T^{4} - \)\(52\!\cdots\!70\)\( T^{6} + \)\(47\!\cdots\!20\)\( T^{8} - \)\(52\!\cdots\!70\)\( p^{16} T^{10} + \)\(47\!\cdots\!73\)\( p^{33} T^{12} - 29569205100777170 p^{48} T^{14} + p^{64} T^{16} \) | |
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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
−5.64466150086337152483465208269, −5.54916194613829693562551945878, −5.34371069149206671315904727901, −4.91542765774100653373474672933, −4.81294893036347996880856739304, −4.58323329201547885768328480838, −4.11299602629671579622778240645, −4.06321134213988433520110903399, −4.02386703040634880841378071536, −3.93466693274527157888500401607, −3.65429613181537156235019108479, −3.59061039250097985671995427081, −3.48905669053704804192191807567, −3.33145433949097105435946120270, −2.73144798370055543249755542046, −2.42747229743590644103152421069, −2.27644034795293500216895783944, −2.14748705475370895761836519408, −2.01672245054869402443433001087, −1.80131567801295619013917513040, −1.48329423374438862053317534778, −0.77236595191753250409894995144, −0.62075137141345265999490784876, −0.12847889786877697682629180807, −0.10936097470449416187678370169, 0.10936097470449416187678370169, 0.12847889786877697682629180807, 0.62075137141345265999490784876, 0.77236595191753250409894995144, 1.48329423374438862053317534778, 1.80131567801295619013917513040, 2.01672245054869402443433001087, 2.14748705475370895761836519408, 2.27644034795293500216895783944, 2.42747229743590644103152421069, 2.73144798370055543249755542046, 3.33145433949097105435946120270, 3.48905669053704804192191807567, 3.59061039250097985671995427081, 3.65429613181537156235019108479, 3.93466693274527157888500401607, 4.02386703040634880841378071536, 4.06321134213988433520110903399, 4.11299602629671579622778240645, 4.58323329201547885768328480838, 4.81294893036347996880856739304, 4.91542765774100653373474672933, 5.34371069149206671315904727901, 5.54916194613829693562551945878, 5.64466150086337152483465208269
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.