Dirichlet series
| L(s) = 1 | + 8.19e3·4-s + 1.95e5·7-s + 1.38e6·9-s − 2.13e5·11-s + 4.19e7·16-s + 1.56e8·23-s + 1.07e9·25-s + 1.59e9·28-s + 3.08e8·29-s + 1.13e10·36-s − 3.24e9·37-s + 2.10e10·43-s − 1.75e9·44-s + 9.41e9·49-s + 1.80e11·53-s + 2.70e11·63-s + 1.71e11·64-s + 3.69e11·67-s + 5.74e11·71-s − 4.17e10·77-s − 6.07e11·79-s + 9.94e11·81-s + 1.28e12·92-s − 2.96e11·99-s + 8.78e12·100-s + 2.16e11·107-s − 1.34e13·109-s + ⋯ |
| L(s) = 1 | + 2·4-s + 1.65·7-s + 2.60·9-s − 0.120·11-s + 5/2·16-s + 1.05·23-s + 4.39·25-s + 3.31·28-s + 0.519·29-s + 5.21·36-s − 1.26·37-s + 3.32·43-s − 0.241·44-s + 0.680·49-s + 8.14·53-s + 4.32·63-s + 5/2·64-s + 4.08·67-s + 4.48·71-s − 0.200·77-s − 2.50·79-s + 3.52·81-s + 2.11·92-s − 0.314·99-s + 8.78·100-s + 0.144·107-s − 8.01·109-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(13-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+6)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.18737\times 10^{8}\) |
| Root analytic conductor: | \(3.57713\) |
| Motivic weight: | \(12\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 7^{8} ,\ ( \ : [6]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{13}{2})\) | \(\approx\) | \(92.94341133\) |
| \(L(\frac12)\) | \(\approx\) | \(92.94341133\) |
| \(L(7)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{11} T^{2} )^{4} \) |
| 7 | \( 1 - 27880 p T + 585165148 p^{2} T^{2} - 34208863720 p^{6} T^{3} + 275471557162 p^{11} T^{4} - 34208863720 p^{18} T^{5} + 585165148 p^{26} T^{6} - 27880 p^{37} T^{7} + p^{48} T^{8} \) | |
| good | 3 | \( 1 - 462104 p T^{2} + 103072783388 p^{2} T^{4} - 291167257255784 p^{7} T^{6} + 774196723541737318 p^{12} T^{8} - 291167257255784 p^{31} T^{10} + 103072783388 p^{50} T^{12} - 462104 p^{73} T^{14} + p^{96} T^{16} \) |
| 5 | \( 1 - 8577448 p^{3} T^{2} + 577562182671894652 T^{4} - \)\(34\!\cdots\!32\)\( p^{4} T^{6} + \)\(15\!\cdots\!94\)\( p^{8} T^{8} - \)\(34\!\cdots\!32\)\( p^{28} T^{10} + 577562182671894652 p^{48} T^{12} - 8577448 p^{75} T^{14} + p^{96} T^{16} \) | |
| 11 | \( ( 1 + 9720 p T + 24309397468 p^{2} T^{2} + 1673698432264200 p^{3} T^{3} + \)\(10\!\cdots\!58\)\( p^{4} T^{4} + 1673698432264200 p^{15} T^{5} + 24309397468 p^{26} T^{6} + 9720 p^{37} T^{7} + p^{48} T^{8} )^{2} \) | |
| 13 | \( 1 - 118483961811080 T^{2} + \)\(72\!\cdots\!20\)\( T^{4} - \)\(21\!\cdots\!60\)\( p T^{6} + \)\(78\!\cdots\!42\)\( T^{8} - \)\(21\!\cdots\!60\)\( p^{25} T^{10} + \)\(72\!\cdots\!20\)\( p^{48} T^{12} - 118483961811080 p^{72} T^{14} + p^{96} T^{16} \) | |
| 17 | \( 1 - 1555137278635016 T^{2} + \)\(11\!\cdots\!96\)\( T^{4} - \)\(73\!\cdots\!92\)\( T^{6} + \)\(44\!\cdots\!26\)\( T^{8} - \)\(73\!\cdots\!92\)\( p^{24} T^{10} + \)\(11\!\cdots\!96\)\( p^{48} T^{12} - 1555137278635016 p^{72} T^{14} + p^{96} T^{16} \) | |
| 19 | \( 1 - 5862118397694152 T^{2} + \)\(21\!\cdots\!92\)\( T^{4} - \)\(55\!\cdots\!08\)\( T^{6} + \)\(12\!\cdots\!38\)\( T^{8} - \)\(55\!\cdots\!08\)\( p^{24} T^{10} + \)\(21\!\cdots\!92\)\( p^{48} T^{12} - 5862118397694152 p^{72} T^{14} + p^{96} T^{16} \) | |
| 23 | \( ( 1 - 78365880 T + 63212424068847100 T^{2} - \)\(29\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!14\)\( T^{4} - \)\(29\!\cdots\!80\)\( p^{12} T^{5} + 63212424068847100 p^{24} T^{6} - 78365880 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 29 | \( ( 1 - 154426824 T + 601673598812273116 T^{2} + \)\(10\!\cdots\!52\)\( T^{3} + \)\(20\!\cdots\!66\)\( T^{4} + \)\(10\!\cdots\!52\)\( p^{12} T^{5} + 601673598812273116 p^{24} T^{6} - 154426824 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 31 | \( 1 - 2302046075483163656 T^{2} + \)\(24\!\cdots\!16\)\( T^{4} - \)\(24\!\cdots\!72\)\( T^{6} + \)\(23\!\cdots\!66\)\( T^{8} - \)\(24\!\cdots\!72\)\( p^{24} T^{10} + \)\(24\!\cdots\!16\)\( p^{48} T^{12} - 2302046075483163656 p^{72} T^{14} + p^{96} T^{16} \) | |
| 37 | \( ( 1 + 1621800440 T + 17056754214865778908 T^{2} + \)\(65\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!30\)\( T^{4} + \)\(65\!\cdots\!40\)\( p^{12} T^{5} + 17056754214865778908 p^{24} T^{6} + 1621800440 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 41 | \( 1 - 68082924589008676616 T^{2} + \)\(24\!\cdots\!56\)\( T^{4} - \)\(60\!\cdots\!12\)\( T^{6} + \)\(13\!\cdots\!66\)\( T^{8} - \)\(60\!\cdots\!12\)\( p^{24} T^{10} + \)\(24\!\cdots\!56\)\( p^{48} T^{12} - 68082924589008676616 p^{72} T^{14} + p^{96} T^{16} \) | |
| 43 | \( ( 1 - 10503151000 T + \)\(19\!\cdots\!16\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!66\)\( T^{4} - \)\(12\!\cdots\!00\)\( p^{12} T^{5} + \)\(19\!\cdots\!16\)\( p^{24} T^{6} - 10503151000 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 47 | \( 1 - \)\(33\!\cdots\!12\)\( T^{2} + \)\(71\!\cdots\!92\)\( T^{4} - \)\(11\!\cdots\!08\)\( T^{6} + \)\(15\!\cdots\!38\)\( T^{8} - \)\(11\!\cdots\!08\)\( p^{24} T^{10} + \)\(71\!\cdots\!92\)\( p^{48} T^{12} - \)\(33\!\cdots\!12\)\( p^{72} T^{14} + p^{96} T^{16} \) | |
| 53 | \( ( 1 - 90222818760 T + \)\(49\!\cdots\!12\)\( T^{2} - \)\(17\!\cdots\!60\)\( T^{3} + \)\(45\!\cdots\!26\)\( T^{4} - \)\(17\!\cdots\!60\)\( p^{12} T^{5} + \)\(49\!\cdots\!12\)\( p^{24} T^{6} - 90222818760 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 59 | \( 1 - \)\(73\!\cdots\!48\)\( T^{2} + \)\(29\!\cdots\!08\)\( T^{4} - \)\(82\!\cdots\!96\)\( T^{6} + \)\(16\!\cdots\!70\)\( T^{8} - \)\(82\!\cdots\!96\)\( p^{24} T^{10} + \)\(29\!\cdots\!08\)\( p^{48} T^{12} - \)\(73\!\cdots\!48\)\( p^{72} T^{14} + p^{96} T^{16} \) | |
| 61 | \( 1 - \)\(12\!\cdots\!52\)\( T^{2} + \)\(69\!\cdots\!72\)\( T^{4} - \)\(24\!\cdots\!28\)\( T^{6} + \)\(70\!\cdots\!98\)\( T^{8} - \)\(24\!\cdots\!28\)\( p^{24} T^{10} + \)\(69\!\cdots\!72\)\( p^{48} T^{12} - \)\(12\!\cdots\!52\)\( p^{72} T^{14} + p^{96} T^{16} \) | |
| 67 | \( ( 1 - 184605629720 T + \)\(42\!\cdots\!72\)\( T^{2} - \)\(46\!\cdots\!20\)\( T^{3} + \)\(56\!\cdots\!06\)\( T^{4} - \)\(46\!\cdots\!20\)\( p^{12} T^{5} + \)\(42\!\cdots\!72\)\( p^{24} T^{6} - 184605629720 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 71 | \( ( 1 - 287029072152 T + \)\(87\!\cdots\!84\)\( T^{2} - \)\(13\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!10\)\( T^{4} - \)\(13\!\cdots\!40\)\( p^{12} T^{5} + \)\(87\!\cdots\!84\)\( p^{24} T^{6} - 287029072152 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 73 | \( 1 - \)\(15\!\cdots\!88\)\( T^{2} + \)\(11\!\cdots\!12\)\( T^{4} - \)\(47\!\cdots\!52\)\( T^{6} + \)\(13\!\cdots\!58\)\( T^{8} - \)\(47\!\cdots\!52\)\( p^{24} T^{10} + \)\(11\!\cdots\!12\)\( p^{48} T^{12} - \)\(15\!\cdots\!88\)\( p^{72} T^{14} + p^{96} T^{16} \) | |
| 79 | \( ( 1 + 303913305064 T + \)\(20\!\cdots\!00\)\( T^{2} + \)\(37\!\cdots\!56\)\( T^{3} + \)\(15\!\cdots\!74\)\( T^{4} + \)\(37\!\cdots\!56\)\( p^{12} T^{5} + \)\(20\!\cdots\!00\)\( p^{24} T^{6} + 303913305064 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 83 | \( 1 - \)\(51\!\cdots\!84\)\( T^{2} + \)\(11\!\cdots\!44\)\( T^{4} - \)\(15\!\cdots\!24\)\( T^{6} + \)\(16\!\cdots\!70\)\( T^{8} - \)\(15\!\cdots\!24\)\( p^{24} T^{10} + \)\(11\!\cdots\!44\)\( p^{48} T^{12} - \)\(51\!\cdots\!84\)\( p^{72} T^{14} + p^{96} T^{16} \) | |
| 89 | \( 1 - \)\(81\!\cdots\!60\)\( T^{2} + \)\(47\!\cdots\!00\)\( T^{4} - \)\(17\!\cdots\!60\)\( T^{6} + \)\(51\!\cdots\!62\)\( T^{8} - \)\(17\!\cdots\!60\)\( p^{24} T^{10} + \)\(47\!\cdots\!00\)\( p^{48} T^{12} - \)\(81\!\cdots\!60\)\( p^{72} T^{14} + p^{96} T^{16} \) | |
| 97 | \( 1 - \)\(32\!\cdots\!40\)\( T^{2} + \)\(47\!\cdots\!60\)\( T^{4} - \)\(44\!\cdots\!40\)\( T^{6} + \)\(32\!\cdots\!22\)\( T^{8} - \)\(44\!\cdots\!40\)\( p^{24} T^{10} + \)\(47\!\cdots\!60\)\( p^{48} T^{12} - \)\(32\!\cdots\!40\)\( p^{72} T^{14} + p^{96} T^{16} \) | |
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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
−6.95390264860430101577309111477, −6.57767940713249198214778754535, −6.52154377695030563136281825494, −6.13266089862179480180297905727, −5.60976235784417813210840043704, −5.28253930833034712581853070817, −5.27767147032163271855782757251, −5.06375493672508534352450625753, −4.98603328929898363400003726986, −4.48562602023509759167347744055, −4.01160572555889778942883489151, −3.96185746391561968547740601369, −3.73322676802597192768627014611, −3.70248593720916924072742691250, −2.70764637171800086815188574881, −2.56032029814957121773463574944, −2.54441916295794901770372679418, −2.53022319685075672309084272612, −2.00586014192355762546262101593, −1.60508614472873612325750055259, −1.26582365170144037925615171057, −1.02894467920572099816435599676, −1.00521955906476333045225683609, −0.875809490556841484550676879137, −0.53178389396024155444550900769, 0.53178389396024155444550900769, 0.875809490556841484550676879137, 1.00521955906476333045225683609, 1.02894467920572099816435599676, 1.26582365170144037925615171057, 1.60508614472873612325750055259, 2.00586014192355762546262101593, 2.53022319685075672309084272612, 2.54441916295794901770372679418, 2.56032029814957121773463574944, 2.70764637171800086815188574881, 3.70248593720916924072742691250, 3.73322676802597192768627014611, 3.96185746391561968547740601369, 4.01160572555889778942883489151, 4.48562602023509759167347744055, 4.98603328929898363400003726986, 5.06375493672508534352450625753, 5.27767147032163271855782757251, 5.28253930833034712581853070817, 5.60976235784417813210840043704, 6.13266089862179480180297905727, 6.52154377695030563136281825494, 6.57767940713249198214778754535, 6.95390264860430101577309111477
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.