Dirichlet series
| L(s) = 1 | − 8·3-s − 5.00e3·5-s − 4.80e3·7-s − 5.63e4·9-s − 6.30e4·11-s + 2.28e5·13-s + 4.00e4·15-s − 7.45e4·17-s − 9.81e5·19-s + 3.84e4·21-s − 1.01e6·23-s + 1.40e7·25-s + 3.00e6·27-s + 3.95e6·29-s + 1.13e7·31-s + 5.04e5·33-s + 2.40e7·35-s + 7.67e6·37-s − 1.82e6·39-s + 5.87e6·41-s + 2.73e7·43-s + 2.81e8·45-s + 6.16e7·47-s − 1.63e8·49-s + 5.96e5·51-s + 1.06e8·53-s + 3.15e8·55-s + ⋯ |
| L(s) = 1 | − 0.0570·3-s − 3.57·5-s − 0.756·7-s − 2.86·9-s − 1.29·11-s + 2.21·13-s + 0.204·15-s − 0.216·17-s − 1.72·19-s + 0.0431·21-s − 0.757·23-s + 36/5·25-s + 1.08·27-s + 1.03·29-s + 2.21·31-s + 0.0740·33-s + 2.70·35-s + 0.673·37-s − 0.126·39-s + 0.324·41-s + 1.22·43-s + 10.2·45-s + 1.84·47-s − 4.04·49-s + 0.0123·51-s + 1.85·53-s + 4.64·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 5^{8} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.03391\times 10^{17}\) |
| Root analytic conductor: | \(11.5719\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{16} \cdot 5^{8} \cdot 13^{8} ,\ ( \ : [9/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + p^{4} T )^{8} \) | |
| 13 | \( ( 1 - p^{4} T )^{8} \) | |
| good | 3 | \( 1 + 8 T + 18808 p T^{2} - 2888 p^{6} T^{3} + 924620 p^{7} T^{4} - 411618584 p^{5} T^{5} + 73063058680 p^{6} T^{6} - 4467715723064 p^{6} T^{7} + 508298427018482 p^{7} T^{8} - 4467715723064 p^{15} T^{9} + 73063058680 p^{24} T^{10} - 411618584 p^{32} T^{11} + 924620 p^{43} T^{12} - 2888 p^{51} T^{13} + 18808 p^{55} T^{14} + 8 p^{63} T^{15} + p^{72} T^{16} \) |
| 7 | \( 1 + 4808 T + 26594424 p T^{2} + 752223548904 T^{3} + 17961928724549916 T^{4} + 8973301368701396856 p T^{5} + \)\(23\!\cdots\!04\)\( p^{2} T^{6} + \)\(10\!\cdots\!72\)\( p^{3} T^{7} + \)\(22\!\cdots\!66\)\( p^{4} T^{8} + \)\(10\!\cdots\!72\)\( p^{12} T^{9} + \)\(23\!\cdots\!04\)\( p^{20} T^{10} + 8973301368701396856 p^{28} T^{11} + 17961928724549916 p^{36} T^{12} + 752223548904 p^{45} T^{13} + 26594424 p^{55} T^{14} + 4808 p^{63} T^{15} + p^{72} T^{16} \) | |
| 11 | \( 1 + 63088 T + 10754505160 T^{2} + 530500300052912 T^{3} + 53153413513018767748 T^{4} + \)\(22\!\cdots\!20\)\( T^{5} + \)\(18\!\cdots\!52\)\( T^{6} + \)\(66\!\cdots\!68\)\( T^{7} + \)\(48\!\cdots\!54\)\( T^{8} + \)\(66\!\cdots\!68\)\( p^{9} T^{9} + \)\(18\!\cdots\!52\)\( p^{18} T^{10} + \)\(22\!\cdots\!20\)\( p^{27} T^{11} + 53153413513018767748 p^{36} T^{12} + 530500300052912 p^{45} T^{13} + 10754505160 p^{54} T^{14} + 63088 p^{63} T^{15} + p^{72} T^{16} \) | |
| 17 | \( 1 + 74512 T + 589644664472 T^{2} + 59149480964612144 T^{3} + \)\(17\!\cdots\!76\)\( T^{4} + \)\(19\!\cdots\!24\)\( T^{5} + \)\(34\!\cdots\!28\)\( T^{6} + \)\(35\!\cdots\!84\)\( T^{7} + \)\(48\!\cdots\!38\)\( T^{8} + \)\(35\!\cdots\!84\)\( p^{9} T^{9} + \)\(34\!\cdots\!28\)\( p^{18} T^{10} + \)\(19\!\cdots\!24\)\( p^{27} T^{11} + \)\(17\!\cdots\!76\)\( p^{36} T^{12} + 59149480964612144 p^{45} T^{13} + 589644664472 p^{54} T^{14} + 74512 p^{63} T^{15} + p^{72} T^{16} \) | |
| 19 | \( 1 + 981248 T + 1674630498216 T^{2} + 1096799990838275168 T^{3} + \)\(11\!\cdots\!08\)\( T^{4} + \)\(61\!\cdots\!76\)\( T^{5} + \)\(29\!\cdots\!76\)\( p T^{6} + \)\(26\!\cdots\!36\)\( T^{7} + \)\(20\!\cdots\!82\)\( T^{8} + \)\(26\!\cdots\!36\)\( p^{9} T^{9} + \)\(29\!\cdots\!76\)\( p^{19} T^{10} + \)\(61\!\cdots\!76\)\( p^{27} T^{11} + \)\(11\!\cdots\!08\)\( p^{36} T^{12} + 1096799990838275168 p^{45} T^{13} + 1674630498216 p^{54} T^{14} + 981248 p^{63} T^{15} + p^{72} T^{16} \) | |
| 23 | \( 1 + 1016136 T + 4947112665592 T^{2} + 9683796821492563640 T^{3} + \)\(16\!\cdots\!60\)\( T^{4} + \)\(34\!\cdots\!36\)\( T^{5} + \)\(50\!\cdots\!64\)\( T^{6} + \)\(73\!\cdots\!32\)\( T^{7} + \)\(11\!\cdots\!82\)\( T^{8} + \)\(73\!\cdots\!32\)\( p^{9} T^{9} + \)\(50\!\cdots\!64\)\( p^{18} T^{10} + \)\(34\!\cdots\!36\)\( p^{27} T^{11} + \)\(16\!\cdots\!60\)\( p^{36} T^{12} + 9683796821492563640 p^{45} T^{13} + 4947112665592 p^{54} T^{14} + 1016136 p^{63} T^{15} + p^{72} T^{16} \) | |
| 29 | \( 1 - 3957952 T + 64043970425192 T^{2} - \)\(23\!\cdots\!96\)\( T^{3} + \)\(20\!\cdots\!36\)\( T^{4} - \)\(66\!\cdots\!48\)\( T^{5} + \)\(44\!\cdots\!56\)\( T^{6} - \)\(12\!\cdots\!00\)\( T^{7} + \)\(72\!\cdots\!50\)\( T^{8} - \)\(12\!\cdots\!00\)\( p^{9} T^{9} + \)\(44\!\cdots\!56\)\( p^{18} T^{10} - \)\(66\!\cdots\!48\)\( p^{27} T^{11} + \)\(20\!\cdots\!36\)\( p^{36} T^{12} - \)\(23\!\cdots\!96\)\( p^{45} T^{13} + 64043970425192 p^{54} T^{14} - 3957952 p^{63} T^{15} + p^{72} T^{16} \) | |
| 31 | \( 1 - 11372736 T + 195115414441336 T^{2} - \)\(17\!\cdots\!76\)\( T^{3} + \)\(16\!\cdots\!56\)\( T^{4} - \)\(11\!\cdots\!52\)\( T^{5} + \)\(86\!\cdots\!36\)\( T^{6} - \)\(48\!\cdots\!68\)\( T^{7} + \)\(90\!\cdots\!50\)\( p T^{8} - \)\(48\!\cdots\!68\)\( p^{9} T^{9} + \)\(86\!\cdots\!36\)\( p^{18} T^{10} - \)\(11\!\cdots\!52\)\( p^{27} T^{11} + \)\(16\!\cdots\!56\)\( p^{36} T^{12} - \)\(17\!\cdots\!76\)\( p^{45} T^{13} + 195115414441336 p^{54} T^{14} - 11372736 p^{63} T^{15} + p^{72} T^{16} \) | |
| 37 | \( 1 - 7674288 T + 506682917186024 T^{2} - \)\(29\!\cdots\!88\)\( T^{3} + \)\(14\!\cdots\!68\)\( T^{4} - \)\(71\!\cdots\!52\)\( T^{5} + \)\(28\!\cdots\!08\)\( T^{6} - \)\(12\!\cdots\!80\)\( T^{7} + \)\(43\!\cdots\!90\)\( T^{8} - \)\(12\!\cdots\!80\)\( p^{9} T^{9} + \)\(28\!\cdots\!08\)\( p^{18} T^{10} - \)\(71\!\cdots\!52\)\( p^{27} T^{11} + \)\(14\!\cdots\!68\)\( p^{36} T^{12} - \)\(29\!\cdots\!88\)\( p^{45} T^{13} + 506682917186024 p^{54} T^{14} - 7674288 p^{63} T^{15} + p^{72} T^{16} \) | |
| 41 | \( 1 - 5873200 T + 2111006432259224 T^{2} - \)\(11\!\cdots\!88\)\( T^{3} + \)\(20\!\cdots\!24\)\( T^{4} - \)\(11\!\cdots\!68\)\( T^{5} + \)\(12\!\cdots\!52\)\( T^{6} - \)\(58\!\cdots\!44\)\( T^{7} + \)\(48\!\cdots\!78\)\( T^{8} - \)\(58\!\cdots\!44\)\( p^{9} T^{9} + \)\(12\!\cdots\!52\)\( p^{18} T^{10} - \)\(11\!\cdots\!68\)\( p^{27} T^{11} + \)\(20\!\cdots\!24\)\( p^{36} T^{12} - \)\(11\!\cdots\!88\)\( p^{45} T^{13} + 2111006432259224 p^{54} T^{14} - 5873200 p^{63} T^{15} + p^{72} T^{16} \) | |
| 43 | \( 1 - 27394472 T + 3705719515456744 T^{2} - \)\(86\!\cdots\!60\)\( T^{3} + \)\(61\!\cdots\!04\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{5} + \)\(58\!\cdots\!24\)\( T^{6} - \)\(96\!\cdots\!00\)\( T^{7} + \)\(36\!\cdots\!74\)\( T^{8} - \)\(96\!\cdots\!00\)\( p^{9} T^{9} + \)\(58\!\cdots\!24\)\( p^{18} T^{10} - \)\(12\!\cdots\!00\)\( p^{27} T^{11} + \)\(61\!\cdots\!04\)\( p^{36} T^{12} - \)\(86\!\cdots\!60\)\( p^{45} T^{13} + 3705719515456744 p^{54} T^{14} - 27394472 p^{63} T^{15} + p^{72} T^{16} \) | |
| 47 | \( 1 - 61633640 T + 4449399332749640 T^{2} - \)\(25\!\cdots\!64\)\( T^{3} + \)\(12\!\cdots\!12\)\( T^{4} - \)\(55\!\cdots\!44\)\( T^{5} + \)\(22\!\cdots\!56\)\( T^{6} - \)\(86\!\cdots\!80\)\( T^{7} + \)\(29\!\cdots\!26\)\( T^{8} - \)\(86\!\cdots\!80\)\( p^{9} T^{9} + \)\(22\!\cdots\!56\)\( p^{18} T^{10} - \)\(55\!\cdots\!44\)\( p^{27} T^{11} + \)\(12\!\cdots\!12\)\( p^{36} T^{12} - \)\(25\!\cdots\!64\)\( p^{45} T^{13} + 4449399332749640 p^{54} T^{14} - 61633640 p^{63} T^{15} + p^{72} T^{16} \) | |
| 53 | \( 1 - 106321168 T + 16359751997030392 T^{2} - \)\(10\!\cdots\!40\)\( T^{3} + \)\(94\!\cdots\!12\)\( T^{4} - \)\(38\!\cdots\!36\)\( T^{5} + \)\(26\!\cdots\!32\)\( T^{6} - \)\(63\!\cdots\!12\)\( T^{7} + \)\(63\!\cdots\!14\)\( T^{8} - \)\(63\!\cdots\!12\)\( p^{9} T^{9} + \)\(26\!\cdots\!32\)\( p^{18} T^{10} - \)\(38\!\cdots\!36\)\( p^{27} T^{11} + \)\(94\!\cdots\!12\)\( p^{36} T^{12} - \)\(10\!\cdots\!40\)\( p^{45} T^{13} + 16359751997030392 p^{54} T^{14} - 106321168 p^{63} T^{15} + p^{72} T^{16} \) | |
| 59 | \( 1 + 40587008 T + 23714929657715208 T^{2} + \)\(17\!\cdots\!40\)\( T^{3} + \)\(41\!\cdots\!48\)\( T^{4} + \)\(32\!\cdots\!20\)\( T^{5} + \)\(48\!\cdots\!48\)\( T^{6} + \)\(40\!\cdots\!84\)\( T^{7} + \)\(47\!\cdots\!50\)\( T^{8} + \)\(40\!\cdots\!84\)\( p^{9} T^{9} + \)\(48\!\cdots\!48\)\( p^{18} T^{10} + \)\(32\!\cdots\!20\)\( p^{27} T^{11} + \)\(41\!\cdots\!48\)\( p^{36} T^{12} + \)\(17\!\cdots\!40\)\( p^{45} T^{13} + 23714929657715208 p^{54} T^{14} + 40587008 p^{63} T^{15} + p^{72} T^{16} \) | |
| 61 | \( 1 + 239058464 T + 58004323720443624 T^{2} + \)\(87\!\cdots\!52\)\( T^{3} + \)\(11\!\cdots\!72\)\( T^{4} + \)\(12\!\cdots\!72\)\( T^{5} + \)\(11\!\cdots\!24\)\( T^{6} + \)\(95\!\cdots\!12\)\( T^{7} + \)\(10\!\cdots\!78\)\( T^{8} + \)\(95\!\cdots\!12\)\( p^{9} T^{9} + \)\(11\!\cdots\!24\)\( p^{18} T^{10} + \)\(12\!\cdots\!72\)\( p^{27} T^{11} + \)\(11\!\cdots\!72\)\( p^{36} T^{12} + \)\(87\!\cdots\!52\)\( p^{45} T^{13} + 58004323720443624 p^{54} T^{14} + 239058464 p^{63} T^{15} + p^{72} T^{16} \) | |
| 67 | \( 1 + 373408008 T + 199074016899954408 T^{2} + \)\(51\!\cdots\!32\)\( T^{3} + \)\(16\!\cdots\!84\)\( T^{4} + \)\(33\!\cdots\!04\)\( T^{5} + \)\(82\!\cdots\!76\)\( T^{6} + \)\(13\!\cdots\!72\)\( T^{7} + \)\(27\!\cdots\!38\)\( T^{8} + \)\(13\!\cdots\!72\)\( p^{9} T^{9} + \)\(82\!\cdots\!76\)\( p^{18} T^{10} + \)\(33\!\cdots\!04\)\( p^{27} T^{11} + \)\(16\!\cdots\!84\)\( p^{36} T^{12} + \)\(51\!\cdots\!32\)\( p^{45} T^{13} + 199074016899954408 p^{54} T^{14} + 373408008 p^{63} T^{15} + p^{72} T^{16} \) | |
| 71 | \( 1 - 20321120 T + 188079128245914616 T^{2} + \)\(20\!\cdots\!28\)\( T^{3} + \)\(20\!\cdots\!32\)\( T^{4} + \)\(27\!\cdots\!52\)\( T^{5} + \)\(14\!\cdots\!16\)\( T^{6} + \)\(23\!\cdots\!56\)\( T^{7} + \)\(79\!\cdots\!58\)\( T^{8} + \)\(23\!\cdots\!56\)\( p^{9} T^{9} + \)\(14\!\cdots\!16\)\( p^{18} T^{10} + \)\(27\!\cdots\!52\)\( p^{27} T^{11} + \)\(20\!\cdots\!32\)\( p^{36} T^{12} + \)\(20\!\cdots\!28\)\( p^{45} T^{13} + 188079128245914616 p^{54} T^{14} - 20321120 p^{63} T^{15} + p^{72} T^{16} \) | |
| 73 | \( 1 + 359059744 T + 315641316940160712 T^{2} + \)\(10\!\cdots\!72\)\( T^{3} + \)\(49\!\cdots\!16\)\( T^{4} + \)\(13\!\cdots\!72\)\( T^{5} + \)\(49\!\cdots\!12\)\( T^{6} + \)\(11\!\cdots\!20\)\( T^{7} + \)\(34\!\cdots\!90\)\( T^{8} + \)\(11\!\cdots\!20\)\( p^{9} T^{9} + \)\(49\!\cdots\!12\)\( p^{18} T^{10} + \)\(13\!\cdots\!72\)\( p^{27} T^{11} + \)\(49\!\cdots\!16\)\( p^{36} T^{12} + \)\(10\!\cdots\!72\)\( p^{45} T^{13} + 315641316940160712 p^{54} T^{14} + 359059744 p^{63} T^{15} + p^{72} T^{16} \) | |
| 79 | \( 1 + 128740816 T + 613861528765706424 T^{2} + \)\(13\!\cdots\!76\)\( T^{3} + \)\(17\!\cdots\!28\)\( T^{4} + \)\(51\!\cdots\!08\)\( T^{5} + \)\(32\!\cdots\!76\)\( T^{6} + \)\(10\!\cdots\!48\)\( T^{7} + \)\(45\!\cdots\!42\)\( T^{8} + \)\(10\!\cdots\!48\)\( p^{9} T^{9} + \)\(32\!\cdots\!76\)\( p^{18} T^{10} + \)\(51\!\cdots\!08\)\( p^{27} T^{11} + \)\(17\!\cdots\!28\)\( p^{36} T^{12} + \)\(13\!\cdots\!76\)\( p^{45} T^{13} + 613861528765706424 p^{54} T^{14} + 128740816 p^{63} T^{15} + p^{72} T^{16} \) | |
| 83 | \( 1 - 343750008 T + 1041335780350288360 T^{2} - \)\(36\!\cdots\!24\)\( T^{3} + \)\(52\!\cdots\!96\)\( T^{4} - \)\(17\!\cdots\!52\)\( T^{5} + \)\(16\!\cdots\!56\)\( T^{6} - \)\(49\!\cdots\!96\)\( T^{7} + \)\(36\!\cdots\!06\)\( T^{8} - \)\(49\!\cdots\!96\)\( p^{9} T^{9} + \)\(16\!\cdots\!56\)\( p^{18} T^{10} - \)\(17\!\cdots\!52\)\( p^{27} T^{11} + \)\(52\!\cdots\!96\)\( p^{36} T^{12} - \)\(36\!\cdots\!24\)\( p^{45} T^{13} + 1041335780350288360 p^{54} T^{14} - 343750008 p^{63} T^{15} + p^{72} T^{16} \) | |
| 89 | \( 1 + 64531728 T + 1865843606808473272 T^{2} - \)\(90\!\cdots\!52\)\( T^{3} + \)\(16\!\cdots\!36\)\( T^{4} - \)\(21\!\cdots\!08\)\( T^{5} + \)\(91\!\cdots\!08\)\( T^{6} - \)\(15\!\cdots\!88\)\( T^{7} + \)\(36\!\cdots\!66\)\( T^{8} - \)\(15\!\cdots\!88\)\( p^{9} T^{9} + \)\(91\!\cdots\!08\)\( p^{18} T^{10} - \)\(21\!\cdots\!08\)\( p^{27} T^{11} + \)\(16\!\cdots\!36\)\( p^{36} T^{12} - \)\(90\!\cdots\!52\)\( p^{45} T^{13} + 1865843606808473272 p^{54} T^{14} + 64531728 p^{63} T^{15} + p^{72} T^{16} \) | |
| 97 | \( 1 + 1380655920 T + 5037518913571607160 T^{2} + \)\(53\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!44\)\( T^{4} + \)\(95\!\cdots\!60\)\( T^{5} + \)\(14\!\cdots\!52\)\( T^{6} + \)\(10\!\cdots\!68\)\( T^{7} + \)\(13\!\cdots\!06\)\( T^{8} + \)\(10\!\cdots\!68\)\( p^{9} T^{9} + \)\(14\!\cdots\!52\)\( p^{18} T^{10} + \)\(95\!\cdots\!60\)\( p^{27} T^{11} + \)\(11\!\cdots\!44\)\( p^{36} T^{12} + \)\(53\!\cdots\!20\)\( p^{45} T^{13} + 5037518913571607160 p^{54} T^{14} + 1380655920 p^{63} T^{15} + p^{72} T^{16} \) | |
| 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
−4.28647942321925648516493029301, −4.18361878322920172651782082919, −4.11642060940259946911138095090, −3.68801017401546209609511600421, −3.66900904212297363349512922847, −3.58882664021896419440207617859, −3.57447343085785287158952168469, −3.54222217847284123611335732642, −3.40431314711915631703056922342, −2.76430396421584230576268265036, −2.76170537588938368574185356747, −2.73580093621013955499051134057, −2.68221365671877516212078857073, −2.66858713569524481942501727079, −2.57762852138956053114767776635, −2.32312649797336759789902366123, −2.29624021664371485381101611460, −1.67369318920752839245126311507, −1.45994111455415988642976305381, −1.40529685465825749538509146381, −1.18276382882582128465677151259, −1.08954722649406818164490317038, −0.977586543838582988094059930544, −0.914769117486973777223197871379, −0.867644046395980513259677992634, 0, 0, 0, 0, 0, 0, 0, 0, 0.867644046395980513259677992634, 0.914769117486973777223197871379, 0.977586543838582988094059930544, 1.08954722649406818164490317038, 1.18276382882582128465677151259, 1.40529685465825749538509146381, 1.45994111455415988642976305381, 1.67369318920752839245126311507, 2.29624021664371485381101611460, 2.32312649797336759789902366123, 2.57762852138956053114767776635, 2.66858713569524481942501727079, 2.68221365671877516212078857073, 2.73580093621013955499051134057, 2.76170537588938368574185356747, 2.76430396421584230576268265036, 3.40431314711915631703056922342, 3.54222217847284123611335732642, 3.57447343085785287158952168469, 3.58882664021896419440207617859, 3.66900904212297363349512922847, 3.68801017401546209609511600421, 4.11642060940259946911138095090, 4.18361878322920172651782082919, 4.28647942321925648516493029301
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.