Dirichlet series
| L(s) = 1 | − 3·3-s + 5.00e3·5-s + 1.92e4·7-s − 5.77e4·9-s − 1.49e4·11-s − 3.43e4·13-s − 1.50e4·15-s + 4.60e5·17-s + 5.77e5·19-s − 5.76e4·21-s + 2.90e6·23-s + 1.40e7·25-s + 9.52e5·27-s + 8.02e6·29-s + 7.10e6·31-s + 4.47e4·33-s + 9.60e7·35-s + 1.37e7·37-s + 1.03e5·39-s + 1.89e7·41-s − 6.85e5·43-s − 2.88e8·45-s − 1.57e7·47-s + 2.07e8·49-s − 1.38e6·51-s − 2.53e7·53-s − 7.46e7·55-s + ⋯ |
| L(s) = 1 | − 0.0213·3-s + 3.57·5-s + 3.02·7-s − 2.93·9-s − 0.307·11-s − 0.333·13-s − 0.0765·15-s + 1.33·17-s + 1.01·19-s − 0.0646·21-s + 2.16·23-s + 36/5·25-s + 0.344·27-s + 2.10·29-s + 1.38·31-s + 0.00657·33-s + 10.8·35-s + 1.20·37-s + 0.00713·39-s + 1.04·41-s − 0.0305·43-s − 10.5·45-s − 0.471·47-s + 36/7·49-s − 0.0285·51-s − 0.441·53-s − 1.10·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{24} \cdot 5^{8} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.87052\times 10^{17}\) |
| Root analytic conductor: | \(12.0087\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{24} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [9/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(165.9758659\) |
| \(L(\frac12)\) | \(\approx\) | \(165.9758659\) |
| \(L(\frac{11}{2})\) | 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 | \( ( 1 - p^{4} T )^{8} \) | |
| 7 | \( ( 1 - p^{4} T )^{8} \) | |
| good | 3 | \( 1 + p T + 57790 T^{2} - 605947 T^{3} + 582029959 p T^{4} - 297746104 p^{2} T^{5} + 1484345002846 p^{3} T^{6} + 11406854189870 p^{4} T^{7} + 3448348449552124 p^{5} T^{8} + 11406854189870 p^{13} T^{9} + 1484345002846 p^{21} T^{10} - 297746104 p^{29} T^{11} + 582029959 p^{37} T^{12} - 605947 p^{45} T^{13} + 57790 p^{54} T^{14} + p^{64} T^{15} + p^{72} T^{16} \) |
| 11 | \( 1 + 14933 T + 5908292398 T^{2} + 15157030272731 T^{3} + 10674881225571403029 T^{4} - \)\(24\!\cdots\!12\)\( T^{5} - \)\(13\!\cdots\!06\)\( T^{6} - \)\(87\!\cdots\!34\)\( p^{2} T^{7} - \)\(26\!\cdots\!16\)\( p^{2} T^{8} - \)\(87\!\cdots\!34\)\( p^{11} T^{9} - \)\(13\!\cdots\!06\)\( p^{18} T^{10} - \)\(24\!\cdots\!12\)\( p^{27} T^{11} + 10674881225571403029 p^{36} T^{12} + 15157030272731 p^{45} T^{13} + 5908292398 p^{54} T^{14} + 14933 p^{63} T^{15} + p^{72} T^{16} \) | |
| 13 | \( 1 + 34351 T + 37571498772 T^{2} + 2384851329208743 T^{3} + \)\(79\!\cdots\!33\)\( T^{4} + \)\(58\!\cdots\!96\)\( T^{5} + \)\(12\!\cdots\!82\)\( T^{6} + \)\(90\!\cdots\!18\)\( T^{7} + \)\(15\!\cdots\!12\)\( T^{8} + \)\(90\!\cdots\!18\)\( p^{9} T^{9} + \)\(12\!\cdots\!82\)\( p^{18} T^{10} + \)\(58\!\cdots\!96\)\( p^{27} T^{11} + \)\(79\!\cdots\!33\)\( p^{36} T^{12} + 2384851329208743 p^{45} T^{13} + 37571498772 p^{54} T^{14} + 34351 p^{63} T^{15} + p^{72} T^{16} \) | |
| 17 | \( 1 - 460131 T + 299901654316 T^{2} - 198130427723423807 T^{3} + \)\(94\!\cdots\!17\)\( T^{4} - \)\(38\!\cdots\!72\)\( T^{5} + \)\(18\!\cdots\!62\)\( T^{6} - \)\(67\!\cdots\!70\)\( T^{7} + \)\(22\!\cdots\!76\)\( T^{8} - \)\(67\!\cdots\!70\)\( p^{9} T^{9} + \)\(18\!\cdots\!62\)\( p^{18} T^{10} - \)\(38\!\cdots\!72\)\( p^{27} T^{11} + \)\(94\!\cdots\!17\)\( p^{36} T^{12} - 198130427723423807 p^{45} T^{13} + 299901654316 p^{54} T^{14} - 460131 p^{63} T^{15} + p^{72} T^{16} \) | |
| 19 | \( 1 - 577890 T + 1606992306712 T^{2} - 440040080892135450 T^{3} + \)\(95\!\cdots\!88\)\( T^{4} + \)\(38\!\cdots\!18\)\( T^{5} + \)\(28\!\cdots\!96\)\( T^{6} + \)\(13\!\cdots\!38\)\( T^{7} + \)\(68\!\cdots\!78\)\( T^{8} + \)\(13\!\cdots\!38\)\( p^{9} T^{9} + \)\(28\!\cdots\!96\)\( p^{18} T^{10} + \)\(38\!\cdots\!18\)\( p^{27} T^{11} + \)\(95\!\cdots\!88\)\( p^{36} T^{12} - 440040080892135450 p^{45} T^{13} + 1606992306712 p^{54} T^{14} - 577890 p^{63} T^{15} + p^{72} T^{16} \) | |
| 23 | \( 1 - 2909230 T + 11929006395728 T^{2} - 27714438782880229470 T^{3} + \)\(67\!\cdots\!40\)\( T^{4} - \)\(12\!\cdots\!86\)\( T^{5} + \)\(22\!\cdots\!00\)\( T^{6} - \)\(33\!\cdots\!42\)\( T^{7} + \)\(50\!\cdots\!22\)\( T^{8} - \)\(33\!\cdots\!42\)\( p^{9} T^{9} + \)\(22\!\cdots\!00\)\( p^{18} T^{10} - \)\(12\!\cdots\!86\)\( p^{27} T^{11} + \)\(67\!\cdots\!40\)\( p^{36} T^{12} - 27714438782880229470 p^{45} T^{13} + 11929006395728 p^{54} T^{14} - 2909230 p^{63} T^{15} + p^{72} T^{16} \) | |
| 29 | \( 1 - 8028925 T + 96060641871340 T^{2} - 21087140480327066825 p T^{3} + \)\(14\!\cdots\!85\)\( p T^{4} - \)\(21\!\cdots\!52\)\( T^{5} + \)\(11\!\cdots\!02\)\( T^{6} - \)\(48\!\cdots\!18\)\( T^{7} + \)\(20\!\cdots\!88\)\( T^{8} - \)\(48\!\cdots\!18\)\( p^{9} T^{9} + \)\(11\!\cdots\!02\)\( p^{18} T^{10} - \)\(21\!\cdots\!52\)\( p^{27} T^{11} + \)\(14\!\cdots\!85\)\( p^{37} T^{12} - 21087140480327066825 p^{46} T^{13} + 96060641871340 p^{54} T^{14} - 8028925 p^{63} T^{15} + p^{72} T^{16} \) | |
| 31 | \( 1 - 7108836 T + 180330448675160 T^{2} - \)\(10\!\cdots\!08\)\( T^{3} + \)\(14\!\cdots\!00\)\( T^{4} - \)\(71\!\cdots\!00\)\( T^{5} + \)\(70\!\cdots\!40\)\( T^{6} - \)\(93\!\cdots\!72\)\( p T^{7} + \)\(22\!\cdots\!38\)\( T^{8} - \)\(93\!\cdots\!72\)\( p^{10} T^{9} + \)\(70\!\cdots\!40\)\( p^{18} T^{10} - \)\(71\!\cdots\!00\)\( p^{27} T^{11} + \)\(14\!\cdots\!00\)\( p^{36} T^{12} - \)\(10\!\cdots\!08\)\( p^{45} T^{13} + 180330448675160 p^{54} T^{14} - 7108836 p^{63} T^{15} + p^{72} T^{16} \) | |
| 37 | \( 1 - 13713844 T + 716312888466848 T^{2} - \)\(98\!\cdots\!04\)\( T^{3} + \)\(25\!\cdots\!16\)\( T^{4} - \)\(31\!\cdots\!32\)\( T^{5} + \)\(58\!\cdots\!72\)\( T^{6} - \)\(62\!\cdots\!28\)\( T^{7} + \)\(92\!\cdots\!14\)\( T^{8} - \)\(62\!\cdots\!28\)\( p^{9} T^{9} + \)\(58\!\cdots\!72\)\( p^{18} T^{10} - \)\(31\!\cdots\!32\)\( p^{27} T^{11} + \)\(25\!\cdots\!16\)\( p^{36} T^{12} - \)\(98\!\cdots\!04\)\( p^{45} T^{13} + 716312888466848 p^{54} T^{14} - 13713844 p^{63} T^{15} + p^{72} T^{16} \) | |
| 41 | \( 1 - 18944338 T + 1548279151289836 T^{2} - \)\(29\!\cdots\!58\)\( T^{3} + \)\(12\!\cdots\!12\)\( T^{4} - \)\(22\!\cdots\!02\)\( T^{5} + \)\(69\!\cdots\!96\)\( T^{6} - \)\(10\!\cdots\!62\)\( T^{7} + \)\(26\!\cdots\!10\)\( T^{8} - \)\(10\!\cdots\!62\)\( p^{9} T^{9} + \)\(69\!\cdots\!96\)\( p^{18} T^{10} - \)\(22\!\cdots\!02\)\( p^{27} T^{11} + \)\(12\!\cdots\!12\)\( p^{36} T^{12} - \)\(29\!\cdots\!58\)\( p^{45} T^{13} + 1548279151289836 p^{54} T^{14} - 18944338 p^{63} T^{15} + p^{72} T^{16} \) | |
| 43 | \( 1 + 685158 T + 14787641546920 T^{2} - \)\(25\!\cdots\!26\)\( T^{3} + \)\(35\!\cdots\!92\)\( T^{4} + \)\(18\!\cdots\!18\)\( T^{5} + \)\(43\!\cdots\!84\)\( T^{6} - \)\(74\!\cdots\!26\)\( T^{7} - \)\(40\!\cdots\!26\)\( T^{8} - \)\(74\!\cdots\!26\)\( p^{9} T^{9} + \)\(43\!\cdots\!84\)\( p^{18} T^{10} + \)\(18\!\cdots\!18\)\( p^{27} T^{11} + \)\(35\!\cdots\!92\)\( p^{36} T^{12} - \)\(25\!\cdots\!26\)\( p^{45} T^{13} + 14787641546920 p^{54} T^{14} + 685158 p^{63} T^{15} + p^{72} T^{16} \) | |
| 47 | \( 1 + 15779003 T + 2672024714789650 T^{2} - \)\(11\!\cdots\!47\)\( T^{3} + \)\(52\!\cdots\!01\)\( T^{4} - \)\(18\!\cdots\!60\)\( T^{5} + \)\(86\!\cdots\!14\)\( T^{6} - \)\(55\!\cdots\!86\)\( T^{7} + \)\(10\!\cdots\!04\)\( T^{8} - \)\(55\!\cdots\!86\)\( p^{9} T^{9} + \)\(86\!\cdots\!14\)\( p^{18} T^{10} - \)\(18\!\cdots\!60\)\( p^{27} T^{11} + \)\(52\!\cdots\!01\)\( p^{36} T^{12} - \)\(11\!\cdots\!47\)\( p^{45} T^{13} + 2672024714789650 p^{54} T^{14} + 15779003 p^{63} T^{15} + p^{72} T^{16} \) | |
| 53 | \( 1 + 25340910 T + 14365095400372884 T^{2} + \)\(26\!\cdots\!58\)\( T^{3} + \)\(96\!\cdots\!80\)\( T^{4} + \)\(86\!\cdots\!38\)\( T^{5} + \)\(41\!\cdots\!64\)\( T^{6} + \)\(38\!\cdots\!14\)\( T^{7} + \)\(14\!\cdots\!86\)\( T^{8} + \)\(38\!\cdots\!14\)\( p^{9} T^{9} + \)\(41\!\cdots\!64\)\( p^{18} T^{10} + \)\(86\!\cdots\!38\)\( p^{27} T^{11} + \)\(96\!\cdots\!80\)\( p^{36} T^{12} + \)\(26\!\cdots\!58\)\( p^{45} T^{13} + 14365095400372884 p^{54} T^{14} + 25340910 p^{63} T^{15} + p^{72} T^{16} \) | |
| 59 | \( 1 + 2602496 T + 30575857756726552 T^{2} - \)\(16\!\cdots\!76\)\( T^{3} + \)\(46\!\cdots\!44\)\( T^{4} + \)\(31\!\cdots\!44\)\( T^{5} + \)\(53\!\cdots\!40\)\( T^{6} + \)\(61\!\cdots\!08\)\( T^{7} + \)\(50\!\cdots\!42\)\( T^{8} + \)\(61\!\cdots\!08\)\( p^{9} T^{9} + \)\(53\!\cdots\!40\)\( p^{18} T^{10} + \)\(31\!\cdots\!44\)\( p^{27} T^{11} + \)\(46\!\cdots\!44\)\( p^{36} T^{12} - \)\(16\!\cdots\!76\)\( p^{45} T^{13} + 30575857756726552 p^{54} T^{14} + 2602496 p^{63} T^{15} + p^{72} T^{16} \) | |
| 61 | \( 1 - 35623262 T + 29598939153537852 T^{2} - \)\(14\!\cdots\!90\)\( T^{3} + \)\(66\!\cdots\!32\)\( T^{4} - \)\(34\!\cdots\!50\)\( T^{5} + \)\(11\!\cdots\!08\)\( T^{6} - \)\(52\!\cdots\!90\)\( T^{7} + \)\(14\!\cdots\!54\)\( T^{8} - \)\(52\!\cdots\!90\)\( p^{9} T^{9} + \)\(11\!\cdots\!08\)\( p^{18} T^{10} - \)\(34\!\cdots\!50\)\( p^{27} T^{11} + \)\(66\!\cdots\!32\)\( p^{36} T^{12} - \)\(14\!\cdots\!90\)\( p^{45} T^{13} + 29598939153537852 p^{54} T^{14} - 35623262 p^{63} T^{15} + p^{72} T^{16} \) | |
| 67 | \( 1 + 148781404 T + 81135986832669032 T^{2} + \)\(15\!\cdots\!12\)\( T^{3} + \)\(46\!\cdots\!04\)\( T^{4} + \)\(74\!\cdots\!76\)\( T^{5} + \)\(19\!\cdots\!88\)\( T^{6} + \)\(28\!\cdots\!80\)\( T^{7} + \)\(57\!\cdots\!78\)\( T^{8} + \)\(28\!\cdots\!80\)\( p^{9} T^{9} + \)\(19\!\cdots\!88\)\( p^{18} T^{10} + \)\(74\!\cdots\!76\)\( p^{27} T^{11} + \)\(46\!\cdots\!04\)\( p^{36} T^{12} + \)\(15\!\cdots\!12\)\( p^{45} T^{13} + 81135986832669032 p^{54} T^{14} + 148781404 p^{63} T^{15} + p^{72} T^{16} \) | |
| 71 | \( 1 - 1278888 T + 217231812382853944 T^{2} + \)\(63\!\cdots\!72\)\( T^{3} + \)\(23\!\cdots\!12\)\( T^{4} + \)\(15\!\cdots\!52\)\( T^{5} + \)\(16\!\cdots\!04\)\( T^{6} - \)\(48\!\cdots\!80\)\( T^{7} + \)\(89\!\cdots\!50\)\( T^{8} - \)\(48\!\cdots\!80\)\( p^{9} T^{9} + \)\(16\!\cdots\!04\)\( p^{18} T^{10} + \)\(15\!\cdots\!52\)\( p^{27} T^{11} + \)\(23\!\cdots\!12\)\( p^{36} T^{12} + \)\(63\!\cdots\!72\)\( p^{45} T^{13} + 217231812382853944 p^{54} T^{14} - 1278888 p^{63} T^{15} + p^{72} T^{16} \) | |
| 73 | \( 1 - 531792684 T + 271333868663607808 T^{2} - \)\(77\!\cdots\!68\)\( T^{3} + \)\(24\!\cdots\!92\)\( T^{4} - \)\(51\!\cdots\!04\)\( T^{5} + \)\(14\!\cdots\!96\)\( T^{6} - \)\(27\!\cdots\!56\)\( T^{7} + \)\(82\!\cdots\!30\)\( T^{8} - \)\(27\!\cdots\!56\)\( p^{9} T^{9} + \)\(14\!\cdots\!96\)\( p^{18} T^{10} - \)\(51\!\cdots\!04\)\( p^{27} T^{11} + \)\(24\!\cdots\!92\)\( p^{36} T^{12} - \)\(77\!\cdots\!68\)\( p^{45} T^{13} + 271333868663607808 p^{54} T^{14} - 531792684 p^{63} T^{15} + p^{72} T^{16} \) | |
| 79 | \( 1 + 103764351 T + 568744754164674322 T^{2} + \)\(45\!\cdots\!33\)\( T^{3} + \)\(16\!\cdots\!17\)\( T^{4} + \)\(85\!\cdots\!76\)\( T^{5} + \)\(30\!\cdots\!78\)\( T^{6} + \)\(10\!\cdots\!38\)\( T^{7} + \)\(42\!\cdots\!00\)\( T^{8} + \)\(10\!\cdots\!38\)\( p^{9} T^{9} + \)\(30\!\cdots\!78\)\( p^{18} T^{10} + \)\(85\!\cdots\!76\)\( p^{27} T^{11} + \)\(16\!\cdots\!17\)\( p^{36} T^{12} + \)\(45\!\cdots\!33\)\( p^{45} T^{13} + 568744754164674322 p^{54} T^{14} + 103764351 p^{63} T^{15} + p^{72} T^{16} \) | |
| 83 | \( 1 + 248063016 T + 557000710463471160 T^{2} + \)\(10\!\cdots\!08\)\( T^{3} + \)\(21\!\cdots\!08\)\( T^{4} + \)\(38\!\cdots\!24\)\( T^{5} + \)\(56\!\cdots\!92\)\( T^{6} + \)\(90\!\cdots\!24\)\( T^{7} + \)\(11\!\cdots\!02\)\( T^{8} + \)\(90\!\cdots\!24\)\( p^{9} T^{9} + \)\(56\!\cdots\!92\)\( p^{18} T^{10} + \)\(38\!\cdots\!24\)\( p^{27} T^{11} + \)\(21\!\cdots\!08\)\( p^{36} T^{12} + \)\(10\!\cdots\!08\)\( p^{45} T^{13} + 557000710463471160 p^{54} T^{14} + 248063016 p^{63} T^{15} + p^{72} T^{16} \) | |
| 89 | \( 1 - 160930110 T + 1962806865093709780 T^{2} - \)\(31\!\cdots\!42\)\( T^{3} + \)\(19\!\cdots\!12\)\( T^{4} - \)\(28\!\cdots\!98\)\( T^{5} + \)\(11\!\cdots\!44\)\( T^{6} - \)\(15\!\cdots\!66\)\( T^{7} + \)\(48\!\cdots\!82\)\( T^{8} - \)\(15\!\cdots\!66\)\( p^{9} T^{9} + \)\(11\!\cdots\!44\)\( p^{18} T^{10} - \)\(28\!\cdots\!98\)\( p^{27} T^{11} + \)\(19\!\cdots\!12\)\( p^{36} T^{12} - \)\(31\!\cdots\!42\)\( p^{45} T^{13} + 1962806865093709780 p^{54} T^{14} - 160930110 p^{63} T^{15} + p^{72} T^{16} \) | |
| 97 | \( 1 - 164147671 T + 1852835479894827620 T^{2} + \)\(43\!\cdots\!49\)\( T^{3} + \)\(16\!\cdots\!69\)\( T^{4} + \)\(10\!\cdots\!44\)\( T^{5} + \)\(16\!\cdots\!10\)\( T^{6} + \)\(10\!\cdots\!38\)\( T^{7} + \)\(14\!\cdots\!12\)\( T^{8} + \)\(10\!\cdots\!38\)\( p^{9} T^{9} + \)\(16\!\cdots\!10\)\( p^{18} T^{10} + \)\(10\!\cdots\!44\)\( p^{27} T^{11} + \)\(16\!\cdots\!69\)\( p^{36} T^{12} + \)\(43\!\cdots\!49\)\( p^{45} T^{13} + 1852835479894827620 p^{54} T^{14} - 164147671 p^{63} T^{15} + p^{72} T^{16} \) | |
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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
−3.72302141890050634063902802756, −3.29487555334772705112788932604, −3.11344031957192319994269817589, −3.09997975833431241685842970441, −3.00493540797107628698092996442, −2.94168462262710818357943412925, −2.85736092808430059483470867590, −2.80182092010497923089113974533, −2.71932489319498936870272632967, −2.30556549313188131116803301810, −2.10098970111437159818359307001, −2.01540906323016769733659973636, −1.98059040081111444598629056090, −1.79658224710635055454190033757, −1.79180717472959508025291097674, −1.63291307912524217989739798758, −1.35215105504757108180780189152, −1.17914720089644017474930282745, −0.970921328745610363423885744529, −0.74381033554870365482641365305, −0.73807183361948389627019070060, −0.69473904287614147826612550098, −0.66656772606958150733547490923, −0.63469587648455599121329380074, −0.14922203273739299691922721213, 0.14922203273739299691922721213, 0.63469587648455599121329380074, 0.66656772606958150733547490923, 0.69473904287614147826612550098, 0.73807183361948389627019070060, 0.74381033554870365482641365305, 0.970921328745610363423885744529, 1.17914720089644017474930282745, 1.35215105504757108180780189152, 1.63291307912524217989739798758, 1.79180717472959508025291097674, 1.79658224710635055454190033757, 1.98059040081111444598629056090, 2.01540906323016769733659973636, 2.10098970111437159818359307001, 2.30556549313188131116803301810, 2.71932489319498936870272632967, 2.80182092010497923089113974533, 2.85736092808430059483470867590, 2.94168462262710818357943412925, 3.00493540797107628698092996442, 3.09997975833431241685842970441, 3.11344031957192319994269817589, 3.29487555334772705112788932604, 3.72302141890050634063902802756
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.