Dirichlet series
| L(s) = 1 | − 84·5-s + 20·7-s + 60·8-s − 288·11-s − 340·13-s + 523·16-s + 900·17-s − 1.56e3·23-s + 2.92e3·25-s − 512·31-s + 420·32-s − 1.68e3·35-s − 3.82e3·37-s − 5.04e3·40-s − 2.71e3·41-s − 1.24e3·43-s + 4.80e3·47-s + 200·49-s + 1.02e3·53-s + 2.41e4·55-s + 1.20e3·56-s − 4.76e3·61-s + 1.80e3·64-s + 2.85e4·65-s − 8.92e3·67-s + 7.53e3·71-s + 1.16e4·73-s + ⋯ |
| L(s) = 1 | − 3.35·5-s + 0.408·7-s + 0.937·8-s − 2.38·11-s − 2.01·13-s + 2.04·16-s + 3.11·17-s − 2.94·23-s + 4.68·25-s − 0.532·31-s + 0.410·32-s − 1.37·35-s − 2.79·37-s − 3.14·40-s − 1.61·41-s − 0.670·43-s + 2.17·47-s + 0.0832·49-s + 0.363·53-s + 7.99·55-s + 0.382·56-s − 1.27·61-s + 0.439·64-s + 6.75·65-s − 1.98·67-s + 1.49·71-s + 2.17·73-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 5^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(33.4106\) |
| Root analytic conductor: | \(1.24521\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 5^{8} ,\ ( \ : [2]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(0.1748437711\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1748437711\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p^{6} T^{4} )^{2} \) |
| 5 | \( 1 + 84 T + 826 p T^{2} + 1164 p^{3} T^{3} + 32298 p^{3} T^{4} + 1164 p^{7} T^{5} + 826 p^{9} T^{6} + 84 p^{12} T^{7} + p^{16} T^{8} \) | |
| good | 2 | \( 1 - 15 p^{2} T^{3} - 523 T^{4} - 105 p^{2} T^{5} + 225 p^{3} T^{6} + 1185 p^{3} T^{7} + 45201 p^{2} T^{8} + 1185 p^{7} T^{9} + 225 p^{11} T^{10} - 105 p^{14} T^{11} - 523 p^{16} T^{12} - 15 p^{22} T^{13} + p^{32} T^{16} \) |
| 7 | \( 1 - 20 T + 200 T^{2} + 199700 T^{3} - 889600 p T^{4} + 79376300 T^{5} + 19597959000 T^{6} - 40112935980 T^{7} + 30640225075198 T^{8} - 40112935980 p^{4} T^{9} + 19597959000 p^{8} T^{10} + 79376300 p^{12} T^{11} - 889600 p^{17} T^{12} + 199700 p^{20} T^{13} + 200 p^{24} T^{14} - 20 p^{28} T^{15} + p^{32} T^{16} \) | |
| 11 | \( ( 1 + 144 T + 3802 p T^{2} + 6150240 T^{3} + 823068186 T^{4} + 6150240 p^{4} T^{5} + 3802 p^{9} T^{6} + 144 p^{12} T^{7} + p^{16} T^{8} )^{2} \) | |
| 13 | \( 1 + 340 T + 57800 T^{2} + 10577660 T^{3} + 3418178672 T^{4} + 809516994020 T^{5} + 133608496263000 T^{6} + 23843087441136780 T^{7} + 4251050341381084894 T^{8} + 23843087441136780 p^{4} T^{9} + 133608496263000 p^{8} T^{10} + 809516994020 p^{12} T^{11} + 3418178672 p^{16} T^{12} + 10577660 p^{20} T^{13} + 57800 p^{24} T^{14} + 340 p^{28} T^{15} + p^{32} T^{16} \) | |
| 17 | \( 1 - 900 T + 405000 T^{2} - 162344700 T^{3} + 70846529936 T^{4} - 1587804473700 p T^{5} + 8778464632575000 T^{6} - 2859179891204281500 T^{7} + \)\(88\!\cdots\!86\)\( T^{8} - 2859179891204281500 p^{4} T^{9} + 8778464632575000 p^{8} T^{10} - 1587804473700 p^{13} T^{11} + 70846529936 p^{16} T^{12} - 162344700 p^{20} T^{13} + 405000 p^{24} T^{14} - 900 p^{28} T^{15} + p^{32} T^{16} \) | |
| 19 | \( 1 - 475280 T^{2} + 118243003996 T^{4} - 19617731389431920 T^{6} + \)\(27\!\cdots\!66\)\( T^{8} - 19617731389431920 p^{8} T^{10} + 118243003996 p^{16} T^{12} - 475280 p^{24} T^{14} + p^{32} T^{16} \) | |
| 23 | \( 1 + 1560 T + 1216800 T^{2} + 797095800 T^{3} + 382745999300 T^{4} + 92493403026600 T^{5} - 3754766037924000 T^{6} - 28706182737150138360 T^{7} - \)\(24\!\cdots\!22\)\( T^{8} - 28706182737150138360 p^{4} T^{9} - 3754766037924000 p^{8} T^{10} + 92493403026600 p^{12} T^{11} + 382745999300 p^{16} T^{12} + 797095800 p^{20} T^{13} + 1216800 p^{24} T^{14} + 1560 p^{28} T^{15} + p^{32} T^{16} \) | |
| 29 | \( 1 - 3993236 T^{2} + 7931637401512 T^{4} - 9869586507702691580 T^{6} + \)\(83\!\cdots\!06\)\( T^{8} - 9869586507702691580 p^{8} T^{10} + 7931637401512 p^{16} T^{12} - 3993236 p^{24} T^{14} + p^{32} T^{16} \) | |
| 31 | \( ( 1 + 256 T + 1349068 T^{2} - 180990272 T^{3} + 1174676381974 T^{4} - 180990272 p^{4} T^{5} + 1349068 p^{8} T^{6} + 256 p^{12} T^{7} + p^{16} T^{8} )^{2} \) | |
| 37 | \( 1 + 3820 T + 7296200 T^{2} + 10974405380 T^{3} + 13446367756400 T^{4} + 10277980840235420 T^{5} + 1373285107740096600 T^{6} - \)\(20\!\cdots\!20\)\( T^{7} - \)\(46\!\cdots\!82\)\( T^{8} - \)\(20\!\cdots\!20\)\( p^{4} T^{9} + 1373285107740096600 p^{8} T^{10} + 10277980840235420 p^{12} T^{11} + 13446367756400 p^{16} T^{12} + 10974405380 p^{20} T^{13} + 7296200 p^{24} T^{14} + 3820 p^{28} T^{15} + p^{32} T^{16} \) | |
| 41 | \( ( 1 + 1356 T + 8325968 T^{2} + 6593365668 T^{3} + 29888933841054 T^{4} + 6593365668 p^{4} T^{5} + 8325968 p^{8} T^{6} + 1356 p^{12} T^{7} + p^{16} T^{8} )^{2} \) | |
| 43 | \( 1 + 1240 T + 768800 T^{2} + 4155585560 T^{3} + 45689035751396 T^{4} + 43012988909739080 T^{5} + 14518562052808800 p^{2} T^{6} + \)\(33\!\cdots\!40\)\( p T^{7} + \)\(79\!\cdots\!06\)\( T^{8} + \)\(33\!\cdots\!40\)\( p^{5} T^{9} + 14518562052808800 p^{10} T^{10} + 43012988909739080 p^{12} T^{11} + 45689035751396 p^{16} T^{12} + 4155585560 p^{20} T^{13} + 768800 p^{24} T^{14} + 1240 p^{28} T^{15} + p^{32} T^{16} \) | |
| 47 | \( 1 - 4800 T + 11520000 T^{2} - 34026427200 T^{3} + 139196198193956 T^{4} - 374777003457297600 T^{5} + \)\(77\!\cdots\!00\)\( T^{6} - \)\(20\!\cdots\!00\)\( T^{7} + \)\(52\!\cdots\!26\)\( T^{8} - \)\(20\!\cdots\!00\)\( p^{4} T^{9} + \)\(77\!\cdots\!00\)\( p^{8} T^{10} - 374777003457297600 p^{12} T^{11} + 139196198193956 p^{16} T^{12} - 34026427200 p^{20} T^{13} + 11520000 p^{24} T^{14} - 4800 p^{28} T^{15} + p^{32} T^{16} \) | |
| 53 | \( 1 - 1020 T + 520200 T^{2} - 3711679620 T^{3} + 27373065594800 T^{4} - 31066466654514780 T^{5} + 24336610065951787800 T^{6} - \)\(12\!\cdots\!80\)\( T^{7} - \)\(51\!\cdots\!42\)\( T^{8} - \)\(12\!\cdots\!80\)\( p^{4} T^{9} + 24336610065951787800 p^{8} T^{10} - 31066466654514780 p^{12} T^{11} + 27373065594800 p^{16} T^{12} - 3711679620 p^{20} T^{13} + 520200 p^{24} T^{14} - 1020 p^{28} T^{15} + p^{32} T^{16} \) | |
| 59 | \( 1 - 68266460 T^{2} + 2308748573454136 T^{4} - \)\(49\!\cdots\!40\)\( T^{6} + \)\(71\!\cdots\!06\)\( T^{8} - \)\(49\!\cdots\!40\)\( p^{8} T^{10} + 2308748573454136 p^{16} T^{12} - 68266460 p^{24} T^{14} + p^{32} T^{16} \) | |
| 61 | \( ( 1 + 2380 T + 35645776 T^{2} + 28234432420 T^{3} + 552631006355806 T^{4} + 28234432420 p^{4} T^{5} + 35645776 p^{8} T^{6} + 2380 p^{12} T^{7} + p^{16} T^{8} )^{2} \) | |
| 67 | \( 1 + 8920 T + 39783200 T^{2} + 240429760280 T^{3} + 743487179691236 T^{4} - 1082655630131333560 T^{5} - \)\(10\!\cdots\!00\)\( T^{6} - \)\(87\!\cdots\!40\)\( T^{7} - \)\(61\!\cdots\!14\)\( T^{8} - \)\(87\!\cdots\!40\)\( p^{4} T^{9} - \)\(10\!\cdots\!00\)\( p^{8} T^{10} - 1082655630131333560 p^{12} T^{11} + 743487179691236 p^{16} T^{12} + 240429760280 p^{20} T^{13} + 39783200 p^{24} T^{14} + 8920 p^{28} T^{15} + p^{32} T^{16} \) | |
| 71 | \( ( 1 - 3768 T + 59990708 T^{2} - 219866356776 T^{3} + 1753029620362470 T^{4} - 219866356776 p^{4} T^{5} + 59990708 p^{8} T^{6} - 3768 p^{12} T^{7} + p^{16} T^{8} )^{2} \) | |
| 73 | \( 1 - 11600 T + 67280000 T^{2} - 460834857520 T^{3} + 3747670062415100 T^{4} - 21684328079009172880 T^{5} + \)\(10\!\cdots\!00\)\( T^{6} - \)\(63\!\cdots\!00\)\( T^{7} + \)\(38\!\cdots\!78\)\( T^{8} - \)\(63\!\cdots\!00\)\( p^{4} T^{9} + \)\(10\!\cdots\!00\)\( p^{8} T^{10} - 21684328079009172880 p^{12} T^{11} + 3747670062415100 p^{16} T^{12} - 460834857520 p^{20} T^{13} + 67280000 p^{24} T^{14} - 11600 p^{28} T^{15} + p^{32} T^{16} \) | |
| 79 | \( 1 - 192979448 T^{2} + 19833375036960508 T^{4} - \)\(13\!\cdots\!96\)\( T^{6} + \)\(60\!\cdots\!70\)\( T^{8} - \)\(13\!\cdots\!96\)\( p^{8} T^{10} + 19833375036960508 p^{16} T^{12} - 192979448 p^{24} T^{14} + p^{32} T^{16} \) | |
| 83 | \( 1 + 32400 T + 524880000 T^{2} + 6014129561040 T^{3} + 60088627758116132 T^{4} + \)\(56\!\cdots\!80\)\( T^{5} + \)\(48\!\cdots\!00\)\( T^{6} + \)\(37\!\cdots\!80\)\( T^{7} + \)\(27\!\cdots\!14\)\( T^{8} + \)\(37\!\cdots\!80\)\( p^{4} T^{9} + \)\(48\!\cdots\!00\)\( p^{8} T^{10} + \)\(56\!\cdots\!80\)\( p^{12} T^{11} + 60088627758116132 p^{16} T^{12} + 6014129561040 p^{20} T^{13} + 524880000 p^{24} T^{14} + 32400 p^{28} T^{15} + p^{32} T^{16} \) | |
| 89 | \( 1 - 372973760 T^{2} + 67192056713379196 T^{4} - \)\(84\!\cdots\!60\)\( p T^{6} + \)\(56\!\cdots\!26\)\( T^{8} - \)\(84\!\cdots\!60\)\( p^{9} T^{10} + 67192056713379196 p^{16} T^{12} - 372973760 p^{24} T^{14} + p^{32} T^{16} \) | |
| 97 | \( 1 - 58640 T + 1719324800 T^{2} - 34474607566960 T^{3} + 547427197512783356 T^{4} - \)\(74\!\cdots\!80\)\( T^{5} + \)\(91\!\cdots\!00\)\( T^{6} - \)\(10\!\cdots\!20\)\( T^{7} + \)\(10\!\cdots\!26\)\( T^{8} - \)\(10\!\cdots\!20\)\( p^{4} T^{9} + \)\(91\!\cdots\!00\)\( p^{8} T^{10} - \)\(74\!\cdots\!80\)\( p^{12} T^{11} + 547427197512783356 p^{16} T^{12} - 34474607566960 p^{20} T^{13} + 1719324800 p^{24} T^{14} - 58640 p^{28} T^{15} + p^{32} T^{16} \) | |
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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
−9.185104374136714160875130453939, −8.724672166095888282844361314084, −8.295989096159804705762791380459, −8.176273455016128514592519486939, −8.144737034280359298080560943850, −8.050361921473410093385612610457, −7.62942490501046037941408650844, −7.53879452563094108768082697690, −7.43025333046922290462064908300, −7.37330813500084200737981940299, −7.18272803020851870699422545652, −6.50061281637004492324347333038, −5.99442174008626375324395730642, −5.78953451208679608495895984839, −5.36473749132794093293437779109, −5.18962143882425666349782604269, −4.97046087383147559606186351794, −4.74065633232060740198419234656, −3.89992702479605424022268000691, −3.87283545399608238211153861663, −3.71299286844569129789295370472, −3.17127249986304817298574641530, −2.69053713140465725591469529110, −1.68035869254979326225587030828, −0.22968909003146704177991085435, 0.22968909003146704177991085435, 1.68035869254979326225587030828, 2.69053713140465725591469529110, 3.17127249986304817298574641530, 3.71299286844569129789295370472, 3.87283545399608238211153861663, 3.89992702479605424022268000691, 4.74065633232060740198419234656, 4.97046087383147559606186351794, 5.18962143882425666349782604269, 5.36473749132794093293437779109, 5.78953451208679608495895984839, 5.99442174008626375324395730642, 6.50061281637004492324347333038, 7.18272803020851870699422545652, 7.37330813500084200737981940299, 7.43025333046922290462064908300, 7.53879452563094108768082697690, 7.62942490501046037941408650844, 8.050361921473410093385612610457, 8.144737034280359298080560943850, 8.176273455016128514592519486939, 8.295989096159804705762791380459, 8.724672166095888282844361314084, 9.185104374136714160875130453939
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.