Dirichlet series
| L(s) = 1 | − 128·2-s + 266·3-s + 6.14e3·4-s + 7.50e3·5-s − 3.40e4·6-s − 4.22e4·7-s + 3.27e5·9-s − 9.60e5·10-s + 2.13e5·11-s + 1.63e6·12-s − 2.60e6·13-s + 5.40e6·14-s + 1.99e6·15-s − 1.57e7·16-s + 8.85e6·17-s − 4.19e7·18-s + 7.23e6·19-s + 4.61e7·20-s − 1.12e7·21-s − 2.72e7·22-s − 1.06e7·23-s + 6.61e7·25-s + 3.34e8·26-s + 1.18e8·27-s − 2.59e8·28-s − 2.21e8·29-s − 2.55e8·30-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 0.631·3-s + 3·4-s + 1.07·5-s − 1.78·6-s − 0.949·7-s + 1.85·9-s − 3.03·10-s + 0.398·11-s + 1.89·12-s − 1.94·13-s + 2.68·14-s + 0.678·15-s − 3.75·16-s + 1.51·17-s − 5.23·18-s + 0.670·19-s + 3.22·20-s − 0.600·21-s − 1.12·22-s − 0.344·23-s + 1.35·25-s + 5.51·26-s + 1.59·27-s − 2.84·28-s − 2.00·29-s − 1.91·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.79252\times 10^{8}\) |
| Root analytic conductor: | \(3.27975\) |
| Motivic weight: | \(11\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 7^{8} ,\ ( \ : [11/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(6)\) | \(\approx\) | \(0.03811181091\) |
| \(L(\frac12)\) | \(\approx\) | \(0.03811181091\) |
| \(L(\frac{13}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{5} T + p^{10} T^{2} )^{4} \) |
| 7 | \( 1 + 6032 p T + 136651276 p^{2} T^{2} + 13729789264 p^{5} T^{3} + 66872750054 p^{10} T^{4} + 13729789264 p^{16} T^{5} + 136651276 p^{24} T^{6} + 6032 p^{34} T^{7} + p^{44} T^{8} \) | |
| good | 3 | \( 1 - 266 T - 257156 T^{2} + 12308576 p T^{3} + 1334453311 p^{2} T^{4} + 43944938660 p^{5} T^{5} - 6032510386856 p^{6} T^{6} - 96956124948802 p^{9} T^{7} + 33644498918829976 p^{10} T^{8} - 96956124948802 p^{20} T^{9} - 6032510386856 p^{28} T^{10} + 43944938660 p^{38} T^{11} + 1334453311 p^{46} T^{12} + 12308576 p^{56} T^{13} - 257156 p^{66} T^{14} - 266 p^{77} T^{15} + p^{88} T^{16} \) |
| 5 | \( 1 - 7504 T - 9797614 T^{2} + 184130226896 p T^{3} - 271326540460139 p^{2} T^{4} + 49585809393668184 p^{3} T^{5} + \)\(27\!\cdots\!02\)\( p^{4} T^{6} - \)\(20\!\cdots\!36\)\( p^{7} T^{7} + \)\(42\!\cdots\!04\)\( p^{6} T^{8} - \)\(20\!\cdots\!36\)\( p^{18} T^{9} + \)\(27\!\cdots\!02\)\( p^{26} T^{10} + 49585809393668184 p^{36} T^{11} - 271326540460139 p^{46} T^{12} + 184130226896 p^{56} T^{13} - 9797614 p^{66} T^{14} - 7504 p^{77} T^{15} + p^{88} T^{16} \) | |
| 11 | \( 1 - 19366 p T - 542750496220 T^{2} + 3403652653052816 p^{2} T^{3} + \)\(11\!\cdots\!55\)\( T^{4} - \)\(13\!\cdots\!32\)\( p^{2} T^{5} + \)\(51\!\cdots\!92\)\( T^{6} + \)\(26\!\cdots\!50\)\( p T^{7} - \)\(24\!\cdots\!56\)\( T^{8} + \)\(26\!\cdots\!50\)\( p^{12} T^{9} + \)\(51\!\cdots\!92\)\( p^{22} T^{10} - \)\(13\!\cdots\!32\)\( p^{35} T^{11} + \)\(11\!\cdots\!55\)\( p^{44} T^{12} + 3403652653052816 p^{57} T^{13} - 542750496220 p^{66} T^{14} - 19366 p^{78} T^{15} + p^{88} T^{16} \) | |
| 13 | \( ( 1 + 1304856 T + 3341702299708 T^{2} + 4230180671651253256 T^{3} + \)\(10\!\cdots\!54\)\( T^{4} + 4230180671651253256 p^{11} T^{5} + 3341702299708 p^{22} T^{6} + 1304856 p^{33} T^{7} + p^{44} T^{8} )^{2} \) | |
| 17 | \( 1 - 8854244 T - 38385748499914 T^{2} + \)\(25\!\cdots\!28\)\( T^{3} + \)\(24\!\cdots\!33\)\( T^{4} - \)\(11\!\cdots\!24\)\( T^{5} - \)\(14\!\cdots\!82\)\( T^{6} + \)\(85\!\cdots\!08\)\( T^{7} + \)\(42\!\cdots\!76\)\( T^{8} + \)\(85\!\cdots\!08\)\( p^{11} T^{9} - \)\(14\!\cdots\!82\)\( p^{22} T^{10} - \)\(11\!\cdots\!24\)\( p^{33} T^{11} + \)\(24\!\cdots\!33\)\( p^{44} T^{12} + \)\(25\!\cdots\!28\)\( p^{55} T^{13} - 38385748499914 p^{66} T^{14} - 8854244 p^{77} T^{15} + p^{88} T^{16} \) | |
| 19 | \( 1 - 380674 p T - 46162362108780 T^{2} - 48760857695754818096 T^{3} - \)\(12\!\cdots\!65\)\( T^{4} + \)\(51\!\cdots\!52\)\( T^{5} + \)\(16\!\cdots\!16\)\( T^{6} - \)\(16\!\cdots\!10\)\( T^{7} - \)\(90\!\cdots\!48\)\( T^{8} - \)\(16\!\cdots\!10\)\( p^{11} T^{9} + \)\(16\!\cdots\!16\)\( p^{22} T^{10} + \)\(51\!\cdots\!52\)\( p^{33} T^{11} - \)\(12\!\cdots\!65\)\( p^{44} T^{12} - 48760857695754818096 p^{55} T^{13} - 46162362108780 p^{66} T^{14} - 380674 p^{78} T^{15} + p^{88} T^{16} \) | |
| 23 | \( 1 + 10649134 T - 2162704629228784 T^{2} - \)\(24\!\cdots\!08\)\( T^{3} + \)\(18\!\cdots\!39\)\( T^{4} + \)\(61\!\cdots\!56\)\( p T^{5} - \)\(24\!\cdots\!36\)\( T^{6} - \)\(66\!\cdots\!58\)\( T^{7} + \)\(32\!\cdots\!68\)\( T^{8} - \)\(66\!\cdots\!58\)\( p^{11} T^{9} - \)\(24\!\cdots\!36\)\( p^{22} T^{10} + \)\(61\!\cdots\!56\)\( p^{34} T^{11} + \)\(18\!\cdots\!39\)\( p^{44} T^{12} - \)\(24\!\cdots\!08\)\( p^{55} T^{13} - 2162704629228784 p^{66} T^{14} + 10649134 p^{77} T^{15} + p^{88} T^{16} \) | |
| 29 | \( ( 1 + 110707288 T + 48591603349178876 T^{2} + \)\(39\!\cdots\!48\)\( T^{3} + \)\(88\!\cdots\!10\)\( T^{4} + \)\(39\!\cdots\!48\)\( p^{11} T^{5} + 48591603349178876 p^{22} T^{6} + 110707288 p^{33} T^{7} + p^{44} T^{8} )^{2} \) | |
| 31 | \( 1 - 486231270 T + 90103516260472424 T^{2} - \)\(99\!\cdots\!56\)\( T^{3} + \)\(11\!\cdots\!91\)\( T^{4} + \)\(20\!\cdots\!76\)\( T^{5} - \)\(47\!\cdots\!64\)\( T^{6} + \)\(76\!\cdots\!54\)\( T^{7} - \)\(85\!\cdots\!76\)\( T^{8} + \)\(76\!\cdots\!54\)\( p^{11} T^{9} - \)\(47\!\cdots\!64\)\( p^{22} T^{10} + \)\(20\!\cdots\!76\)\( p^{33} T^{11} + \)\(11\!\cdots\!91\)\( p^{44} T^{12} - \)\(99\!\cdots\!56\)\( p^{55} T^{13} + 90103516260472424 p^{66} T^{14} - 486231270 p^{77} T^{15} + p^{88} T^{16} \) | |
| 37 | \( 1 - 463131040 T - 156603083235116718 T^{2} - \)\(10\!\cdots\!92\)\( T^{3} + \)\(78\!\cdots\!17\)\( T^{4} + \)\(20\!\cdots\!76\)\( T^{5} + \)\(59\!\cdots\!54\)\( T^{6} - \)\(48\!\cdots\!16\)\( T^{7} - \)\(13\!\cdots\!56\)\( T^{8} - \)\(48\!\cdots\!16\)\( p^{11} T^{9} + \)\(59\!\cdots\!54\)\( p^{22} T^{10} + \)\(20\!\cdots\!76\)\( p^{33} T^{11} + \)\(78\!\cdots\!17\)\( p^{44} T^{12} - \)\(10\!\cdots\!92\)\( p^{55} T^{13} - 156603083235116718 p^{66} T^{14} - 463131040 p^{77} T^{15} + p^{88} T^{16} \) | |
| 41 | \( ( 1 + 808861704 T + 720256975042877324 T^{2} + \)\(43\!\cdots\!56\)\( T^{3} + \)\(99\!\cdots\!10\)\( T^{4} + \)\(43\!\cdots\!56\)\( p^{11} T^{5} + 720256975042877324 p^{22} T^{6} + 808861704 p^{33} T^{7} + p^{44} T^{8} )^{2} \) | |
| 43 | \( ( 1 - 1463448176 T + 2195888618193448108 T^{2} - \)\(26\!\cdots\!36\)\( T^{3} + \)\(26\!\cdots\!14\)\( T^{4} - \)\(26\!\cdots\!36\)\( p^{11} T^{5} + 2195888618193448108 p^{22} T^{6} - 1463448176 p^{33} T^{7} + p^{44} T^{8} )^{2} \) | |
| 47 | \( 1 + 894091254 T - 6677695988453882792 T^{2} - \)\(44\!\cdots\!08\)\( T^{3} + \)\(24\!\cdots\!63\)\( T^{4} + \)\(82\!\cdots\!52\)\( T^{5} - \)\(18\!\cdots\!12\)\( p T^{6} - \)\(48\!\cdots\!94\)\( T^{7} + \)\(26\!\cdots\!48\)\( T^{8} - \)\(48\!\cdots\!94\)\( p^{11} T^{9} - \)\(18\!\cdots\!12\)\( p^{23} T^{10} + \)\(82\!\cdots\!52\)\( p^{33} T^{11} + \)\(24\!\cdots\!63\)\( p^{44} T^{12} - \)\(44\!\cdots\!08\)\( p^{55} T^{13} - 6677695988453882792 p^{66} T^{14} + 894091254 p^{77} T^{15} + p^{88} T^{16} \) | |
| 53 | \( 1 + 1448863512 T - 16284707098415834486 T^{2} - \)\(11\!\cdots\!36\)\( T^{3} + \)\(14\!\cdots\!73\)\( T^{4} + \)\(59\!\cdots\!72\)\( T^{5} + \)\(26\!\cdots\!42\)\( T^{6} + \)\(20\!\cdots\!04\)\( T^{7} - \)\(98\!\cdots\!44\)\( T^{8} + \)\(20\!\cdots\!04\)\( p^{11} T^{9} + \)\(26\!\cdots\!42\)\( p^{22} T^{10} + \)\(59\!\cdots\!72\)\( p^{33} T^{11} + \)\(14\!\cdots\!73\)\( p^{44} T^{12} - \)\(11\!\cdots\!36\)\( p^{55} T^{13} - 16284707098415834486 p^{66} T^{14} + 1448863512 p^{77} T^{15} + p^{88} T^{16} \) | |
| 59 | \( 1 - 14386900738 T + 12385622377665699956 T^{2} + \)\(12\!\cdots\!08\)\( T^{3} + \)\(67\!\cdots\!79\)\( T^{4} - \)\(39\!\cdots\!20\)\( T^{5} - \)\(93\!\cdots\!36\)\( T^{6} - \)\(46\!\cdots\!34\)\( T^{7} + \)\(12\!\cdots\!72\)\( T^{8} - \)\(46\!\cdots\!34\)\( p^{11} T^{9} - \)\(93\!\cdots\!36\)\( p^{22} T^{10} - \)\(39\!\cdots\!20\)\( p^{33} T^{11} + \)\(67\!\cdots\!79\)\( p^{44} T^{12} + \)\(12\!\cdots\!08\)\( p^{55} T^{13} + 12385622377665699956 p^{66} T^{14} - 14386900738 p^{77} T^{15} + p^{88} T^{16} \) | |
| 61 | \( 1 - 10854402216 T + 11197091040561526106 T^{2} - \)\(44\!\cdots\!72\)\( T^{3} + \)\(49\!\cdots\!77\)\( T^{4} + \)\(29\!\cdots\!64\)\( T^{5} + \)\(86\!\cdots\!82\)\( T^{6} - \)\(71\!\cdots\!32\)\( T^{7} - \)\(44\!\cdots\!24\)\( T^{8} - \)\(71\!\cdots\!32\)\( p^{11} T^{9} + \)\(86\!\cdots\!82\)\( p^{22} T^{10} + \)\(29\!\cdots\!64\)\( p^{33} T^{11} + \)\(49\!\cdots\!77\)\( p^{44} T^{12} - \)\(44\!\cdots\!72\)\( p^{55} T^{13} + 11197091040561526106 p^{66} T^{14} - 10854402216 p^{77} T^{15} + p^{88} T^{16} \) | |
| 67 | \( 1 - 19629545546 T + 75687379635761819196 T^{2} - \)\(84\!\cdots\!24\)\( T^{3} + \)\(17\!\cdots\!79\)\( T^{4} - \)\(56\!\cdots\!48\)\( T^{5} + \)\(13\!\cdots\!40\)\( T^{6} - \)\(88\!\cdots\!46\)\( T^{7} - \)\(33\!\cdots\!00\)\( T^{8} - \)\(88\!\cdots\!46\)\( p^{11} T^{9} + \)\(13\!\cdots\!40\)\( p^{22} T^{10} - \)\(56\!\cdots\!48\)\( p^{33} T^{11} + \)\(17\!\cdots\!79\)\( p^{44} T^{12} - \)\(84\!\cdots\!24\)\( p^{55} T^{13} + 75687379635761819196 p^{66} T^{14} - 19629545546 p^{77} T^{15} + p^{88} T^{16} \) | |
| 71 | \( ( 1 - 737402464 T + \)\(21\!\cdots\!12\)\( T^{2} + \)\(28\!\cdots\!72\)\( T^{3} - \)\(46\!\cdots\!82\)\( T^{4} + \)\(28\!\cdots\!72\)\( p^{11} T^{5} + \)\(21\!\cdots\!12\)\( p^{22} T^{6} - 737402464 p^{33} T^{7} + p^{44} T^{8} )^{2} \) | |
| 73 | \( 1 - 21420158732 T - \)\(77\!\cdots\!50\)\( T^{2} + \)\(97\!\cdots\!00\)\( T^{3} + \)\(56\!\cdots\!65\)\( T^{4} - \)\(35\!\cdots\!12\)\( T^{5} - \)\(25\!\cdots\!30\)\( T^{6} + \)\(41\!\cdots\!04\)\( T^{7} + \)\(93\!\cdots\!56\)\( T^{8} + \)\(41\!\cdots\!04\)\( p^{11} T^{9} - \)\(25\!\cdots\!30\)\( p^{22} T^{10} - \)\(35\!\cdots\!12\)\( p^{33} T^{11} + \)\(56\!\cdots\!65\)\( p^{44} T^{12} + \)\(97\!\cdots\!00\)\( p^{55} T^{13} - \)\(77\!\cdots\!50\)\( p^{66} T^{14} - 21420158732 p^{77} T^{15} + p^{88} T^{16} \) | |
| 79 | \( 1 + 60246238086 T - \)\(32\!\cdots\!80\)\( T^{2} - \)\(36\!\cdots\!48\)\( T^{3} + \)\(26\!\cdots\!63\)\( T^{4} + \)\(72\!\cdots\!88\)\( T^{5} - \)\(17\!\cdots\!28\)\( T^{6} + \)\(86\!\cdots\!86\)\( T^{7} + \)\(25\!\cdots\!36\)\( T^{8} + \)\(86\!\cdots\!86\)\( p^{11} T^{9} - \)\(17\!\cdots\!28\)\( p^{22} T^{10} + \)\(72\!\cdots\!88\)\( p^{33} T^{11} + \)\(26\!\cdots\!63\)\( p^{44} T^{12} - \)\(36\!\cdots\!48\)\( p^{55} T^{13} - \)\(32\!\cdots\!80\)\( p^{66} T^{14} + 60246238086 p^{77} T^{15} + p^{88} T^{16} \) | |
| 83 | \( ( 1 + 69946791152 T + \)\(60\!\cdots\!00\)\( T^{2} + \)\(25\!\cdots\!44\)\( T^{3} + \)\(12\!\cdots\!38\)\( T^{4} + \)\(25\!\cdots\!44\)\( p^{11} T^{5} + \)\(60\!\cdots\!00\)\( p^{22} T^{6} + 69946791152 p^{33} T^{7} + p^{44} T^{8} )^{2} \) | |
| 89 | \( 1 - 126354105612 T - \)\(86\!\cdots\!14\)\( T^{2} + \)\(19\!\cdots\!76\)\( T^{3} + \)\(57\!\cdots\!89\)\( T^{4} - \)\(28\!\cdots\!60\)\( T^{5} - \)\(12\!\cdots\!10\)\( T^{6} - \)\(20\!\cdots\!28\)\( T^{7} + \)\(89\!\cdots\!92\)\( T^{8} - \)\(20\!\cdots\!28\)\( p^{11} T^{9} - \)\(12\!\cdots\!10\)\( p^{22} T^{10} - \)\(28\!\cdots\!60\)\( p^{33} T^{11} + \)\(57\!\cdots\!89\)\( p^{44} T^{12} + \)\(19\!\cdots\!76\)\( p^{55} T^{13} - \)\(86\!\cdots\!14\)\( p^{66} T^{14} - 126354105612 p^{77} T^{15} + p^{88} T^{16} \) | |
| 97 | \( ( 1 + 269091442088 T + \)\(35\!\cdots\!52\)\( T^{2} + \)\(29\!\cdots\!28\)\( T^{3} + \)\(22\!\cdots\!94\)\( T^{4} + \)\(29\!\cdots\!28\)\( p^{11} T^{5} + \)\(35\!\cdots\!52\)\( p^{22} T^{6} + 269091442088 p^{33} T^{7} + p^{44} T^{8} )^{2} \) | |
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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
−7.16016018801116547648372978031, −6.96830805140757458290120707423, −6.61596829530497141909373675048, −6.54560669900045933075864428663, −6.32693243609154122973787998788, −5.62387379605224122958712243577, −5.48889291722345986632416675528, −5.41527725785846239971294482756, −4.97059982550709340124337903828, −4.75827108675730193671468201866, −4.33239168363233163803346682112, −4.14005260317639810023622168779, −3.98502386243268892342293191685, −3.54226460854227878467960621946, −2.92309111117513032671440476578, −2.89931392877282963731889207696, −2.70651237357779342933101771980, −2.04613389530688819008975471976, −1.98132651661050457366425651786, −1.66197439626150281184855403833, −1.10316622732432792463937293850, −1.03687875556164256421848047273, −0.993314441776735800178329016825, −0.52671418108601694980555588643, −0.03732175591088562613839110927, 0.03732175591088562613839110927, 0.52671418108601694980555588643, 0.993314441776735800178329016825, 1.03687875556164256421848047273, 1.10316622732432792463937293850, 1.66197439626150281184855403833, 1.98132651661050457366425651786, 2.04613389530688819008975471976, 2.70651237357779342933101771980, 2.89931392877282963731889207696, 2.92309111117513032671440476578, 3.54226460854227878467960621946, 3.98502386243268892342293191685, 4.14005260317639810023622168779, 4.33239168363233163803346682112, 4.75827108675730193671468201866, 4.97059982550709340124337903828, 5.41527725785846239971294482756, 5.48889291722345986632416675528, 5.62387379605224122958712243577, 6.32693243609154122973787998788, 6.54560669900045933075864428663, 6.61596829530497141909373675048, 6.96830805140757458290120707423, 7.16016018801116547648372978031
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.