Dirichlet series
| L(s) = 1 | + 2.19e6·3-s − 1.34e8·4-s + 1.83e11·7-s − 1.81e10·9-s − 2.94e14·12-s + 5.12e14·13-s + 1.12e16·16-s + 1.32e16·19-s + 4.02e17·21-s + 3.14e18·25-s + 1.50e17·27-s − 2.45e19·28-s − 3.12e19·31-s + 2.43e18·36-s − 8.84e20·37-s + 1.12e21·39-s − 5.16e21·43-s + 2.47e22·48-s − 3.57e22·49-s − 6.87e22·52-s + 2.90e22·57-s + 2.35e23·61-s − 3.32e21·63-s − 7.55e23·64-s + 4.21e24·67-s + 1.64e24·73-s + 6.91e24·75-s + ⋯ |
| L(s) = 1 | + 1.37·3-s − 2·4-s + 1.89·7-s − 0.00713·9-s − 2.75·12-s + 1.69·13-s + 5/2·16-s + 0.315·19-s + 2.60·21-s + 2.11·25-s + 0.0370·27-s − 3.78·28-s − 1.28·31-s + 0.0142·36-s − 3.63·37-s + 2.32·39-s − 3.00·43-s + 3.44·48-s − 3.80·49-s − 3.38·52-s + 0.433·57-s + 1.45·61-s − 0.0134·63-s − 5/2·64-s + 7.69·67-s + 0.986·73-s + 2.90·75-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \, \Lambda(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+13)^{8} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(1679616\) = \(2^{8} \cdot 3^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.90167\times 10^{11}\) |
| Root analytic conductor: | \(5.06927\) |
| Motivic weight: | \(26\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 1679616,\ (\ :[13]^{8}),\ 1)\) |
Particular Values
| \(L(\frac{27}{2})\) | \(\approx\) | \(13.15653800\) |
| \(L(\frac12)\) | \(\approx\) | \(13.15653800\) |
| \(L(14)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{25} T^{2} )^{4} \) |
| 3 | \( 1 - 243880 p^{2} T + 737052668 p^{8} T^{2} - 250988792600 p^{16} T^{3} + 7120264260022 p^{26} T^{4} - 250988792600 p^{42} T^{5} + 737052668 p^{60} T^{6} - 243880 p^{80} T^{7} + p^{104} T^{8} \) | |
| good | 5 | \( 1 - 25188389500581736 p^{3} T^{2} + \)\(10\!\cdots\!76\)\( p^{4} T^{4} - \)\(61\!\cdots\!48\)\( p^{9} T^{6} + \)\(28\!\cdots\!26\)\( p^{14} T^{8} - \)\(61\!\cdots\!48\)\( p^{61} T^{10} + \)\(10\!\cdots\!76\)\( p^{108} T^{12} - 25188389500581736 p^{159} T^{14} + p^{208} T^{16} \) |
| 7 | \( ( 1 - 13086518840 p T + \)\(43\!\cdots\!44\)\( p T^{2} - \)\(68\!\cdots\!00\)\( p^{3} T^{3} + \)\(33\!\cdots\!62\)\( p^{6} T^{4} - \)\(68\!\cdots\!00\)\( p^{29} T^{5} + \)\(43\!\cdots\!44\)\( p^{53} T^{6} - 13086518840 p^{79} T^{7} + p^{104} T^{8} )^{2} \) | |
| 11 | \( 1 - \)\(34\!\cdots\!80\)\( T^{2} + \)\(71\!\cdots\!04\)\( p^{2} T^{4} - \)\(10\!\cdots\!40\)\( p^{4} T^{6} + \)\(95\!\cdots\!66\)\( p^{8} T^{8} - \)\(10\!\cdots\!40\)\( p^{56} T^{10} + \)\(71\!\cdots\!04\)\( p^{106} T^{12} - \)\(34\!\cdots\!80\)\( p^{156} T^{14} + p^{208} T^{16} \) | |
| 13 | \( ( 1 - 19708808559560 p T + \)\(24\!\cdots\!72\)\( p T^{2} - \)\(40\!\cdots\!40\)\( p^{2} T^{3} + \)\(24\!\cdots\!94\)\( p^{2} T^{4} - \)\(40\!\cdots\!40\)\( p^{28} T^{5} + \)\(24\!\cdots\!72\)\( p^{53} T^{6} - 19708808559560 p^{79} T^{7} + p^{104} T^{8} )^{2} \) | |
| 17 | \( 1 - \)\(56\!\cdots\!00\)\( T^{2} + \)\(14\!\cdots\!04\)\( T^{4} - \)\(46\!\cdots\!00\)\( p^{3} T^{6} + \)\(31\!\cdots\!26\)\( p^{4} T^{8} - \)\(46\!\cdots\!00\)\( p^{55} T^{10} + \)\(14\!\cdots\!04\)\( p^{104} T^{12} - \)\(56\!\cdots\!00\)\( p^{156} T^{14} + p^{208} T^{16} \) | |
| 19 | \( ( 1 - 6625251412866152 T + \)\(63\!\cdots\!92\)\( p T^{2} + \)\(58\!\cdots\!56\)\( p^{2} T^{3} + \)\(34\!\cdots\!70\)\( p^{3} T^{4} + \)\(58\!\cdots\!56\)\( p^{28} T^{5} + \)\(63\!\cdots\!92\)\( p^{53} T^{6} - 6625251412866152 p^{78} T^{7} + p^{104} T^{8} )^{2} \) | |
| 23 | \( 1 - \)\(10\!\cdots\!80\)\( T^{2} + \)\(12\!\cdots\!36\)\( p^{2} T^{4} - \)\(97\!\cdots\!40\)\( p^{4} T^{6} + \)\(55\!\cdots\!94\)\( p^{6} T^{8} - \)\(97\!\cdots\!40\)\( p^{56} T^{10} + \)\(12\!\cdots\!36\)\( p^{106} T^{12} - \)\(10\!\cdots\!80\)\( p^{156} T^{14} + p^{208} T^{16} \) | |
| 29 | \( 1 - \)\(30\!\cdots\!08\)\( T^{2} + \)\(67\!\cdots\!88\)\( T^{4} - \)\(10\!\cdots\!16\)\( T^{6} + \)\(12\!\cdots\!70\)\( T^{8} - \)\(10\!\cdots\!16\)\( p^{52} T^{10} + \)\(67\!\cdots\!88\)\( p^{104} T^{12} - \)\(30\!\cdots\!08\)\( p^{156} T^{14} + p^{208} T^{16} \) | |
| 31 | \( ( 1 + 15642204647001784888 T - \)\(30\!\cdots\!72\)\( T^{2} - \)\(11\!\cdots\!24\)\( T^{3} - \)\(15\!\cdots\!30\)\( T^{4} - \)\(11\!\cdots\!24\)\( p^{26} T^{5} - \)\(30\!\cdots\!72\)\( p^{52} T^{6} + 15642204647001784888 p^{78} T^{7} + p^{104} T^{8} )^{2} \) | |
| 37 | \( ( 1 + \)\(44\!\cdots\!00\)\( T + \)\(25\!\cdots\!28\)\( T^{2} + \)\(77\!\cdots\!60\)\( T^{3} + \)\(22\!\cdots\!78\)\( T^{4} + \)\(77\!\cdots\!60\)\( p^{26} T^{5} + \)\(25\!\cdots\!28\)\( p^{52} T^{6} + \)\(44\!\cdots\!00\)\( p^{78} T^{7} + p^{104} T^{8} )^{2} \) | |
| 41 | \( 1 - \)\(27\!\cdots\!60\)\( T^{2} + \)\(44\!\cdots\!64\)\( T^{4} - \)\(53\!\cdots\!80\)\( T^{6} + \)\(50\!\cdots\!46\)\( T^{8} - \)\(53\!\cdots\!80\)\( p^{52} T^{10} + \)\(44\!\cdots\!64\)\( p^{104} T^{12} - \)\(27\!\cdots\!60\)\( p^{156} T^{14} + p^{208} T^{16} \) | |
| 43 | \( ( 1 + \)\(25\!\cdots\!20\)\( T + \)\(45\!\cdots\!24\)\( T^{2} - \)\(11\!\cdots\!60\)\( T^{3} - \)\(67\!\cdots\!34\)\( T^{4} - \)\(11\!\cdots\!60\)\( p^{26} T^{5} + \)\(45\!\cdots\!24\)\( p^{52} T^{6} + \)\(25\!\cdots\!20\)\( p^{78} T^{7} + p^{104} T^{8} )^{2} \) | |
| 47 | \( 1 - \)\(18\!\cdots\!32\)\( T^{2} + \)\(15\!\cdots\!48\)\( T^{4} - \)\(78\!\cdots\!84\)\( T^{6} + \)\(27\!\cdots\!70\)\( T^{8} - \)\(78\!\cdots\!84\)\( p^{52} T^{10} + \)\(15\!\cdots\!48\)\( p^{104} T^{12} - \)\(18\!\cdots\!32\)\( p^{156} T^{14} + p^{208} T^{16} \) | |
| 53 | \( 1 - \)\(36\!\cdots\!80\)\( T^{2} + \)\(67\!\cdots\!24\)\( T^{4} - \)\(79\!\cdots\!40\)\( T^{6} + \)\(64\!\cdots\!06\)\( T^{8} - \)\(79\!\cdots\!40\)\( p^{52} T^{10} + \)\(67\!\cdots\!24\)\( p^{104} T^{12} - \)\(36\!\cdots\!80\)\( p^{156} T^{14} + p^{208} T^{16} \) | |
| 59 | \( 1 - \)\(34\!\cdots\!80\)\( p T^{2} + \)\(30\!\cdots\!84\)\( T^{4} - \)\(47\!\cdots\!60\)\( T^{6} + \)\(60\!\cdots\!86\)\( T^{8} - \)\(47\!\cdots\!60\)\( p^{52} T^{10} + \)\(30\!\cdots\!84\)\( p^{104} T^{12} - \)\(34\!\cdots\!80\)\( p^{157} T^{14} + p^{208} T^{16} \) | |
| 61 | \( ( 1 - \)\(19\!\cdots\!56\)\( p T + \)\(98\!\cdots\!80\)\( T^{2} - \)\(84\!\cdots\!04\)\( T^{3} + \)\(38\!\cdots\!14\)\( T^{4} - \)\(84\!\cdots\!04\)\( p^{26} T^{5} + \)\(98\!\cdots\!80\)\( p^{52} T^{6} - \)\(19\!\cdots\!56\)\( p^{79} T^{7} + p^{104} T^{8} )^{2} \) | |
| 67 | \( ( 1 - \)\(21\!\cdots\!40\)\( T + \)\(26\!\cdots\!88\)\( T^{2} - \)\(22\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!38\)\( T^{4} - \)\(22\!\cdots\!80\)\( p^{26} T^{5} + \)\(26\!\cdots\!88\)\( p^{52} T^{6} - \)\(21\!\cdots\!40\)\( p^{78} T^{7} + p^{104} T^{8} )^{2} \) | |
| 71 | \( 1 - \)\(34\!\cdots\!08\)\( T^{2} + \)\(64\!\cdots\!88\)\( T^{4} - \)\(12\!\cdots\!16\)\( T^{6} + \)\(19\!\cdots\!70\)\( T^{8} - \)\(12\!\cdots\!16\)\( p^{52} T^{10} + \)\(64\!\cdots\!88\)\( p^{104} T^{12} - \)\(34\!\cdots\!08\)\( p^{156} T^{14} + p^{208} T^{16} \) | |
| 73 | \( ( 1 - \)\(82\!\cdots\!00\)\( T + \)\(25\!\cdots\!96\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!46\)\( T^{4} - \)\(13\!\cdots\!00\)\( p^{26} T^{5} + \)\(25\!\cdots\!96\)\( p^{52} T^{6} - \)\(82\!\cdots\!00\)\( p^{78} T^{7} + p^{104} T^{8} )^{2} \) | |
| 79 | \( ( 1 + \)\(46\!\cdots\!12\)\( T + \)\(51\!\cdots\!48\)\( T^{2} + \)\(16\!\cdots\!24\)\( T^{3} + \)\(12\!\cdots\!30\)\( T^{4} + \)\(16\!\cdots\!24\)\( p^{26} T^{5} + \)\(51\!\cdots\!48\)\( p^{52} T^{6} + \)\(46\!\cdots\!12\)\( p^{78} T^{7} + p^{104} T^{8} )^{2} \) | |
| 83 | \( 1 - \)\(12\!\cdots\!84\)\( T^{2} + \)\(76\!\cdots\!80\)\( T^{4} - \)\(46\!\cdots\!96\)\( T^{6} + \)\(29\!\cdots\!14\)\( T^{8} - \)\(46\!\cdots\!96\)\( p^{52} T^{10} + \)\(76\!\cdots\!80\)\( p^{104} T^{12} - \)\(12\!\cdots\!84\)\( p^{156} T^{14} + p^{208} T^{16} \) | |
| 89 | \( 1 - \)\(22\!\cdots\!80\)\( T^{2} + \)\(25\!\cdots\!44\)\( T^{4} - \)\(19\!\cdots\!40\)\( T^{6} + \)\(10\!\cdots\!66\)\( T^{8} - \)\(19\!\cdots\!40\)\( p^{52} T^{10} + \)\(25\!\cdots\!44\)\( p^{104} T^{12} - \)\(22\!\cdots\!80\)\( p^{156} T^{14} + p^{208} T^{16} \) | |
| 97 | \( ( 1 + \)\(46\!\cdots\!40\)\( T + \)\(11\!\cdots\!64\)\( T^{2} + \)\(42\!\cdots\!20\)\( T^{3} + \)\(73\!\cdots\!86\)\( T^{4} + \)\(42\!\cdots\!20\)\( p^{26} T^{5} + \)\(11\!\cdots\!64\)\( p^{52} T^{6} + \)\(46\!\cdots\!40\)\( p^{78} T^{7} + p^{104} T^{8} )^{2} \) | |
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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
−5.67679321088721489871938049568, −5.16623390261639371964886930587, −5.12943523461531689497741424172, −5.07765717399630931535945358078, −4.89439105405847051435505365445, −4.77189572110779406848904622100, −4.75697089245883205568186985839, −4.04879988531831466614207952585, −3.61676421690774760987391098739, −3.56525154183647456625362633788, −3.56079650884158242099086206872, −3.53952159733582160286083128312, −3.26333835483676900168983272453, −2.82165637318965270851144526243, −2.55424317825278415833738360212, −2.18655651932216561706715907988, −1.99157556790693684811760554959, −1.83740488275707405602613440527, −1.44790994241176929788526488852, −1.33719430065491625225697319358, −1.24426973634046856035387547989, −0.931360643907596556733814780499, −0.45021001871316084996659327434, −0.38574444123283862280937051296, −0.33471400306106074123376869757, 0.33471400306106074123376869757, 0.38574444123283862280937051296, 0.45021001871316084996659327434, 0.931360643907596556733814780499, 1.24426973634046856035387547989, 1.33719430065491625225697319358, 1.44790994241176929788526488852, 1.83740488275707405602613440527, 1.99157556790693684811760554959, 2.18655651932216561706715907988, 2.55424317825278415833738360212, 2.82165637318965270851144526243, 3.26333835483676900168983272453, 3.53952159733582160286083128312, 3.56079650884158242099086206872, 3.56525154183647456625362633788, 3.61676421690774760987391098739, 4.04879988531831466614207952585, 4.75697089245883205568186985839, 4.77189572110779406848904622100, 4.89439105405847051435505365445, 5.07765717399630931535945358078, 5.12943523461531689497741424172, 5.16623390261639371964886930587, 5.67679321088721489871938049568
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.