Dirichlet series
| L(s) = 1 | + 4.09e3·4-s + 6.82e4·7-s − 7.34e6·13-s + 4.19e6·16-s + 2.40e7·19-s − 6.55e8·25-s + 2.79e8·28-s − 3.97e9·31-s + 1.08e10·37-s − 1.10e10·43-s + 3.20e10·49-s − 3.00e10·52-s + 1.75e11·61-s − 1.71e10·64-s − 8.80e10·67-s + 3.78e11·73-s + 9.85e10·76-s + 4.16e11·79-s − 5.01e11·91-s + 3.52e11·97-s − 2.68e12·100-s − 4.35e12·103-s + 1.68e12·109-s + 2.86e11·112-s − 1.01e13·121-s − 1.62e13·124-s + 127-s + ⋯ |
| L(s) = 1 | + 4-s + 0.580·7-s − 1.52·13-s + 1/4·16-s + 0.511·19-s − 2.68·25-s + 0.580·28-s − 4.47·31-s + 4.23·37-s − 1.74·43-s + 2.31·49-s − 1.52·52-s + 3.40·61-s − 1/4·64-s − 0.973·67-s + 2.50·73-s + 0.511·76-s + 1.71·79-s − 0.882·91-s + 0.422·97-s − 2.68·100-s − 3.64·103-s + 1.00·109-s + 0.145·112-s − 3.23·121-s − 4.47·124-s + 0.296·133-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(13-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+6)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.31028\times 10^{17}\) |
| Root analytic conductor: | \(12.1682\) |
| Motivic weight: | \(12\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{32} ,\ ( \ : [6]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{13}{2})\) | \(\approx\) | \(0.2346404233\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2346404233\) |
| \(L(7)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 + 26213812 p^{2} T^{2} + 331868663597194 p^{4} T^{4} + \)\(17\!\cdots\!92\)\( p^{8} T^{6} + \)\(82\!\cdots\!31\)\( p^{12} T^{8} + \)\(17\!\cdots\!92\)\( p^{32} T^{10} + 331868663597194 p^{52} T^{12} + 26213812 p^{74} T^{14} + p^{96} T^{16} \) |
| 7 | \( ( 1 - 34142 T - 14255240543 T^{2} + 59805359574670 p T^{3} + 614060859885333316 p^{2} T^{4} + 59805359574670 p^{13} T^{5} - 14255240543 p^{24} T^{6} - 34142 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 11 | \( 1 + 83824141492 p^{2} T^{2} + \)\(39\!\cdots\!82\)\( p^{4} T^{4} + \)\(14\!\cdots\!60\)\( p^{6} T^{6} + \)\(41\!\cdots\!59\)\( p^{8} T^{8} + \)\(14\!\cdots\!60\)\( p^{30} T^{10} + \)\(39\!\cdots\!82\)\( p^{52} T^{12} + 83824141492 p^{74} T^{14} + p^{96} T^{16} \) | |
| 13 | \( ( 1 + 3671518 T + 7211123191297 T^{2} - \)\(14\!\cdots\!30\)\( T^{3} - \)\(81\!\cdots\!76\)\( T^{4} - \)\(14\!\cdots\!30\)\( p^{12} T^{5} + 7211123191297 p^{24} T^{6} + 3671518 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 17 | \( ( 1 - 186365459324884 T^{2} + \)\(48\!\cdots\!42\)\( T^{4} - 186365459324884 p^{24} T^{6} + p^{48} T^{8} )^{2} \) | |
| 19 | \( ( 1 - 6015298 T + 4387421763666339 T^{2} - 6015298 p^{12} T^{3} + p^{24} T^{4} )^{4} \) | |
| 23 | \( 1 + 57623193481093204 T^{2} + \)\(15\!\cdots\!94\)\( T^{4} + \)\(46\!\cdots\!60\)\( T^{6} + \)\(13\!\cdots\!47\)\( T^{8} + \)\(46\!\cdots\!60\)\( p^{24} T^{10} + \)\(15\!\cdots\!94\)\( p^{48} T^{12} + 57623193481093204 p^{72} T^{14} + p^{96} T^{16} \) | |
| 29 | \( 1 + 327209185044691204 T^{2} - \)\(12\!\cdots\!66\)\( T^{4} - \)\(57\!\cdots\!20\)\( T^{6} + \)\(25\!\cdots\!67\)\( T^{8} - \)\(57\!\cdots\!20\)\( p^{24} T^{10} - \)\(12\!\cdots\!66\)\( p^{48} T^{12} + 327209185044691204 p^{72} T^{14} + p^{96} T^{16} \) | |
| 31 | \( ( 1 + 1986916972 T + 1482848302161475882 T^{2} + \)\(57\!\cdots\!60\)\( p T^{3} + \)\(24\!\cdots\!19\)\( p^{2} T^{4} + \)\(57\!\cdots\!60\)\( p^{13} T^{5} + 1482848302161475882 p^{24} T^{6} + 1986916972 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 37 | \( ( 1 - 2714888110 T + 12010167810377243187 T^{2} - 2714888110 p^{12} T^{3} + p^{24} T^{4} )^{4} \) | |
| 41 | \( 1 + 89666171762954058436 T^{2} + \)\(50\!\cdots\!86\)\( T^{4} + \)\(18\!\cdots\!68\)\( T^{6} + \)\(48\!\cdots\!23\)\( T^{8} + \)\(18\!\cdots\!68\)\( p^{24} T^{10} + \)\(50\!\cdots\!86\)\( p^{48} T^{12} + 89666171762954058436 p^{72} T^{14} + p^{96} T^{16} \) | |
| 43 | \( ( 1 + 5512774252 T - 46487411842141900790 T^{2} - \)\(16\!\cdots\!16\)\( T^{3} + \)\(30\!\cdots\!31\)\( T^{4} - \)\(16\!\cdots\!16\)\( p^{12} T^{5} - 46487411842141900790 p^{24} T^{6} + 5512774252 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 47 | \( 1 + \)\(35\!\cdots\!76\)\( T^{2} + \)\(67\!\cdots\!06\)\( T^{4} + \)\(10\!\cdots\!08\)\( T^{6} + \)\(14\!\cdots\!83\)\( T^{8} + \)\(10\!\cdots\!08\)\( p^{24} T^{10} + \)\(67\!\cdots\!06\)\( p^{48} T^{12} + \)\(35\!\cdots\!76\)\( p^{72} T^{14} + p^{96} T^{16} \) | |
| 53 | \( ( 1 + \)\(53\!\cdots\!32\)\( T^{2} + \)\(52\!\cdots\!18\)\( T^{4} + \)\(53\!\cdots\!32\)\( p^{24} T^{6} + p^{48} T^{8} )^{2} \) | |
| 59 | \( 1 + \)\(23\!\cdots\!60\)\( T^{2} + \)\(75\!\cdots\!78\)\( T^{4} - \)\(34\!\cdots\!00\)\( T^{6} - \)\(39\!\cdots\!37\)\( T^{8} - \)\(34\!\cdots\!00\)\( p^{24} T^{10} + \)\(75\!\cdots\!78\)\( p^{48} T^{12} + \)\(23\!\cdots\!60\)\( p^{72} T^{14} + p^{96} T^{16} \) | |
| 61 | \( ( 1 - 87767669282 T + \)\(16\!\cdots\!37\)\( T^{2} - \)\(65\!\cdots\!90\)\( T^{3} + \)\(91\!\cdots\!24\)\( T^{4} - \)\(65\!\cdots\!90\)\( p^{12} T^{5} + \)\(16\!\cdots\!37\)\( p^{24} T^{6} - 87767669282 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 67 | \( ( 1 + 44042724850 T - \)\(65\!\cdots\!83\)\( T^{2} - \)\(34\!\cdots\!50\)\( T^{3} - \)\(10\!\cdots\!32\)\( T^{4} - \)\(34\!\cdots\!50\)\( p^{12} T^{5} - \)\(65\!\cdots\!83\)\( p^{24} T^{6} + 44042724850 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 71 | \( ( 1 - \)\(41\!\cdots\!08\)\( T^{2} + \)\(89\!\cdots\!78\)\( T^{4} - \)\(41\!\cdots\!08\)\( p^{24} T^{6} + p^{48} T^{8} )^{2} \) | |
| 73 | \( ( 1 - 94702762174 T + \)\(33\!\cdots\!75\)\( T^{2} - 94702762174 p^{12} T^{3} + p^{24} T^{4} )^{4} \) | |
| 79 | \( ( 1 - 208160642318 T - \)\(47\!\cdots\!75\)\( T^{2} + \)\(14\!\cdots\!94\)\( T^{3} - \)\(37\!\cdots\!64\)\( T^{4} + \)\(14\!\cdots\!94\)\( p^{12} T^{5} - \)\(47\!\cdots\!75\)\( p^{24} T^{6} - 208160642318 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 83 | \( 1 + \)\(23\!\cdots\!24\)\( T^{2} + \)\(22\!\cdots\!74\)\( T^{4} + \)\(22\!\cdots\!40\)\( T^{6} + \)\(29\!\cdots\!67\)\( T^{8} + \)\(22\!\cdots\!40\)\( p^{24} T^{10} + \)\(22\!\cdots\!74\)\( p^{48} T^{12} + \)\(23\!\cdots\!24\)\( p^{72} T^{14} + p^{96} T^{16} \) | |
| 89 | \( ( 1 - \)\(83\!\cdots\!20\)\( T^{2} + \)\(28\!\cdots\!82\)\( T^{4} - \)\(83\!\cdots\!20\)\( p^{24} T^{6} + p^{48} T^{8} )^{2} \) | |
| 97 | \( ( 1 - 176152203122 T - \)\(96\!\cdots\!43\)\( T^{2} + \)\(69\!\cdots\!10\)\( T^{3} + \)\(48\!\cdots\!04\)\( T^{4} + \)\(69\!\cdots\!10\)\( p^{12} T^{5} - \)\(96\!\cdots\!43\)\( p^{24} T^{6} - 176152203122 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
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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
−3.80312260224076504217983631412, −3.78324345454265166939780155137, −3.47367372466640351428276785090, −3.23864744296179730414934920562, −3.11961300147669387750426102441, −3.00823761076878033495305927651, −2.99881016575929830943240862039, −2.44477815940087771142388648226, −2.42451891212778444079943185661, −2.29583275527815517349303638179, −2.25950753911267581211383581979, −2.11886919349970679287884212709, −2.05391442139081130188312516922, −2.03392179139912151990392606819, −1.78594490072493027710263599131, −1.43756221910242428193920547595, −1.17690244999554686253362022145, −1.17472127130855732882441392609, −1.10899783514109308031148424658, −0.994651682828196762215657289408, −0.76292966084875809951400785502, −0.61905218699799616560750825882, −0.15629730143518682526025110721, −0.12168003394888198800786021689, −0.091941464627681397408321308208, 0.091941464627681397408321308208, 0.12168003394888198800786021689, 0.15629730143518682526025110721, 0.61905218699799616560750825882, 0.76292966084875809951400785502, 0.994651682828196762215657289408, 1.10899783514109308031148424658, 1.17472127130855732882441392609, 1.17690244999554686253362022145, 1.43756221910242428193920547595, 1.78594490072493027710263599131, 2.03392179139912151990392606819, 2.05391442139081130188312516922, 2.11886919349970679287884212709, 2.25950753911267581211383581979, 2.29583275527815517349303638179, 2.42451891212778444079943185661, 2.44477815940087771142388648226, 2.99881016575929830943240862039, 3.00823761076878033495305927651, 3.11961300147669387750426102441, 3.23864744296179730414934920562, 3.47367372466640351428276785090, 3.78324345454265166939780155137, 3.80312260224076504217983631412
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.