Dirichlet series
| L(s) = 1 | − 2.04e3·4-s + 4.51e4·7-s − 7.21e5·13-s + 2.62e6·16-s − 7.31e5·19-s + 5.28e7·25-s − 9.23e7·28-s + 7.25e7·31-s − 1.66e8·37-s − 1.25e7·43-s − 1.83e7·49-s + 1.47e9·52-s − 1.78e9·61-s − 2.68e9·64-s + 1.08e8·67-s + 5.10e9·73-s + 1.49e9·76-s − 9.24e9·79-s − 3.25e10·91-s + 2.24e9·97-s − 1.08e11·100-s + 6.13e10·103-s − 1.17e11·109-s + 1.18e11·112-s + 1.75e11·121-s − 1.48e11·124-s + 127-s + ⋯ |
| L(s) = 1 | − 2·4-s + 2.68·7-s − 1.94·13-s + 5/2·16-s − 0.295·19-s + 5.41·25-s − 5.36·28-s + 2.53·31-s − 2.40·37-s − 0.0851·43-s − 0.0649·49-s + 3.88·52-s − 2.11·61-s − 5/2·64-s + 0.0805·67-s + 2.46·73-s + 0.590·76-s − 3.00·79-s − 5.21·91-s + 0.261·97-s − 10.8·100-s + 5.29·103-s − 7.64·109-s + 6.71·112-s + 6.77·121-s − 5.06·124-s − 0.792·133-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+5)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.25969\times 10^{16}\) |
| Root analytic conductor: | \(10.1453\) |
| Motivic weight: | \(10\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{32} ,\ ( \ : [5]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{11}{2})\) | \(\approx\) | \(0.0002989952097\) |
| \(L(\frac12)\) | \(\approx\) | \(0.0002989952097\) |
| \(L(6)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{9} T^{2} )^{4} \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 - 10571104 p T^{2} + 55030668906286 p^{2} T^{4} - \)\(18\!\cdots\!64\)\( p^{3} T^{6} + \)\(42\!\cdots\!31\)\( p^{4} T^{8} - \)\(18\!\cdots\!64\)\( p^{23} T^{10} + 55030668906286 p^{42} T^{12} - 10571104 p^{61} T^{14} + p^{80} T^{16} \) |
| 7 | \( ( 1 - 22556 T + 772331668 T^{2} - 1917011915108 p T^{3} + 6643521062567722 p^{2} T^{4} - 1917011915108 p^{11} T^{5} + 772331668 p^{20} T^{6} - 22556 p^{30} T^{7} + p^{40} T^{8} )^{2} \) | |
| 11 | \( 1 - 175732299920 T^{2} + \)\(14\!\cdots\!64\)\( T^{4} - \)\(68\!\cdots\!60\)\( T^{6} + \)\(21\!\cdots\!26\)\( T^{8} - \)\(68\!\cdots\!60\)\( p^{20} T^{10} + \)\(14\!\cdots\!64\)\( p^{40} T^{12} - 175732299920 p^{60} T^{14} + p^{80} T^{16} \) | |
| 13 | \( ( 1 + 27748 p T + 331408182946 T^{2} + 118099918472653600 T^{3} + \)\(58\!\cdots\!75\)\( T^{4} + 118099918472653600 p^{10} T^{5} + 331408182946 p^{20} T^{6} + 27748 p^{31} T^{7} + p^{40} T^{8} )^{2} \) | |
| 17 | \( 1 - 9754668435248 T^{2} + \)\(47\!\cdots\!34\)\( T^{4} - \)\(14\!\cdots\!20\)\( T^{6} + \)\(34\!\cdots\!35\)\( T^{8} - \)\(14\!\cdots\!20\)\( p^{20} T^{10} + \)\(47\!\cdots\!34\)\( p^{40} T^{12} - 9754668435248 p^{60} T^{14} + p^{80} T^{16} \) | |
| 19 | \( ( 1 + 365596 T + 10628096140324 T^{2} + 15723599245461508156 T^{3} + \)\(72\!\cdots\!70\)\( T^{4} + 15723599245461508156 p^{10} T^{5} + 10628096140324 p^{20} T^{6} + 365596 p^{30} T^{7} + p^{40} T^{8} )^{2} \) | |
| 23 | \( 1 - 167180527959872 T^{2} + \)\(16\!\cdots\!48\)\( T^{4} - \)\(10\!\cdots\!44\)\( T^{6} + \)\(51\!\cdots\!70\)\( T^{8} - \)\(10\!\cdots\!44\)\( p^{20} T^{10} + \)\(16\!\cdots\!48\)\( p^{40} T^{12} - 167180527959872 p^{60} T^{14} + p^{80} T^{16} \) | |
| 29 | \( 1 - 2322362447160176 T^{2} + \)\(26\!\cdots\!30\)\( T^{4} - \)\(19\!\cdots\!44\)\( T^{6} + \)\(95\!\cdots\!39\)\( T^{8} - \)\(19\!\cdots\!44\)\( p^{20} T^{10} + \)\(26\!\cdots\!30\)\( p^{40} T^{12} - 2322362447160176 p^{60} T^{14} + p^{80} T^{16} \) | |
| 31 | \( ( 1 - 36251264 T + 2335436830550956 T^{2} - \)\(62\!\cdots\!88\)\( T^{3} + \)\(25\!\cdots\!06\)\( T^{4} - \)\(62\!\cdots\!88\)\( p^{10} T^{5} + 2335436830550956 p^{20} T^{6} - 36251264 p^{30} T^{7} + p^{40} T^{8} )^{2} \) | |
| 37 | \( ( 1 + 83476120 T + 15137019401494870 T^{2} + \)\(11\!\cdots\!80\)\( T^{3} + \)\(98\!\cdots\!67\)\( T^{4} + \)\(11\!\cdots\!80\)\( p^{10} T^{5} + 15137019401494870 p^{20} T^{6} + 83476120 p^{30} T^{7} + p^{40} T^{8} )^{2} \) | |
| 41 | \( 1 - 52900132959553136 T^{2} + \)\(13\!\cdots\!84\)\( T^{4} - \)\(25\!\cdots\!16\)\( T^{6} + \)\(37\!\cdots\!90\)\( T^{8} - \)\(25\!\cdots\!16\)\( p^{20} T^{10} + \)\(13\!\cdots\!84\)\( p^{40} T^{12} - 52900132959553136 p^{60} T^{14} + p^{80} T^{16} \) | |
| 43 | \( ( 1 + 6255220 T + 59946472373598052 T^{2} + \)\(52\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!58\)\( T^{4} + \)\(52\!\cdots\!80\)\( p^{10} T^{5} + 59946472373598052 p^{20} T^{6} + 6255220 p^{30} T^{7} + p^{40} T^{8} )^{2} \) | |
| 47 | \( 1 - 305022268387356968 T^{2} + \)\(42\!\cdots\!04\)\( T^{4} - \)\(36\!\cdots\!40\)\( T^{6} + \)\(22\!\cdots\!30\)\( T^{8} - \)\(36\!\cdots\!40\)\( p^{20} T^{10} + \)\(42\!\cdots\!04\)\( p^{40} T^{12} - 305022268387356968 p^{60} T^{14} + p^{80} T^{16} \) | |
| 53 | \( 1 - 842830763760485360 T^{2} + \)\(36\!\cdots\!88\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{6} + \)\(21\!\cdots\!78\)\( T^{8} - \)\(10\!\cdots\!80\)\( p^{20} T^{10} + \)\(36\!\cdots\!88\)\( p^{40} T^{12} - 842830763760485360 p^{60} T^{14} + p^{80} T^{16} \) | |
| 59 | \( 1 - 2467177976527842344 T^{2} + \)\(31\!\cdots\!16\)\( T^{4} - \)\(25\!\cdots\!28\)\( T^{6} + \)\(15\!\cdots\!86\)\( T^{8} - \)\(25\!\cdots\!28\)\( p^{20} T^{10} + \)\(31\!\cdots\!16\)\( p^{40} T^{12} - 2467177976527842344 p^{60} T^{14} + p^{80} T^{16} \) | |
| 61 | \( ( 1 + 891136624 T + 959003895617693566 T^{2} + \)\(76\!\cdots\!68\)\( T^{3} + \)\(10\!\cdots\!11\)\( T^{4} + \)\(76\!\cdots\!68\)\( p^{10} T^{5} + 959003895617693566 p^{20} T^{6} + 891136624 p^{30} T^{7} + p^{40} T^{8} )^{2} \) | |
| 67 | \( ( 1 - 54371828 T + 6949081761198061684 T^{2} - \)\(18\!\cdots\!24\)\( T^{3} + \)\(18\!\cdots\!26\)\( T^{4} - \)\(18\!\cdots\!24\)\( p^{10} T^{5} + 6949081761198061684 p^{20} T^{6} - 54371828 p^{30} T^{7} + p^{40} T^{8} )^{2} \) | |
| 71 | \( 1 - 11772016379311621760 T^{2} + \)\(64\!\cdots\!88\)\( T^{4} - \)\(21\!\cdots\!00\)\( T^{6} + \)\(64\!\cdots\!98\)\( T^{8} - \)\(21\!\cdots\!00\)\( p^{20} T^{10} + \)\(64\!\cdots\!88\)\( p^{40} T^{12} - 11772016379311621760 p^{60} T^{14} + p^{80} T^{16} \) | |
| 73 | \( ( 1 - 2551117184 T + 12044333407858330222 T^{2} - \)\(22\!\cdots\!52\)\( T^{3} + \)\(66\!\cdots\!15\)\( T^{4} - \)\(22\!\cdots\!52\)\( p^{10} T^{5} + 12044333407858330222 p^{20} T^{6} - 2551117184 p^{30} T^{7} + p^{40} T^{8} )^{2} \) | |
| 79 | \( ( 1 + 4623100876 T + 21165890035194757924 T^{2} + \)\(72\!\cdots\!16\)\( T^{3} + \)\(24\!\cdots\!30\)\( T^{4} + \)\(72\!\cdots\!16\)\( p^{10} T^{5} + 21165890035194757924 p^{20} T^{6} + 4623100876 p^{30} T^{7} + p^{40} T^{8} )^{2} \) | |
| 83 | \( 1 - 17141683756914722216 T^{2} + \)\(23\!\cdots\!76\)\( T^{4} - \)\(25\!\cdots\!12\)\( T^{6} + \)\(17\!\cdots\!26\)\( T^{8} - \)\(25\!\cdots\!12\)\( p^{20} T^{10} + \)\(23\!\cdots\!76\)\( p^{40} T^{12} - 17141683756914722216 p^{60} T^{14} + p^{80} T^{16} \) | |
| 89 | \( 1 - \)\(16\!\cdots\!12\)\( T^{2} + \)\(14\!\cdots\!54\)\( T^{4} - \)\(74\!\cdots\!20\)\( T^{6} + \)\(27\!\cdots\!95\)\( T^{8} - \)\(74\!\cdots\!20\)\( p^{20} T^{10} + \)\(14\!\cdots\!54\)\( p^{40} T^{12} - \)\(16\!\cdots\!12\)\( p^{60} T^{14} + p^{80} T^{16} \) | |
| 97 | \( ( 1 - 1123887968 T + \)\(21\!\cdots\!40\)\( T^{2} - \)\(44\!\cdots\!88\)\( T^{3} + \)\(20\!\cdots\!54\)\( T^{4} - \)\(44\!\cdots\!88\)\( p^{10} T^{5} + \)\(21\!\cdots\!40\)\( p^{20} T^{6} - 1123887968 p^{30} T^{7} + p^{40} T^{8} )^{2} \) | |
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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
−4.22835432149511970652131759568, −4.04451182093044245398413798917, −3.59927257851468378198523269186, −3.45050839901050357018106956739, −3.42724637947953735886102569459, −3.22844415429425033090082597003, −2.96830060594818683465361730485, −2.93207183613246722697477143223, −2.79843545760809728281894393042, −2.70842344162123111006552747335, −2.44501343455951481776251578389, −2.14848638209012520324366958524, −2.12757607146850460455735613146, −1.73961856689841864253023174937, −1.71966932186845210141534304448, −1.58998980324730076137633710451, −1.40306977421509167833218306824, −1.18321391278827217632757820930, −1.12651742934974951314401690492, −0.972698377568837740727091101106, −0.64655762079178956917357384698, −0.61633936159866120402352347755, −0.55649728237376193553810989305, −0.24318338738249075595810190803, −0.00101148887324556246473785412, 0.00101148887324556246473785412, 0.24318338738249075595810190803, 0.55649728237376193553810989305, 0.61633936159866120402352347755, 0.64655762079178956917357384698, 0.972698377568837740727091101106, 1.12651742934974951314401690492, 1.18321391278827217632757820930, 1.40306977421509167833218306824, 1.58998980324730076137633710451, 1.71966932186845210141534304448, 1.73961856689841864253023174937, 2.12757607146850460455735613146, 2.14848638209012520324366958524, 2.44501343455951481776251578389, 2.70842344162123111006552747335, 2.79843545760809728281894393042, 2.93207183613246722697477143223, 2.96830060594818683465361730485, 3.22844415429425033090082597003, 3.42724637947953735886102569459, 3.45050839901050357018106956739, 3.59927257851468378198523269186, 4.04451182093044245398413798917, 4.22835432149511970652131759568
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.