Dirichlet series
| L(s) = 1 | − 4·2-s + 4·4-s − 5·5-s − 7-s + 20·10-s + 155·11-s + 7·13-s + 4·14-s − 161·17-s − 272·19-s − 20·20-s − 620·22-s − 628·23-s + 129·25-s − 28·26-s − 4·28-s − 33·29-s + 323·31-s + 64·32-s + 644·34-s + 5·35-s + 49·37-s + 1.08e3·38-s − 361·41-s + 1.44e3·43-s + 620·44-s + 2.51e3·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 0.447·5-s − 0.0539·7-s + 0.632·10-s + 4.24·11-s + 0.149·13-s + 0.0763·14-s − 2.29·17-s − 3.28·19-s − 0.223·20-s − 6.00·22-s − 5.69·23-s + 1.03·25-s − 0.211·26-s − 0.0269·28-s − 0.211·29-s + 1.87·31-s + 0.353·32-s + 3.24·34-s + 0.0241·35-s + 0.217·37-s + 4.64·38-s − 1.37·41-s + 5.11·43-s + 2.12·44-s + 8.05·46-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{16} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.46936\times 10^{8}\) |
| Root analytic conductor: | \(3.41794\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{16} \cdot 11^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.6325204435\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6325204435\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 3 | \( 1 \) | |
| 11 | \( 1 - 155 T + 13111 T^{2} - 68645 p T^{3} + 264456 p^{2} T^{4} - 68645 p^{4} T^{5} + 13111 p^{6} T^{6} - 155 p^{9} T^{7} + p^{12} T^{8} \) | |
| good | 5 | \( 1 + p T - 104 T^{2} + 326 p T^{3} + 10006 T^{4} - 56777 p T^{5} + 2835571 T^{6} + 1265534 p^{2} T^{7} - 387953844 T^{8} + 1265534 p^{5} T^{9} + 2835571 p^{6} T^{10} - 56777 p^{10} T^{11} + 10006 p^{12} T^{12} + 326 p^{16} T^{13} - 104 p^{18} T^{14} + p^{22} T^{15} + p^{24} T^{16} \) |
| 7 | \( 1 + T + 12 T^{2} - 1104 T^{3} - 87888 T^{4} - 922401 T^{5} + 28062801 T^{6} + 211635072 T^{7} + 13215739812 T^{8} + 211635072 p^{3} T^{9} + 28062801 p^{6} T^{10} - 922401 p^{9} T^{11} - 87888 p^{12} T^{12} - 1104 p^{15} T^{13} + 12 p^{18} T^{14} + p^{21} T^{15} + p^{24} T^{16} \) | |
| 13 | \( 1 - 7 T + 1930 T^{2} + 79384 T^{3} + 10224596 T^{4} + 326819897 T^{5} + 769971197 p T^{6} + 1235524128730 T^{7} + 47709276128944 T^{8} + 1235524128730 p^{3} T^{9} + 769971197 p^{7} T^{10} + 326819897 p^{9} T^{11} + 10224596 p^{12} T^{12} + 79384 p^{15} T^{13} + 1930 p^{18} T^{14} - 7 p^{21} T^{15} + p^{24} T^{16} \) | |
| 17 | \( 1 + 161 T + 5020 T^{2} - 211188 T^{3} + 21130156 T^{4} + 1323457999 T^{5} - 265862283281 T^{6} - 778694329080 p T^{7} + 277440196391544 T^{8} - 778694329080 p^{4} T^{9} - 265862283281 p^{6} T^{10} + 1323457999 p^{9} T^{11} + 21130156 p^{12} T^{12} - 211188 p^{15} T^{13} + 5020 p^{18} T^{14} + 161 p^{21} T^{15} + p^{24} T^{16} \) | |
| 19 | \( 1 + 272 T + 33201 T^{2} + 3039584 T^{3} + 292957015 T^{4} + 21884364992 T^{5} + 909384182595 T^{6} + 32108296725040 T^{7} + 2715838419478696 T^{8} + 32108296725040 p^{3} T^{9} + 909384182595 p^{6} T^{10} + 21884364992 p^{9} T^{11} + 292957015 p^{12} T^{12} + 3039584 p^{15} T^{13} + 33201 p^{18} T^{14} + 272 p^{21} T^{15} + p^{24} T^{16} \) | |
| 23 | \( ( 1 + 314 T + 61816 T^{2} + 8042722 T^{3} + 950275950 T^{4} + 8042722 p^{3} T^{5} + 61816 p^{6} T^{6} + 314 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 29 | \( 1 + 33 T + 14176 T^{2} - 2091144 T^{3} + 419909220 T^{4} - 81926257017 T^{5} + 7198320043835 T^{6} - 2719607629962300 T^{7} + 63828590693142176 T^{8} - 2719607629962300 p^{3} T^{9} + 7198320043835 p^{6} T^{10} - 81926257017 p^{9} T^{11} + 419909220 p^{12} T^{12} - 2091144 p^{15} T^{13} + 14176 p^{18} T^{14} + 33 p^{21} T^{15} + p^{24} T^{16} \) | |
| 31 | \( 1 - 323 T - 4098 T^{2} + 5799192 T^{3} + 305184598 T^{4} - 93424183073 T^{5} + 1165032156515 p T^{6} - 3925905573337100 T^{7} - 564631402757797204 T^{8} - 3925905573337100 p^{3} T^{9} + 1165032156515 p^{7} T^{10} - 93424183073 p^{9} T^{11} + 305184598 p^{12} T^{12} + 5799192 p^{15} T^{13} - 4098 p^{18} T^{14} - 323 p^{21} T^{15} + p^{24} T^{16} \) | |
| 37 | \( 1 - 49 T - 7868 T^{2} + 4260746 T^{3} - 566286138 T^{4} + 1042572020189 T^{5} + 91349107260671 T^{6} - 14416097444880138 T^{7} + 5705343181934634812 T^{8} - 14416097444880138 p^{3} T^{9} + 91349107260671 p^{6} T^{10} + 1042572020189 p^{9} T^{11} - 566286138 p^{12} T^{12} + 4260746 p^{15} T^{13} - 7868 p^{18} T^{14} - 49 p^{21} T^{15} + p^{24} T^{16} \) | |
| 41 | \( 1 + 361 T - 19140 T^{2} - 12862580 T^{3} + 3853225260 T^{4} + 605378813703 T^{5} - 283363657590617 T^{6} - 99783171614693080 T^{7} - 22290482644766813160 T^{8} - 99783171614693080 p^{3} T^{9} - 283363657590617 p^{6} T^{10} + 605378813703 p^{9} T^{11} + 3853225260 p^{12} T^{12} - 12862580 p^{15} T^{13} - 19140 p^{18} T^{14} + 361 p^{21} T^{15} + p^{24} T^{16} \) | |
| 43 | \( ( 1 - 721 T + 420117 T^{2} - 154459221 T^{3} + 51447883420 T^{4} - 154459221 p^{3} T^{5} + 420117 p^{6} T^{6} - 721 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 47 | \( 1 - 1069 T + 301070 T^{2} + 43368042 T^{3} - 24642745634 T^{4} - 6223420821981 T^{5} + 4635059307315839 T^{6} - 746453682627128500 T^{7} + 29698413240018564184 T^{8} - 746453682627128500 p^{3} T^{9} + 4635059307315839 p^{6} T^{10} - 6223420821981 p^{9} T^{11} - 24642745634 p^{12} T^{12} + 43368042 p^{15} T^{13} + 301070 p^{18} T^{14} - 1069 p^{21} T^{15} + p^{24} T^{16} \) | |
| 53 | \( 1 - 281 T - 343802 T^{2} + 60864958 T^{3} + 33213391926 T^{4} - 4932564267181 T^{5} + 8874412003121445 T^{6} - 107096647673733916 T^{7} - \)\(25\!\cdots\!56\)\( T^{8} - 107096647673733916 p^{3} T^{9} + 8874412003121445 p^{6} T^{10} - 4932564267181 p^{9} T^{11} + 33213391926 p^{12} T^{12} + 60864958 p^{15} T^{13} - 343802 p^{18} T^{14} - 281 p^{21} T^{15} + p^{24} T^{16} \) | |
| 59 | \( 1 - 128 T - 87119 T^{2} + 50674984 T^{3} + 39669673335 T^{4} - 496570131448 T^{5} - 4049042036547885 T^{6} + 22790010341236160 T^{7} + \)\(27\!\cdots\!36\)\( T^{8} + 22790010341236160 p^{3} T^{9} - 4049042036547885 p^{6} T^{10} - 496570131448 p^{9} T^{11} + 39669673335 p^{12} T^{12} + 50674984 p^{15} T^{13} - 87119 p^{18} T^{14} - 128 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 + 617 T - 60798 T^{2} - 112995618 T^{3} + 22253617738 T^{4} + 1960695737 p^{2} T^{5} - 13214872371636415 T^{6} + 3429112403997460700 T^{7} + \)\(68\!\cdots\!36\)\( T^{8} + 3429112403997460700 p^{3} T^{9} - 13214872371636415 p^{6} T^{10} + 1960695737 p^{11} T^{11} + 22253617738 p^{12} T^{12} - 112995618 p^{15} T^{13} - 60798 p^{18} T^{14} + 617 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( ( 1 - 289 T + 686735 T^{2} - 243308709 T^{3} + 267199450216 T^{4} - 243308709 p^{3} T^{5} + 686735 p^{6} T^{6} - 289 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 71 | \( 1 + 115 T - 138428 T^{2} - 42960000 T^{3} + 77925723838 T^{4} + 71653925274035 T^{5} + 12191505393571699 T^{6} - 3285417296443548450 T^{7} + \)\(11\!\cdots\!80\)\( T^{8} - 3285417296443548450 p^{3} T^{9} + 12191505393571699 p^{6} T^{10} + 71653925274035 p^{9} T^{11} + 77925723838 p^{12} T^{12} - 42960000 p^{15} T^{13} - 138428 p^{18} T^{14} + 115 p^{21} T^{15} + p^{24} T^{16} \) | |
| 73 | \( 1 + 1487 T + 142728 T^{2} - 14316264 p T^{3} - 569519703768 T^{4} + 366060407928453 T^{5} + 393291607153115139 T^{6} - 68019158899540158384 T^{7} - \)\(19\!\cdots\!28\)\( T^{8} - 68019158899540158384 p^{3} T^{9} + 393291607153115139 p^{6} T^{10} + 366060407928453 p^{9} T^{11} - 569519703768 p^{12} T^{12} - 14316264 p^{16} T^{13} + 142728 p^{18} T^{14} + 1487 p^{21} T^{15} + p^{24} T^{16} \) | |
| 79 | \( 1 - 71 T - 500222 T^{2} - 73013624 T^{3} + 350790961998 T^{4} + 97550067497959 T^{5} - 161731343227399015 T^{6} + 8534351363848547400 T^{7} + \)\(59\!\cdots\!76\)\( T^{8} + 8534351363848547400 p^{3} T^{9} - 161731343227399015 p^{6} T^{10} + 97550067497959 p^{9} T^{11} + 350790961998 p^{12} T^{12} - 73013624 p^{15} T^{13} - 500222 p^{18} T^{14} - 71 p^{21} T^{15} + p^{24} T^{16} \) | |
| 83 | \( 1 + 1942 T + 875395 T^{2} - 857035884 T^{3} - 790010610449 T^{4} + 618864428088728 T^{5} + 845503237658086801 T^{6} - \)\(17\!\cdots\!30\)\( T^{7} - \)\(59\!\cdots\!96\)\( T^{8} - \)\(17\!\cdots\!30\)\( p^{3} T^{9} + 845503237658086801 p^{6} T^{10} + 618864428088728 p^{9} T^{11} - 790010610449 p^{12} T^{12} - 857035884 p^{15} T^{13} + 875395 p^{18} T^{14} + 1942 p^{21} T^{15} + p^{24} T^{16} \) | |
| 89 | \( ( 1 - 1101 T + 2406895 T^{2} - 1914547747 T^{3} + 2334666485008 T^{4} - 1914547747 p^{3} T^{5} + 2406895 p^{6} T^{6} - 1101 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 97 | \( 1 + 5128 T + 10343177 T^{2} + 9500759764 T^{3} + 1479496697751 T^{4} - 4959875190278012 T^{5} - 1475957592224572125 T^{6} + \)\(96\!\cdots\!12\)\( T^{7} + \)\(15\!\cdots\!64\)\( T^{8} + \)\(96\!\cdots\!12\)\( p^{3} T^{9} - 1475957592224572125 p^{6} T^{10} - 4959875190278012 p^{9} T^{11} + 1479496697751 p^{12} T^{12} + 9500759764 p^{15} T^{13} + 10343177 p^{18} T^{14} + 5128 p^{21} T^{15} + p^{24} T^{16} \) | |
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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
−5.19468886477606543335421489984, −4.67485596278319877328169084163, −4.64101402438889301911726390728, −4.46283447349523431190264329834, −4.40571185273252384607696928085, −4.20960322749867423597688744890, −4.20810190088723532540820292354, −4.00078545056211054302871338701, −3.90073815411787567737681251929, −3.85845056805569284350356103262, −3.76554483297691817402730714377, −3.46049062598469595599816983887, −2.85762738684233419360763824539, −2.62837264606473019680998892951, −2.56419730493745183683138542707, −2.40218465819802130724660017286, −2.39928628298257780809585660593, −1.80581440364272826723430779815, −1.75699250519106437388691991434, −1.51536303468452797985591739803, −1.43510482163779866977397970884, −0.862200880216326687201293337108, −0.75316968661260623291676758183, −0.33461357162972588458734610786, −0.19172092051388117392342167545, 0.19172092051388117392342167545, 0.33461357162972588458734610786, 0.75316968661260623291676758183, 0.862200880216326687201293337108, 1.43510482163779866977397970884, 1.51536303468452797985591739803, 1.75699250519106437388691991434, 1.80581440364272826723430779815, 2.39928628298257780809585660593, 2.40218465819802130724660017286, 2.56419730493745183683138542707, 2.62837264606473019680998892951, 2.85762738684233419360763824539, 3.46049062598469595599816983887, 3.76554483297691817402730714377, 3.85845056805569284350356103262, 3.90073815411787567737681251929, 4.00078545056211054302871338701, 4.20810190088723532540820292354, 4.20960322749867423597688744890, 4.40571185273252384607696928085, 4.46283447349523431190264329834, 4.64101402438889301911726390728, 4.67485596278319877328169084163, 5.19468886477606543335421489984
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.