Dirichlet series
| L(s) = 1 | − 2·2-s + 8·4-s + 16·5-s + 14·7-s − 16·8-s − 6·9-s − 32·10-s − 14·11-s + 2·13-s − 28·14-s + 25·16-s + 18·17-s + 12·18-s − 94·19-s + 128·20-s + 28·22-s − 30·23-s + 128·25-s − 4·26-s + 112·28-s − 64·29-s + 80·31-s − 56·32-s − 36·34-s + 224·35-s − 48·36-s + 110·37-s + ⋯ |
| L(s) = 1 | − 2-s + 2·4-s + 16/5·5-s + 2·7-s − 2·8-s − 2/3·9-s − 3.19·10-s − 1.27·11-s + 2/13·13-s − 2·14-s + 1.56·16-s + 1.05·17-s + 2/3·18-s − 4.94·19-s + 32/5·20-s + 1.27·22-s − 1.30·23-s + 5.11·25-s − 0.153·26-s + 4·28-s − 2.20·29-s + 2.58·31-s − 7/4·32-s − 1.05·34-s + 32/5·35-s − 4/3·36-s + 2.97·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.62628\) |
| Root analytic conductor: | \(1.03086\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 13^{8} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(1.720507347\) |
| \(L(\frac12)\) | \(\approx\) | \(1.720507347\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | \( 1 - 2 T + 352 T^{2} + 64 p T^{3} + 31 p^{3} T^{4} + 64 p^{3} T^{5} + 352 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} \) | |
| good | 2 | \( 1 + p T - p^{2} T^{2} - p^{3} T^{3} + 23 T^{4} + 13 p^{2} T^{5} - 3 p^{4} T^{6} - 9 p T^{7} + 409 T^{8} - 9 p^{3} T^{9} - 3 p^{8} T^{10} + 13 p^{8} T^{11} + 23 p^{8} T^{12} - p^{13} T^{13} - p^{14} T^{14} + p^{15} T^{15} + p^{16} T^{16} \) |
| 5 | \( 1 - 16 T + 128 T^{2} - 848 T^{3} + 5594 T^{4} - 34208 T^{5} + 190848 T^{6} - 1044576 T^{7} + 5483131 T^{8} - 1044576 p^{2} T^{9} + 190848 p^{4} T^{10} - 34208 p^{6} T^{11} + 5594 p^{8} T^{12} - 848 p^{10} T^{13} + 128 p^{12} T^{14} - 16 p^{14} T^{15} + p^{16} T^{16} \) | |
| 7 | \( 1 - 2 p T + 104 T^{2} - 508 T^{3} + 464 p T^{4} - 35458 T^{5} + 305892 T^{6} - 2169234 T^{7} + 15335923 T^{8} - 2169234 p^{2} T^{9} + 305892 p^{4} T^{10} - 35458 p^{6} T^{11} + 464 p^{9} T^{12} - 508 p^{10} T^{13} + 104 p^{12} T^{14} - 2 p^{15} T^{15} + p^{16} T^{16} \) | |
| 11 | \( 1 + 14 T - 4 T^{2} - 2408 T^{3} - 25264 T^{4} - 87122 T^{5} + 115308 p T^{6} + 24412374 T^{7} + 295012723 T^{8} + 24412374 p^{2} T^{9} + 115308 p^{5} T^{10} - 87122 p^{6} T^{11} - 25264 p^{8} T^{12} - 2408 p^{10} T^{13} - 4 p^{12} T^{14} + 14 p^{14} T^{15} + p^{16} T^{16} \) | |
| 17 | \( 1 - 18 T + 1084 T^{2} - 17568 T^{3} + 643213 T^{4} - 10164852 T^{5} + 272868280 T^{6} - 3989091474 T^{7} + 87854539888 T^{8} - 3989091474 p^{2} T^{9} + 272868280 p^{4} T^{10} - 10164852 p^{6} T^{11} + 643213 p^{8} T^{12} - 17568 p^{10} T^{13} + 1084 p^{12} T^{14} - 18 p^{14} T^{15} + p^{16} T^{16} \) | |
| 19 | \( 1 + 94 T + 4976 T^{2} + 170216 T^{3} + 4086560 T^{4} + 63491486 T^{5} + 409228740 T^{6} - 9450263490 T^{7} - 315850663997 T^{8} - 9450263490 p^{2} T^{9} + 409228740 p^{4} T^{10} + 63491486 p^{6} T^{11} + 4086560 p^{8} T^{12} + 170216 p^{10} T^{13} + 4976 p^{12} T^{14} + 94 p^{14} T^{15} + p^{16} T^{16} \) | |
| 23 | \( 1 + 30 T + 1072 T^{2} + 23160 T^{3} + 186352 T^{4} + 7099002 T^{5} + 239637352 T^{6} + 9579372978 T^{7} + 309063720787 T^{8} + 9579372978 p^{2} T^{9} + 239637352 p^{4} T^{10} + 7099002 p^{6} T^{11} + 186352 p^{8} T^{12} + 23160 p^{10} T^{13} + 1072 p^{12} T^{14} + 30 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( 1 + 64 T + 570 T^{2} - 23104 T^{3} + 14077 p T^{4} + 23009952 T^{5} - 564914390 T^{6} - 34531584032 T^{7} - 917264410092 T^{8} - 34531584032 p^{2} T^{9} - 564914390 p^{4} T^{10} + 23009952 p^{6} T^{11} + 14077 p^{9} T^{12} - 23104 p^{10} T^{13} + 570 p^{12} T^{14} + 64 p^{14} T^{15} + p^{16} T^{16} \) | |
| 31 | \( 1 - 80 T + 3200 T^{2} - 135280 T^{3} + 4242692 T^{4} - 52834960 T^{5} - 199478400 T^{6} + 65576969040 T^{7} - 3589529796602 T^{8} + 65576969040 p^{2} T^{9} - 199478400 p^{4} T^{10} - 52834960 p^{6} T^{11} + 4242692 p^{8} T^{12} - 135280 p^{10} T^{13} + 3200 p^{12} T^{14} - 80 p^{14} T^{15} + p^{16} T^{16} \) | |
| 37 | \( 1 - 110 T + 5036 T^{2} - 73276 T^{3} - 3853975 T^{4} + 260028560 T^{5} - 6673799184 T^{6} + 61429700466 T^{7} + 420091956616 T^{8} + 61429700466 p^{2} T^{9} - 6673799184 p^{4} T^{10} + 260028560 p^{6} T^{11} - 3853975 p^{8} T^{12} - 73276 p^{10} T^{13} + 5036 p^{12} T^{14} - 110 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( 1 - 22 T - 316 T^{2} + 57268 T^{3} - 5345971 T^{4} + 114872692 T^{5} + 2159280720 T^{6} - 211854113778 T^{7} + 16381346614048 T^{8} - 211854113778 p^{2} T^{9} + 2159280720 p^{4} T^{10} + 114872692 p^{6} T^{11} - 5345971 p^{8} T^{12} + 57268 p^{10} T^{13} - 316 p^{12} T^{14} - 22 p^{14} T^{15} + p^{16} T^{16} \) | |
| 43 | \( 1 + 54 T + 4336 T^{2} + 181656 T^{3} + 11624608 T^{4} + 550335870 T^{5} + 29853321952 T^{6} + 1300891274154 T^{7} + 57482863306483 T^{8} + 1300891274154 p^{2} T^{9} + 29853321952 p^{4} T^{10} + 550335870 p^{6} T^{11} + 11624608 p^{8} T^{12} + 181656 p^{10} T^{13} + 4336 p^{12} T^{14} + 54 p^{14} T^{15} + p^{16} T^{16} \) | |
| 47 | \( 1 + 332 T + 55112 T^{2} + 6410488 T^{3} + 594684584 T^{4} + 45938277352 T^{5} + 3024429486528 T^{6} + 173231732518956 T^{7} + 8698084578326674 T^{8} + 173231732518956 p^{2} T^{9} + 3024429486528 p^{4} T^{10} + 45938277352 p^{6} T^{11} + 594684584 p^{8} T^{12} + 6410488 p^{10} T^{13} + 55112 p^{12} T^{14} + 332 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( ( 1 - 16 T + 8014 T^{2} - 15880 T^{3} + 28210279 T^{4} - 15880 p^{2} T^{5} + 8014 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 59 | \( 1 - 52 T - 5488 T^{2} + 261172 T^{3} + 16427102 T^{4} - 567494336 T^{5} - 36066454200 T^{6} + 583606781256 T^{7} + 74252077939459 T^{8} + 583606781256 p^{2} T^{9} - 36066454200 p^{4} T^{10} - 567494336 p^{6} T^{11} + 16427102 p^{8} T^{12} + 261172 p^{10} T^{13} - 5488 p^{12} T^{14} - 52 p^{14} T^{15} + p^{16} T^{16} \) | |
| 61 | \( 1 - 46 T - 6672 T^{2} + 610192 T^{3} + 11688581 T^{4} - 2461656624 T^{5} + 42881771812 T^{6} + 3796382369486 T^{7} - 218746580552784 T^{8} + 3796382369486 p^{2} T^{9} + 42881771812 p^{4} T^{10} - 2461656624 p^{6} T^{11} + 11688581 p^{8} T^{12} + 610192 p^{10} T^{13} - 6672 p^{12} T^{14} - 46 p^{14} T^{15} + p^{16} T^{16} \) | |
| 67 | \( 1 - 86 T + 764 T^{2} - 20620 T^{3} + 12967832 T^{4} - 137049826 T^{5} - 31634618220 T^{6} + 5310424618446 T^{7} - 487922992236125 T^{8} + 5310424618446 p^{2} T^{9} - 31634618220 p^{4} T^{10} - 137049826 p^{6} T^{11} + 12967832 p^{8} T^{12} - 20620 p^{10} T^{13} + 764 p^{12} T^{14} - 86 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( 1 - 94 T + 8372 T^{2} + 213148 T^{3} - 29995288 T^{4} + 4215412522 T^{5} - 6456617604 T^{6} + 2247155123694 T^{7} + 1054875342767587 T^{8} + 2247155123694 p^{2} T^{9} - 6456617604 p^{4} T^{10} + 4215412522 p^{6} T^{11} - 29995288 p^{8} T^{12} + 213148 p^{10} T^{13} + 8372 p^{12} T^{14} - 94 p^{14} T^{15} + p^{16} T^{16} \) | |
| 73 | \( 1 - 56 T + 1568 T^{2} + 119384 T^{3} - 4266814 T^{4} - 10588768 p T^{5} + 10715616 p^{2} T^{6} + 13696800 p^{3} T^{7} - 45661085 p^{4} T^{8} + 13696800 p^{5} T^{9} + 10715616 p^{6} T^{10} - 10588768 p^{7} T^{11} - 4266814 p^{8} T^{12} + 119384 p^{10} T^{13} + 1568 p^{12} T^{14} - 56 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( ( 1 + 40 T + 5812 T^{2} - 1472 T^{3} + 25997374 T^{4} - 1472 p^{2} T^{5} + 5812 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 83 | \( 1 - 136 T + 9248 T^{2} + 229660 T^{3} + 13723208 T^{4} - 7779217124 T^{5} + 957433159080 T^{6} + 1699662332136 T^{7} - 3192170311751342 T^{8} + 1699662332136 p^{2} T^{9} + 957433159080 p^{4} T^{10} - 7779217124 p^{6} T^{11} + 13723208 p^{8} T^{12} + 229660 p^{10} T^{13} + 9248 p^{12} T^{14} - 136 p^{14} T^{15} + p^{16} T^{16} \) | |
| 89 | \( 1 + 128 T - 28 T^{2} + 594748 T^{3} + 192061778 T^{4} + 13599254908 T^{5} + 757532650320 T^{6} + 109535143452204 T^{7} + 12735179269105603 T^{8} + 109535143452204 p^{2} T^{9} + 757532650320 p^{4} T^{10} + 13599254908 p^{6} T^{11} + 192061778 p^{8} T^{12} + 594748 p^{10} T^{13} - 28 p^{12} T^{14} + 128 p^{14} T^{15} + p^{16} T^{16} \) | |
| 97 | \( 1 + 40 T - 232 T^{2} - 433444 T^{3} - 74341426 T^{4} - 2637932212 T^{5} + 90237844536 T^{6} + 37927168744092 T^{7} - 2020519333655021 T^{8} + 37927168744092 p^{2} T^{9} + 90237844536 p^{4} T^{10} - 2637932212 p^{6} T^{11} - 74341426 p^{8} T^{12} - 433444 p^{10} T^{13} - 232 p^{12} T^{14} + 40 p^{14} T^{15} + p^{16} T^{16} \) | |
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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
−8.040767165938691909339371245142, −7.54921854326466228988720089802, −7.53521561206887819578670225885, −7.07906771723573799069626000657, −6.88425018168525994849957676679, −6.58690827256320507364615294741, −6.52827343986116804744454988478, −6.27988546117402270706333354727, −6.24408346412871547384512598894, −5.95982108526250435752845165636, −5.88460057450082241779309133835, −5.75404185204452917209957699487, −5.18822426000018497585025072046, −5.07246054828626169489668773777, −4.89449874160980558785888368432, −4.79039634929267519553093428192, −4.20404966055548007345096185389, −4.15706526152139468190056892150, −3.56769911569006616679307730878, −2.92169342248754909725123137733, −2.79778612842431276013033908534, −2.24419999336455408904465669134, −2.00100864246532679916516909613, −1.95061897208858984015443582517, −1.80196690425333050960417183134, 1.80196690425333050960417183134, 1.95061897208858984015443582517, 2.00100864246532679916516909613, 2.24419999336455408904465669134, 2.79778612842431276013033908534, 2.92169342248754909725123137733, 3.56769911569006616679307730878, 4.15706526152139468190056892150, 4.20404966055548007345096185389, 4.79039634929267519553093428192, 4.89449874160980558785888368432, 5.07246054828626169489668773777, 5.18822426000018497585025072046, 5.75404185204452917209957699487, 5.88460057450082241779309133835, 5.95982108526250435752845165636, 6.24408346412871547384512598894, 6.27988546117402270706333354727, 6.52827343986116804744454988478, 6.58690827256320507364615294741, 6.88425018168525994849957676679, 7.07906771723573799069626000657, 7.53521561206887819578670225885, 7.54921854326466228988720089802, 8.040767165938691909339371245142
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.