Dirichlet series
| L(s) = 1 | − 512·4-s − 8.87e3·7-s − 1.17e5·13-s + 1.63e5·16-s + 2.70e5·19-s + 6.61e5·25-s + 4.54e6·28-s − 3.93e5·31-s + 1.83e6·37-s + 1.11e7·43-s + 9.53e6·49-s + 6.00e7·52-s + 1.21e7·61-s − 4.19e7·64-s − 8.03e7·67-s + 1.97e8·73-s − 1.38e8·76-s + 8.44e7·79-s + 1.04e9·91-s + 3.41e8·97-s − 3.38e8·100-s − 8.87e8·103-s + 6.17e8·109-s − 1.45e9·112-s + 6.31e8·121-s + 2.01e8·124-s + 127-s + ⋯ |
| L(s) = 1 | − 2·4-s − 3.69·7-s − 4.10·13-s + 5/2·16-s + 2.07·19-s + 1.69·25-s + 7.39·28-s − 0.425·31-s + 0.976·37-s + 3.25·43-s + 1.65·49-s + 8.21·52-s + 0.879·61-s − 5/2·64-s − 3.98·67-s + 6.94·73-s − 4.14·76-s + 2.16·79-s + 15.1·91-s + 3.85·97-s − 3.38·100-s − 7.88·103-s + 4.37·109-s − 9.24·112-s + 2.94·121-s + 0.851·124-s − 7.66·133-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+4)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.59837\times 10^{14}\) |
| Root analytic conductor: | \(8.12375\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{32} ,\ ( \ : [4]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(0.6285662210\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6285662210\) |
| \(L(5)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 - 661664 T^{2} + 594467354974 T^{4} - 8936656184743424 p^{2} T^{6} + \)\(19\!\cdots\!99\)\( p^{4} T^{8} - 8936656184743424 p^{18} T^{10} + 594467354974 p^{32} T^{12} - 661664 p^{48} T^{14} + p^{64} T^{16} \) |
| 7 | \( ( 1 + 634 p T + 505594 p^{2} T^{2} + 199152070 p^{3} T^{3} + 89403523714 p^{4} T^{4} + 199152070 p^{11} T^{5} + 505594 p^{18} T^{6} + 634 p^{25} T^{7} + p^{32} T^{8} )^{2} \) | |
| 11 | \( 1 - 631184252 T^{2} + 210122515694973400 T^{4} - \)\(56\!\cdots\!96\)\( T^{6} + \)\(13\!\cdots\!58\)\( T^{8} - \)\(56\!\cdots\!96\)\( p^{16} T^{10} + 210122515694973400 p^{32} T^{12} - 631184252 p^{48} T^{14} + p^{64} T^{16} \) | |
| 13 | \( ( 1 + 58690 T + 267688822 p T^{2} + 10999268277952 p T^{3} + 4346410063669029907 T^{4} + 10999268277952 p^{9} T^{5} + 267688822 p^{17} T^{6} + 58690 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 17 | \( 1 - 15649060760 T^{2} + \)\(19\!\cdots\!34\)\( T^{4} - \)\(13\!\cdots\!72\)\( T^{6} + \)\(10\!\cdots\!19\)\( T^{8} - \)\(13\!\cdots\!72\)\( p^{16} T^{10} + \)\(19\!\cdots\!34\)\( p^{32} T^{12} - 15649060760 p^{48} T^{14} + p^{64} T^{16} \) | |
| 19 | \( ( 1 - 135110 T + 32254494706 T^{2} - 3766954271134970 T^{3} + \)\(78\!\cdots\!54\)\( T^{4} - 3766954271134970 p^{8} T^{5} + 32254494706 p^{16} T^{6} - 135110 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 23 | \( 1 - 342953508980 T^{2} + \)\(62\!\cdots\!16\)\( T^{4} - \)\(78\!\cdots\!20\)\( T^{6} + \)\(71\!\cdots\!06\)\( T^{8} - \)\(78\!\cdots\!20\)\( p^{16} T^{10} + \)\(62\!\cdots\!16\)\( p^{32} T^{12} - 342953508980 p^{48} T^{14} + p^{64} T^{16} \) | |
| 29 | \( 1 - 2324399132840 T^{2} + \)\(95\!\cdots\!78\)\( p T^{4} - \)\(22\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!43\)\( T^{8} - \)\(22\!\cdots\!80\)\( p^{16} T^{10} + \)\(95\!\cdots\!78\)\( p^{33} T^{12} - 2324399132840 p^{48} T^{14} + p^{64} T^{16} \) | |
| 31 | \( ( 1 + 196672 T + 1670996532184 T^{2} - 399656795912615936 T^{3} + \)\(14\!\cdots\!22\)\( T^{4} - 399656795912615936 p^{8} T^{5} + 1670996532184 p^{16} T^{6} + 196672 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 37 | \( ( 1 - 915494 T + 13290230542654 T^{2} - 9321069227721324824 T^{3} + \)\(68\!\cdots\!07\)\( T^{4} - 9321069227721324824 p^{8} T^{5} + 13290230542654 p^{16} T^{6} - 915494 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 41 | \( 1 - 21476609469848 T^{2} + \)\(23\!\cdots\!84\)\( T^{4} - \)\(20\!\cdots\!20\)\( T^{6} + \)\(16\!\cdots\!42\)\( T^{8} - \)\(20\!\cdots\!20\)\( p^{16} T^{10} + \)\(23\!\cdots\!84\)\( p^{32} T^{12} - 21476609469848 p^{48} T^{14} + p^{64} T^{16} \) | |
| 43 | \( ( 1 - 5567618 T + 27250065112594 T^{2} - 74047286684777602430 T^{3} + \)\(22\!\cdots\!22\)\( T^{4} - 74047286684777602430 p^{8} T^{5} + 27250065112594 p^{16} T^{6} - 5567618 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 47 | \( 1 - 2719825992808 p T^{2} + \)\(76\!\cdots\!16\)\( T^{4} - \)\(28\!\cdots\!88\)\( T^{6} + \)\(78\!\cdots\!90\)\( T^{8} - \)\(28\!\cdots\!88\)\( p^{16} T^{10} + \)\(76\!\cdots\!16\)\( p^{32} T^{12} - 2719825992808 p^{49} T^{14} + p^{64} T^{16} \) | |
| 53 | \( 1 - 209196847256696 T^{2} + \)\(25\!\cdots\!48\)\( T^{4} - \)\(22\!\cdots\!60\)\( T^{6} + \)\(15\!\cdots\!98\)\( T^{8} - \)\(22\!\cdots\!60\)\( p^{16} T^{10} + \)\(25\!\cdots\!48\)\( p^{32} T^{12} - 209196847256696 p^{48} T^{14} + p^{64} T^{16} \) | |
| 59 | \( 1 - 1088674734907640 T^{2} + \)\(53\!\cdots\!52\)\( T^{4} - \)\(15\!\cdots\!28\)\( T^{6} + \)\(27\!\cdots\!54\)\( T^{8} - \)\(15\!\cdots\!28\)\( p^{16} T^{10} + \)\(53\!\cdots\!52\)\( p^{32} T^{12} - 1088674734907640 p^{48} T^{14} + p^{64} T^{16} \) | |
| 61 | \( ( 1 - 6092102 T + 458913069050674 T^{2} - \)\(92\!\cdots\!24\)\( T^{3} + \)\(10\!\cdots\!67\)\( T^{4} - \)\(92\!\cdots\!24\)\( p^{8} T^{5} + 458913069050674 p^{16} T^{6} - 6092102 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 67 | \( ( 1 + 40177858 T + 27341072812750 p T^{2} + \)\(43\!\cdots\!70\)\( T^{3} + \)\(11\!\cdots\!06\)\( T^{4} + \)\(43\!\cdots\!70\)\( p^{8} T^{5} + 27341072812750 p^{17} T^{6} + 40177858 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 71 | \( 1 - 2956882545670292 T^{2} + \)\(42\!\cdots\!12\)\( T^{4} - \)\(41\!\cdots\!20\)\( T^{6} + \)\(30\!\cdots\!34\)\( T^{8} - \)\(41\!\cdots\!20\)\( p^{16} T^{10} + \)\(42\!\cdots\!12\)\( p^{32} T^{12} - 2956882545670292 p^{48} T^{14} + p^{64} T^{16} \) | |
| 73 | \( ( 1 - 98542880 T + 5718687406904854 T^{2} - \)\(23\!\cdots\!12\)\( T^{3} + \)\(73\!\cdots\!99\)\( T^{4} - \)\(23\!\cdots\!12\)\( p^{8} T^{5} + 5718687406904854 p^{16} T^{6} - 98542880 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 79 | \( ( 1 - 42225926 T + 1514385882457330 T^{2} - \)\(75\!\cdots\!34\)\( T^{3} + \)\(52\!\cdots\!10\)\( T^{4} - \)\(75\!\cdots\!34\)\( p^{8} T^{5} + 1514385882457330 p^{16} T^{6} - 42225926 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 83 | \( 1 - 6987969045312440 T^{2} + \)\(21\!\cdots\!32\)\( T^{4} - \)\(34\!\cdots\!28\)\( T^{6} + \)\(50\!\cdots\!14\)\( T^{8} - \)\(34\!\cdots\!28\)\( p^{16} T^{10} + \)\(21\!\cdots\!32\)\( p^{32} T^{12} - 6987969045312440 p^{48} T^{14} + p^{64} T^{16} \) | |
| 89 | \( 1 - 12389815547838512 T^{2} + \)\(99\!\cdots\!74\)\( T^{4} - \)\(58\!\cdots\!60\)\( T^{6} + \)\(26\!\cdots\!95\)\( T^{8} - \)\(58\!\cdots\!60\)\( p^{16} T^{10} + \)\(99\!\cdots\!74\)\( p^{32} T^{12} - 12389815547838512 p^{48} T^{14} + p^{64} T^{16} \) | |
| 97 | \( ( 1 - 170568464 T + 22467183167929432 T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!30\)\( T^{4} - \)\(19\!\cdots\!80\)\( p^{8} T^{5} + 22467183167929432 p^{16} T^{6} - 170568464 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
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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
−4.43329902519457220339823658467, −4.06858272618333753978385298087, −3.82029644935601849876109611816, −3.74253719530711999031067419731, −3.65585765890095961972491350134, −3.49317481692258953464687564948, −3.35835542406641963367878152238, −3.16305467996213042064260113876, −2.99068925370474074380008724316, −2.82393714169690310587027056598, −2.61435416719910056702796121793, −2.55611855536053374170899808270, −2.55137787745017680312204722275, −2.32445794704013231947651634398, −1.94660556983890851511802763683, −1.73063820956850096207760301638, −1.53092308033709128563808825005, −1.35074560142666889127378069781, −0.903684851125420247907695008225, −0.844315332310920484888870373235, −0.66397986049757119928875975777, −0.65684166529317716810486083706, −0.35595863417613802871883126837, −0.23278412989715698207983892120, −0.14843035047486259924333793619, 0.14843035047486259924333793619, 0.23278412989715698207983892120, 0.35595863417613802871883126837, 0.65684166529317716810486083706, 0.66397986049757119928875975777, 0.844315332310920484888870373235, 0.903684851125420247907695008225, 1.35074560142666889127378069781, 1.53092308033709128563808825005, 1.73063820956850096207760301638, 1.94660556983890851511802763683, 2.32445794704013231947651634398, 2.55137787745017680312204722275, 2.55611855536053374170899808270, 2.61435416719910056702796121793, 2.82393714169690310587027056598, 2.99068925370474074380008724316, 3.16305467996213042064260113876, 3.35835542406641963367878152238, 3.49317481692258953464687564948, 3.65585765890095961972491350134, 3.74253719530711999031067419731, 3.82029644935601849876109611816, 4.06858272618333753978385298087, 4.43329902519457220339823658467
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.