Dirichlet series
| L(s) = 1 | + 39·2-s + 4.37e3·3-s − 1.37e4·4-s + 9.37e4·5-s + 1.70e5·6-s − 7.05e5·7-s − 7.57e5·8-s + 1.11e7·9-s + 3.65e6·10-s − 1.28e7·11-s − 6.03e7·12-s − 6.95e6·13-s − 2.75e7·14-s + 4.10e8·15-s + 1.89e7·16-s − 4.90e7·17-s + 4.35e8·18-s + 3.47e7·19-s − 1.29e9·20-s − 3.08e9·21-s − 5.02e8·22-s − 7.25e8·23-s − 3.31e9·24-s + 5.12e9·25-s − 2.71e8·26-s + 2.16e10·27-s + 9.73e9·28-s + ⋯ |
| L(s) = 1 | + 0.430·2-s + 3.46·3-s − 1.68·4-s + 2.68·5-s + 1.49·6-s − 2.26·7-s − 1.02·8-s + 7·9-s + 1.15·10-s − 2.19·11-s − 5.83·12-s − 0.399·13-s − 0.977·14-s + 9.29·15-s + 0.282·16-s − 0.493·17-s + 3.01·18-s + 0.169·19-s − 4.51·20-s − 7.85·21-s − 0.944·22-s − 1.02·23-s − 3.54·24-s + 21/5·25-s − 0.172·26-s + 10.7·27-s + 3.81·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(14-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+13/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(3^{6} \cdot 5^{6} \cdot 7^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.03730\times 10^{12}\) |
| Root analytic conductor: | \(10.6109\) |
| Motivic weight: | \(13\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 3^{6} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [13/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(7)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{15}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - p^{6} T )^{6} \) |
| 5 | \( ( 1 - p^{6} T )^{6} \) | |
| 7 | \( ( 1 + p^{6} T )^{6} \) | |
| good | 2 | \( 1 - 39 T + 7659 p T^{2} - 47203 p^{3} T^{3} + 5547687 p^{5} T^{4} - 48986613 p^{7} T^{5} + 909927441 p^{11} T^{6} - 48986613 p^{20} T^{7} + 5547687 p^{31} T^{8} - 47203 p^{42} T^{9} + 7659 p^{53} T^{10} - 39 p^{65} T^{11} + p^{78} T^{12} \) |
| 11 | \( 1 + 1171260 p T + 1142316729270 p^{2} T^{2} + 1042362027567008356 p^{3} T^{3} + \)\(73\!\cdots\!15\)\( p^{4} T^{4} + \)\(46\!\cdots\!00\)\( p^{5} T^{5} + \)\(25\!\cdots\!56\)\( p^{7} T^{6} + \)\(46\!\cdots\!00\)\( p^{18} T^{7} + \)\(73\!\cdots\!15\)\( p^{30} T^{8} + 1042362027567008356 p^{42} T^{9} + 1142316729270 p^{54} T^{10} + 1171260 p^{66} T^{11} + p^{78} T^{12} \) | |
| 13 | \( 1 + 534624 p T + 1134059040788502 T^{2} - 20838103945097015744 p T^{3} + \)\(51\!\cdots\!19\)\( T^{4} - \)\(25\!\cdots\!16\)\( p T^{5} + \)\(15\!\cdots\!92\)\( T^{6} - \)\(25\!\cdots\!16\)\( p^{14} T^{7} + \)\(51\!\cdots\!19\)\( p^{26} T^{8} - 20838103945097015744 p^{40} T^{9} + 1134059040788502 p^{52} T^{10} + 534624 p^{66} T^{11} + p^{78} T^{12} \) | |
| 17 | \( 1 + 49082100 T + 28098723448151586 T^{2} + \)\(19\!\cdots\!24\)\( T^{3} + \)\(35\!\cdots\!31\)\( T^{4} + \)\(37\!\cdots\!32\)\( T^{5} + \)\(35\!\cdots\!76\)\( T^{6} + \)\(37\!\cdots\!32\)\( p^{13} T^{7} + \)\(35\!\cdots\!31\)\( p^{26} T^{8} + \)\(19\!\cdots\!24\)\( p^{39} T^{9} + 28098723448151586 p^{52} T^{10} + 49082100 p^{65} T^{11} + p^{78} T^{12} \) | |
| 19 | \( 1 - 34744140 T + 168911469635919798 T^{2} - \)\(95\!\cdots\!24\)\( T^{3} + \)\(14\!\cdots\!03\)\( T^{4} - \)\(73\!\cdots\!36\)\( T^{5} + \)\(74\!\cdots\!60\)\( T^{6} - \)\(73\!\cdots\!36\)\( p^{13} T^{7} + \)\(14\!\cdots\!03\)\( p^{26} T^{8} - \)\(95\!\cdots\!24\)\( p^{39} T^{9} + 168911469635919798 p^{52} T^{10} - 34744140 p^{65} T^{11} + p^{78} T^{12} \) | |
| 23 | \( 1 + 725199552 T + 1774235240787001434 T^{2} + \)\(14\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!87\)\( T^{4} + \)\(12\!\cdots\!20\)\( T^{5} + \)\(10\!\cdots\!64\)\( T^{6} + \)\(12\!\cdots\!20\)\( p^{13} T^{7} + \)\(17\!\cdots\!87\)\( p^{26} T^{8} + \)\(14\!\cdots\!40\)\( p^{39} T^{9} + 1774235240787001434 p^{52} T^{10} + 725199552 p^{65} T^{11} + p^{78} T^{12} \) | |
| 29 | \( 1 + 5126842092 T + 60099486552728963274 T^{2} + \)\(22\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!15\)\( T^{4} + \)\(41\!\cdots\!72\)\( T^{5} + \)\(19\!\cdots\!04\)\( T^{6} + \)\(41\!\cdots\!72\)\( p^{13} T^{7} + \)\(14\!\cdots\!15\)\( p^{26} T^{8} + \)\(22\!\cdots\!40\)\( p^{39} T^{9} + 60099486552728963274 p^{52} T^{10} + 5126842092 p^{65} T^{11} + p^{78} T^{12} \) | |
| 31 | \( 1 + 14501085108 T + \)\(16\!\cdots\!22\)\( T^{2} + \)\(41\!\cdots\!60\)\( p T^{3} + \)\(29\!\cdots\!89\)\( p T^{4} + \)\(53\!\cdots\!40\)\( T^{5} + \)\(29\!\cdots\!16\)\( T^{6} + \)\(53\!\cdots\!40\)\( p^{13} T^{7} + \)\(29\!\cdots\!89\)\( p^{27} T^{8} + \)\(41\!\cdots\!60\)\( p^{40} T^{9} + \)\(16\!\cdots\!22\)\( p^{52} T^{10} + 14501085108 p^{65} T^{11} + p^{78} T^{12} \) | |
| 37 | \( 1 + 28916102700 T + \)\(10\!\cdots\!26\)\( T^{2} + \)\(22\!\cdots\!76\)\( T^{3} + \)\(52\!\cdots\!51\)\( T^{4} + \)\(92\!\cdots\!08\)\( T^{5} + \)\(15\!\cdots\!76\)\( T^{6} + \)\(92\!\cdots\!08\)\( p^{13} T^{7} + \)\(52\!\cdots\!51\)\( p^{26} T^{8} + \)\(22\!\cdots\!76\)\( p^{39} T^{9} + \)\(10\!\cdots\!26\)\( p^{52} T^{10} + 28916102700 p^{65} T^{11} + p^{78} T^{12} \) | |
| 41 | \( 1 + 34779091956 T + \)\(27\!\cdots\!62\)\( T^{2} + \)\(89\!\cdots\!92\)\( T^{3} + \)\(43\!\cdots\!83\)\( T^{4} + \)\(12\!\cdots\!72\)\( T^{5} + \)\(49\!\cdots\!08\)\( T^{6} + \)\(12\!\cdots\!72\)\( p^{13} T^{7} + \)\(43\!\cdots\!83\)\( p^{26} T^{8} + \)\(89\!\cdots\!92\)\( p^{39} T^{9} + \)\(27\!\cdots\!62\)\( p^{52} T^{10} + 34779091956 p^{65} T^{11} + p^{78} T^{12} \) | |
| 43 | \( 1 + 65008551744 T + \)\(29\!\cdots\!14\)\( T^{2} + \)\(36\!\cdots\!44\)\( T^{3} + \)\(37\!\cdots\!03\)\( T^{4} + \)\(25\!\cdots\!76\)\( T^{5} + \)\(17\!\cdots\!40\)\( T^{6} + \)\(25\!\cdots\!76\)\( p^{13} T^{7} + \)\(37\!\cdots\!03\)\( p^{26} T^{8} + \)\(36\!\cdots\!44\)\( p^{39} T^{9} + \)\(29\!\cdots\!14\)\( p^{52} T^{10} + 65008551744 p^{65} T^{11} + p^{78} T^{12} \) | |
| 47 | \( 1 + 132188553792 T + \)\(23\!\cdots\!18\)\( T^{2} + \)\(24\!\cdots\!40\)\( T^{3} + \)\(27\!\cdots\!03\)\( T^{4} + \)\(22\!\cdots\!00\)\( T^{5} + \)\(19\!\cdots\!84\)\( T^{6} + \)\(22\!\cdots\!00\)\( p^{13} T^{7} + \)\(27\!\cdots\!03\)\( p^{26} T^{8} + \)\(24\!\cdots\!40\)\( p^{39} T^{9} + \)\(23\!\cdots\!18\)\( p^{52} T^{10} + 132188553792 p^{65} T^{11} + p^{78} T^{12} \) | |
| 53 | \( 1 + 33706005720 T + \)\(68\!\cdots\!02\)\( T^{2} + \)\(13\!\cdots\!76\)\( T^{3} + \)\(28\!\cdots\!47\)\( T^{4} + \)\(15\!\cdots\!76\)\( T^{5} + \)\(82\!\cdots\!44\)\( T^{6} + \)\(15\!\cdots\!76\)\( p^{13} T^{7} + \)\(28\!\cdots\!47\)\( p^{26} T^{8} + \)\(13\!\cdots\!76\)\( p^{39} T^{9} + \)\(68\!\cdots\!02\)\( p^{52} T^{10} + 33706005720 p^{65} T^{11} + p^{78} T^{12} \) | |
| 59 | \( 1 - 702982926888 T + \)\(40\!\cdots\!78\)\( T^{2} - \)\(14\!\cdots\!92\)\( T^{3} + \)\(58\!\cdots\!55\)\( T^{4} - \)\(27\!\cdots\!96\)\( p T^{5} + \)\(59\!\cdots\!12\)\( T^{6} - \)\(27\!\cdots\!96\)\( p^{14} T^{7} + \)\(58\!\cdots\!55\)\( p^{26} T^{8} - \)\(14\!\cdots\!92\)\( p^{39} T^{9} + \)\(40\!\cdots\!78\)\( p^{52} T^{10} - 702982926888 p^{65} T^{11} + p^{78} T^{12} \) | |
| 61 | \( 1 - 22471378668 T + \)\(44\!\cdots\!66\)\( T^{2} + \)\(11\!\cdots\!80\)\( T^{3} + \)\(84\!\cdots\!55\)\( T^{4} + \)\(46\!\cdots\!12\)\( T^{5} + \)\(13\!\cdots\!64\)\( T^{6} + \)\(46\!\cdots\!12\)\( p^{13} T^{7} + \)\(84\!\cdots\!55\)\( p^{26} T^{8} + \)\(11\!\cdots\!80\)\( p^{39} T^{9} + \)\(44\!\cdots\!66\)\( p^{52} T^{10} - 22471378668 p^{65} T^{11} + p^{78} T^{12} \) | |
| 67 | \( 1 + 463665460224 T + \)\(19\!\cdots\!42\)\( T^{2} + \)\(77\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!15\)\( T^{4} + \)\(68\!\cdots\!44\)\( T^{5} + \)\(13\!\cdots\!96\)\( T^{6} + \)\(68\!\cdots\!44\)\( p^{13} T^{7} + \)\(20\!\cdots\!15\)\( p^{26} T^{8} + \)\(77\!\cdots\!00\)\( p^{39} T^{9} + \)\(19\!\cdots\!42\)\( p^{52} T^{10} + 463665460224 p^{65} T^{11} + p^{78} T^{12} \) | |
| 71 | \( 1 + 1379022870660 T + \)\(58\!\cdots\!06\)\( T^{2} + \)\(68\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!75\)\( T^{4} + \)\(14\!\cdots\!00\)\( T^{5} + \)\(23\!\cdots\!60\)\( T^{6} + \)\(14\!\cdots\!00\)\( p^{13} T^{7} + \)\(15\!\cdots\!75\)\( p^{26} T^{8} + \)\(68\!\cdots\!00\)\( p^{39} T^{9} + \)\(58\!\cdots\!06\)\( p^{52} T^{10} + 1379022870660 p^{65} T^{11} + p^{78} T^{12} \) | |
| 73 | \( 1 + 1291647842112 T + \)\(78\!\cdots\!14\)\( T^{2} + \)\(86\!\cdots\!56\)\( T^{3} + \)\(27\!\cdots\!83\)\( T^{4} + \)\(25\!\cdots\!24\)\( T^{5} + \)\(57\!\cdots\!24\)\( T^{6} + \)\(25\!\cdots\!24\)\( p^{13} T^{7} + \)\(27\!\cdots\!83\)\( p^{26} T^{8} + \)\(86\!\cdots\!56\)\( p^{39} T^{9} + \)\(78\!\cdots\!14\)\( p^{52} T^{10} + 1291647842112 p^{65} T^{11} + p^{78} T^{12} \) | |
| 79 | \( 1 - 2800267319784 T + \)\(22\!\cdots\!34\)\( T^{2} - \)\(42\!\cdots\!20\)\( T^{3} + \)\(19\!\cdots\!15\)\( T^{4} - \)\(28\!\cdots\!24\)\( T^{5} + \)\(10\!\cdots\!76\)\( T^{6} - \)\(28\!\cdots\!24\)\( p^{13} T^{7} + \)\(19\!\cdots\!15\)\( p^{26} T^{8} - \)\(42\!\cdots\!20\)\( p^{39} T^{9} + \)\(22\!\cdots\!34\)\( p^{52} T^{10} - 2800267319784 p^{65} T^{11} + p^{78} T^{12} \) | |
| 83 | \( 1 - 223155398232 T + \)\(35\!\cdots\!58\)\( T^{2} - \)\(26\!\cdots\!00\)\( T^{3} + \)\(60\!\cdots\!15\)\( T^{4} - \)\(55\!\cdots\!92\)\( T^{5} + \)\(65\!\cdots\!04\)\( T^{6} - \)\(55\!\cdots\!92\)\( p^{13} T^{7} + \)\(60\!\cdots\!15\)\( p^{26} T^{8} - \)\(26\!\cdots\!00\)\( p^{39} T^{9} + \)\(35\!\cdots\!58\)\( p^{52} T^{10} - 223155398232 p^{65} T^{11} + p^{78} T^{12} \) | |
| 89 | \( 1 + 2140993348620 T + \)\(40\!\cdots\!78\)\( T^{2} + \)\(13\!\cdots\!44\)\( T^{3} + \)\(67\!\cdots\!43\)\( T^{4} + \)\(41\!\cdots\!36\)\( T^{5} + \)\(55\!\cdots\!40\)\( T^{6} + \)\(41\!\cdots\!36\)\( p^{13} T^{7} + \)\(67\!\cdots\!43\)\( p^{26} T^{8} + \)\(13\!\cdots\!44\)\( p^{39} T^{9} + \)\(40\!\cdots\!78\)\( p^{52} T^{10} + 2140993348620 p^{65} T^{11} + p^{78} T^{12} \) | |
| 97 | \( 1 + 7021989071064 T + \)\(28\!\cdots\!46\)\( T^{2} + \)\(84\!\cdots\!28\)\( T^{3} + \)\(30\!\cdots\!23\)\( T^{4} + \)\(53\!\cdots\!68\)\( T^{5} + \)\(21\!\cdots\!56\)\( T^{6} + \)\(53\!\cdots\!68\)\( p^{13} T^{7} + \)\(30\!\cdots\!23\)\( p^{26} T^{8} + \)\(84\!\cdots\!28\)\( p^{39} T^{9} + \)\(28\!\cdots\!46\)\( p^{52} T^{10} + 7021989071064 p^{65} T^{11} + p^{78} T^{12} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.98093837544150372516405371059, −5.37281277103056831803848649584, −5.19631312944154406673635365168, −5.07005080353169235995657034413, −5.02771961792071346906006778530, −5.01923676277257363457444530558, −4.88796036160463100324437167616, −3.99080075197093407644648220849, −3.97032158451737739576646960525, −3.87154383058396401427593258092, −3.80974900952462380775513675355, −3.63626614696211847119663956297, −3.45655909981198644120710921233, −2.93540759586204674023901785569, −2.89678999909050029922466528605, −2.71587681163086187400632809586, −2.64768455609074187706504715223, −2.26146477321996367445336907078, −2.24382002269991873190825019818, −2.22507722207397670416052361264, −1.81777471352151602831940619194, −1.44065259980506304045119583506, −1.28962438100710499591237049133, −1.25682318986671194548746052866, −1.20943480194484558400336304033, 0, 0, 0, 0, 0, 0, 1.20943480194484558400336304033, 1.25682318986671194548746052866, 1.28962438100710499591237049133, 1.44065259980506304045119583506, 1.81777471352151602831940619194, 2.22507722207397670416052361264, 2.24382002269991873190825019818, 2.26146477321996367445336907078, 2.64768455609074187706504715223, 2.71587681163086187400632809586, 2.89678999909050029922466528605, 2.93540759586204674023901785569, 3.45655909981198644120710921233, 3.63626614696211847119663956297, 3.80974900952462380775513675355, 3.87154383058396401427593258092, 3.97032158451737739576646960525, 3.99080075197093407644648220849, 4.88796036160463100324437167616, 5.01923676277257363457444530558, 5.02771961792071346906006778530, 5.07005080353169235995657034413, 5.19631312944154406673635365168, 5.37281277103056831803848649584, 5.98093837544150372516405371059
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.