Dirichlet series
| L(s) = 1 | + 2·2-s + 2·4-s − 220·5-s − 2.35e3·7-s + 2.91e3·8-s − 440·10-s − 2.31e4·11-s − 1.19e5·13-s − 4.70e3·14-s + 6.27e4·16-s + 2.65e5·17-s − 440·20-s − 4.63e4·22-s − 2.88e4·23-s − 1.45e5·25-s − 2.38e5·26-s − 4.70e3·28-s − 7.47e5·31-s − 9.07e5·32-s + 5.31e5·34-s + 5.17e5·35-s − 4.54e5·37-s − 6.40e5·40-s − 2.48e6·41-s + 7.92e5·43-s − 4.63e4·44-s − 5.77e4·46-s + ⋯ |
| L(s) = 1 | + 1/8·2-s + 0.00781·4-s − 0.351·5-s − 0.979·7-s + 0.710·8-s − 0.0439·10-s − 1.58·11-s − 4.17·13-s − 0.122·14-s + 0.957·16-s + 3.17·17-s − 0.00274·20-s − 0.198·22-s − 0.103·23-s − 0.373·25-s − 0.521·26-s − 0.00765·28-s − 0.809·31-s − 0.865·32-s + 0.397·34-s + 0.344·35-s − 0.242·37-s − 0.250·40-s − 0.880·41-s + 0.231·43-s − 0.0123·44-s − 0.0129·46-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+4)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(3^{12} \cdot 5^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.79545\times 10^{7}\) |
| Root analytic conductor: | \(4.28159\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 3^{12} \cdot 5^{6} ,\ ( \ : [4]^{6} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(3.328738899\) |
| \(L(\frac12)\) | \(\approx\) | \(3.328738899\) |
| \(L(5)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 44 p T + 311 p^{4} T^{2} + 2552 p^{7} T^{3} + 311 p^{12} T^{4} + 44 p^{17} T^{5} + p^{24} T^{6} \) | |
| good | 2 | \( 1 - p T + p T^{2} - 91 p^{5} T^{3} - 799 p^{6} T^{4} + 8873 p^{7} T^{5} + 16177 p^{7} T^{6} + 8873 p^{15} T^{7} - 799 p^{22} T^{8} - 91 p^{29} T^{9} + p^{33} T^{10} - p^{41} T^{11} + p^{48} T^{12} \) |
| 7 | \( 1 + 48 p^{2} T + 1152 p^{4} T^{2} - 6884936 p^{3} T^{3} + 2619602799 p^{4} T^{4} + 7645236875928 p^{5} T^{5} + 2444629423375904 p^{6} T^{6} + 7645236875928 p^{13} T^{7} + 2619602799 p^{20} T^{8} - 6884936 p^{27} T^{9} + 1152 p^{36} T^{10} + 48 p^{42} T^{11} + p^{48} T^{12} \) | |
| 11 | \( ( 1 + 11596 T + 493098215 T^{2} + 5106545516920 T^{3} + 493098215 p^{8} T^{4} + 11596 p^{16} T^{5} + p^{24} T^{6} )^{2} \) | |
| 13 | \( 1 + 119142 T + 7097408082 T^{2} + 332686420223782 T^{3} + 13680551086291514559 T^{4} + \)\(46\!\cdots\!56\)\( T^{5} + \)\(13\!\cdots\!76\)\( T^{6} + \)\(46\!\cdots\!56\)\( p^{8} T^{7} + 13680551086291514559 p^{16} T^{8} + 332686420223782 p^{24} T^{9} + 7097408082 p^{32} T^{10} + 119142 p^{40} T^{11} + p^{48} T^{12} \) | |
| 17 | \( 1 - 265502 T + 35245656002 T^{2} - 3505767378301982 T^{3} + \)\(37\!\cdots\!19\)\( T^{4} - \)\(40\!\cdots\!76\)\( T^{5} + \)\(37\!\cdots\!76\)\( T^{6} - \)\(40\!\cdots\!76\)\( p^{8} T^{7} + \)\(37\!\cdots\!19\)\( p^{16} T^{8} - 3505767378301982 p^{24} T^{9} + 35245656002 p^{32} T^{10} - 265502 p^{40} T^{11} + p^{48} T^{12} \) | |
| 19 | \( 1 - 66391003446 T^{2} + \)\(21\!\cdots\!15\)\( T^{4} - \)\(42\!\cdots\!20\)\( T^{6} + \)\(21\!\cdots\!15\)\( p^{16} T^{8} - 66391003446 p^{32} T^{10} + p^{48} T^{12} \) | |
| 23 | \( 1 + 1256 p T + 788768 p^{2} T^{2} + 2207314762520128 T^{3} + \)\(86\!\cdots\!39\)\( T^{4} + \)\(83\!\cdots\!64\)\( T^{5} + \)\(12\!\cdots\!16\)\( T^{6} + \)\(83\!\cdots\!64\)\( p^{8} T^{7} + \)\(86\!\cdots\!39\)\( p^{16} T^{8} + 2207314762520128 p^{24} T^{9} + 788768 p^{34} T^{10} + 1256 p^{41} T^{11} + p^{48} T^{12} \) | |
| 29 | \( 1 - 1726260912966 T^{2} + \)\(13\!\cdots\!15\)\( T^{4} - \)\(77\!\cdots\!20\)\( T^{6} + \)\(13\!\cdots\!15\)\( p^{16} T^{8} - 1726260912966 p^{32} T^{10} + p^{48} T^{12} \) | |
| 31 | \( ( 1 + 373824 T + 1438821265815 T^{2} + 103488660558742480 T^{3} + 1438821265815 p^{8} T^{4} + 373824 p^{16} T^{5} + p^{24} T^{6} )^{2} \) | |
| 37 | \( 1 + 454002 T + 103058908002 T^{2} + 1591257258997412242 T^{3} + \)\(36\!\cdots\!59\)\( T^{4} + \)\(11\!\cdots\!36\)\( T^{5} + \)\(25\!\cdots\!36\)\( T^{6} + \)\(11\!\cdots\!36\)\( p^{8} T^{7} + \)\(36\!\cdots\!59\)\( p^{16} T^{8} + 1591257258997412242 p^{24} T^{9} + 103058908002 p^{32} T^{10} + 454002 p^{40} T^{11} + p^{48} T^{12} \) | |
| 41 | \( ( 1 + 1244716 T + 19779856453415 T^{2} + 17348876621060070520 T^{3} + 19779856453415 p^{8} T^{4} + 1244716 p^{16} T^{5} + p^{24} T^{6} )^{2} \) | |
| 43 | \( 1 - 792648 T + 314145425952 T^{2} - 6701894073526462448 T^{3} + \)\(27\!\cdots\!99\)\( T^{4} - \)\(18\!\cdots\!04\)\( T^{5} + \)\(86\!\cdots\!96\)\( T^{6} - \)\(18\!\cdots\!04\)\( p^{8} T^{7} + \)\(27\!\cdots\!99\)\( p^{16} T^{8} - 6701894073526462448 p^{24} T^{9} + 314145425952 p^{32} T^{10} - 792648 p^{40} T^{11} + p^{48} T^{12} \) | |
| 47 | \( 1 - 325816 p T + 53078032928 p^{2} T^{2} - \)\(85\!\cdots\!72\)\( T^{3} + \)\(63\!\cdots\!79\)\( T^{4} - \)\(35\!\cdots\!16\)\( T^{5} + \)\(17\!\cdots\!16\)\( T^{6} - \)\(35\!\cdots\!16\)\( p^{8} T^{7} + \)\(63\!\cdots\!79\)\( p^{16} T^{8} - \)\(85\!\cdots\!72\)\( p^{24} T^{9} + 53078032928 p^{34} T^{10} - 325816 p^{41} T^{11} + p^{48} T^{12} \) | |
| 53 | \( 1 - 13509122 T + 91248188605442 T^{2} - \)\(98\!\cdots\!42\)\( T^{3} + \)\(11\!\cdots\!79\)\( T^{4} - \)\(81\!\cdots\!76\)\( T^{5} + \)\(50\!\cdots\!36\)\( T^{6} - \)\(81\!\cdots\!76\)\( p^{8} T^{7} + \)\(11\!\cdots\!79\)\( p^{16} T^{8} - \)\(98\!\cdots\!42\)\( p^{24} T^{9} + 91248188605442 p^{32} T^{10} - 13509122 p^{40} T^{11} + p^{48} T^{12} \) | |
| 59 | \( 1 - 413223229068726 T^{2} + \)\(98\!\cdots\!15\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(98\!\cdots\!15\)\( p^{16} T^{8} - 413223229068726 p^{32} T^{10} + p^{48} T^{12} \) | |
| 61 | \( ( 1 - 12055596 T + 200152007609415 T^{2} + \)\(24\!\cdots\!80\)\( T^{3} + 200152007609415 p^{8} T^{4} - 12055596 p^{16} T^{5} + p^{24} T^{6} )^{2} \) | |
| 67 | \( 1 + 32827752 T + 538830650686752 T^{2} + \)\(16\!\cdots\!32\)\( T^{3} + \)\(54\!\cdots\!19\)\( T^{4} + \)\(88\!\cdots\!76\)\( T^{5} + \)\(12\!\cdots\!76\)\( T^{6} + \)\(88\!\cdots\!76\)\( p^{8} T^{7} + \)\(54\!\cdots\!19\)\( p^{16} T^{8} + \)\(16\!\cdots\!32\)\( p^{24} T^{9} + 538830650686752 p^{32} T^{10} + 32827752 p^{40} T^{11} + p^{48} T^{12} \) | |
| 71 | \( ( 1 - 6996464 T + 1071469029384215 T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + 1071469029384215 p^{8} T^{4} - 6996464 p^{16} T^{5} + p^{24} T^{6} )^{2} \) | |
| 73 | \( 1 - 111859638 T + 6256289306745522 T^{2} - \)\(29\!\cdots\!78\)\( T^{3} + \)\(12\!\cdots\!39\)\( T^{4} - \)\(42\!\cdots\!64\)\( T^{5} + \)\(12\!\cdots\!16\)\( T^{6} - \)\(42\!\cdots\!64\)\( p^{8} T^{7} + \)\(12\!\cdots\!39\)\( p^{16} T^{8} - \)\(29\!\cdots\!78\)\( p^{24} T^{9} + 6256289306745522 p^{32} T^{10} - 111859638 p^{40} T^{11} + p^{48} T^{12} \) | |
| 79 | \( 1 - 4185433961698566 T^{2} + \)\(55\!\cdots\!15\)\( T^{4} - \)\(46\!\cdots\!20\)\( T^{6} + \)\(55\!\cdots\!15\)\( p^{16} T^{8} - 4185433961698566 p^{32} T^{10} + p^{48} T^{12} \) | |
| 83 | \( 1 - 14768432 T + 109053291869312 T^{2} - \)\(28\!\cdots\!12\)\( T^{3} + \)\(10\!\cdots\!19\)\( T^{4} - \)\(10\!\cdots\!16\)\( T^{5} + \)\(83\!\cdots\!56\)\( T^{6} - \)\(10\!\cdots\!16\)\( p^{8} T^{7} + \)\(10\!\cdots\!19\)\( p^{16} T^{8} - \)\(28\!\cdots\!12\)\( p^{24} T^{9} + 109053291869312 p^{32} T^{10} - 14768432 p^{40} T^{11} + p^{48} T^{12} \) | |
| 89 | \( 1 - 8165894455313286 T^{2} + \)\(62\!\cdots\!15\)\( T^{4} - \)\(26\!\cdots\!20\)\( T^{6} + \)\(62\!\cdots\!15\)\( p^{16} T^{8} - 8165894455313286 p^{32} T^{10} + p^{48} T^{12} \) | |
| 97 | \( 1 + 186656202 T + 17420268872532402 T^{2} + \)\(93\!\cdots\!22\)\( T^{3} + \)\(66\!\cdots\!79\)\( T^{4} + \)\(11\!\cdots\!16\)\( T^{5} + \)\(13\!\cdots\!16\)\( T^{6} + \)\(11\!\cdots\!16\)\( p^{8} T^{7} + \)\(66\!\cdots\!79\)\( p^{16} T^{8} + \)\(93\!\cdots\!22\)\( p^{24} T^{9} + 17420268872532402 p^{32} T^{10} + 186656202 p^{40} T^{11} + p^{48} T^{12} \) | |
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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
−7.31431689890558990453278439310, −7.12122331507014765639175697130, −6.85135918345786827736108307359, −6.68470865271232699048368026348, −6.36200958481848116998214941467, −5.54465065492804848449689145104, −5.50262297780090319601002982099, −5.40026336373054847065532106255, −5.40009883975456987706913488993, −5.31020829373764732595388023127, −4.70008257319482315888031536596, −4.39879984731979531386814861837, −4.08022893680600692059266699379, −3.76864501468675446529485174879, −3.53974041598075630496011032943, −3.19390325903650332216816295279, −2.73843146706331489213997769811, −2.68103376054177015661813175788, −2.39141781716257731087265727384, −2.06603728370764696427572965762, −1.68034801586190557266335142379, −1.17055407090526799329798797296, −0.67754169716033985306631437015, −0.43502008096455994607391502958, −0.35384879631331589134279140242, 0.35384879631331589134279140242, 0.43502008096455994607391502958, 0.67754169716033985306631437015, 1.17055407090526799329798797296, 1.68034801586190557266335142379, 2.06603728370764696427572965762, 2.39141781716257731087265727384, 2.68103376054177015661813175788, 2.73843146706331489213997769811, 3.19390325903650332216816295279, 3.53974041598075630496011032943, 3.76864501468675446529485174879, 4.08022893680600692059266699379, 4.39879984731979531386814861837, 4.70008257319482315888031536596, 5.31020829373764732595388023127, 5.40009883975456987706913488993, 5.40026336373054847065532106255, 5.50262297780090319601002982099, 5.54465065492804848449689145104, 6.36200958481848116998214941467, 6.68470865271232699048368026348, 6.85135918345786827736108307359, 7.12122331507014765639175697130, 7.31431689890558990453278439310
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.