Dirichlet series
| L(s) = 1 | − 5.15e10·2-s + 1.54e16·3-s + 1.77e21·4-s + 2.01e24·5-s − 7.97e26·6-s − 2.09e29·7-s − 5.07e31·8-s + 5.24e32·9-s − 1.03e35·10-s − 2.70e35·11-s + 2.74e37·12-s − 8.39e38·13-s + 1.07e40·14-s + 3.11e40·15-s + 1.30e42·16-s + 1.92e41·17-s − 2.70e43·18-s + 8.82e43·19-s + 3.56e45·20-s − 3.23e45·21-s + 1.39e46·22-s − 2.00e47·23-s − 7.85e47·24-s − 6.56e47·25-s + 4.32e49·26-s + 8.58e48·27-s − 3.70e50·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 0.535·3-s + 3·4-s + 1.54·5-s − 1.13·6-s − 1.46·7-s − 3.53·8-s + 0.628·9-s − 3.28·10-s − 0.319·11-s + 1.60·12-s − 3.11·13-s + 3.09·14-s + 0.829·15-s + 15/4·16-s + 0.0683·17-s − 1.33·18-s + 0.674·19-s + 4.64·20-s − 0.782·21-s + 0.677·22-s − 2.10·23-s − 1.89·24-s − 0.387·25-s + 6.59·26-s + 0.356·27-s − 4.38·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(70-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+69/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(6\) |
| Conductor: | \(8\) = \(2^{3}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(219288.\) |
| Root analytic conductor: | \(7.76550\) |
| Motivic weight: | \(69\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(3\) |
| Selberg data: | \((6,\ 8,\ (\ :69/2, 69/2, 69/2),\ -1)\) |
Particular Values
| \(L(35)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{71}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| bad | 2 | $C_1$ | \( ( 1 + p^{34} T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 21238244672828 p^{6} T - \)\(59\!\cdots\!87\)\( p^{14} T^{2} + \)\(19\!\cdots\!56\)\( p^{30} T^{3} - \)\(59\!\cdots\!87\)\( p^{83} T^{4} - 21238244672828 p^{144} T^{5} + p^{207} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - \)\(40\!\cdots\!34\)\( p T + \)\(15\!\cdots\!03\)\( p^{5} T^{2} - \)\(41\!\cdots\!16\)\( p^{13} T^{3} + \)\(15\!\cdots\!03\)\( p^{74} T^{4} - \)\(40\!\cdots\!34\)\( p^{139} T^{5} + p^{207} T^{6} \) | |
| 7 | $S_4\times C_2$ | \( 1 + \)\(29\!\cdots\!28\)\( p T + \)\(18\!\cdots\!57\)\( p^{6} T^{2} + \)\(23\!\cdots\!16\)\( p^{13} T^{3} + \)\(18\!\cdots\!57\)\( p^{75} T^{4} + \)\(29\!\cdots\!28\)\( p^{139} T^{5} + p^{207} T^{6} \) | |
| 11 | $S_4\times C_2$ | \( 1 + \)\(24\!\cdots\!64\)\( p T + \)\(72\!\cdots\!95\)\( p^{5} T^{2} + \)\(27\!\cdots\!60\)\( p^{9} T^{3} + \)\(72\!\cdots\!95\)\( p^{74} T^{4} + \)\(24\!\cdots\!64\)\( p^{139} T^{5} + p^{207} T^{6} \) | |
| 13 | $S_4\times C_2$ | \( 1 + \)\(83\!\cdots\!58\)\( T + \)\(25\!\cdots\!03\)\( p^{2} T^{2} + \)\(29\!\cdots\!36\)\( p^{6} T^{3} + \)\(25\!\cdots\!03\)\( p^{71} T^{4} + \)\(83\!\cdots\!58\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 17 | $S_4\times C_2$ | \( 1 - \)\(11\!\cdots\!62\)\( p T + \)\(24\!\cdots\!51\)\( p^{3} T^{2} - \)\(10\!\cdots\!44\)\( p^{5} T^{3} + \)\(24\!\cdots\!51\)\( p^{72} T^{4} - \)\(11\!\cdots\!62\)\( p^{139} T^{5} + p^{207} T^{6} \) | |
| 19 | $S_4\times C_2$ | \( 1 - \)\(46\!\cdots\!20\)\( p T + \)\(29\!\cdots\!43\)\( p^{3} T^{2} + \)\(45\!\cdots\!60\)\( p^{6} T^{3} + \)\(29\!\cdots\!43\)\( p^{72} T^{4} - \)\(46\!\cdots\!20\)\( p^{139} T^{5} + p^{207} T^{6} \) | |
| 23 | $S_4\times C_2$ | \( 1 + \)\(87\!\cdots\!96\)\( p T + \)\(65\!\cdots\!13\)\( p^{2} T^{2} + \)\(27\!\cdots\!92\)\( p^{3} T^{3} + \)\(65\!\cdots\!13\)\( p^{71} T^{4} + \)\(87\!\cdots\!96\)\( p^{139} T^{5} + p^{207} T^{6} \) | |
| 29 | $S_4\times C_2$ | \( 1 - \)\(35\!\cdots\!70\)\( p T + \)\(83\!\cdots\!83\)\( p T^{2} - \)\(67\!\cdots\!60\)\( p^{3} T^{3} + \)\(83\!\cdots\!83\)\( p^{70} T^{4} - \)\(35\!\cdots\!70\)\( p^{139} T^{5} + p^{207} T^{6} \) | |
| 31 | $S_4\times C_2$ | \( 1 - \)\(12\!\cdots\!76\)\( p T + \)\(11\!\cdots\!25\)\( p^{2} T^{2} - \)\(12\!\cdots\!60\)\( p^{4} T^{3} + \)\(11\!\cdots\!25\)\( p^{71} T^{4} - \)\(12\!\cdots\!76\)\( p^{139} T^{5} + p^{207} T^{6} \) | |
| 37 | $S_4\times C_2$ | \( 1 + \)\(41\!\cdots\!26\)\( T + \)\(42\!\cdots\!23\)\( T^{2} + \)\(31\!\cdots\!16\)\( p T^{3} + \)\(42\!\cdots\!23\)\( p^{69} T^{4} + \)\(41\!\cdots\!26\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 41 | $S_4\times C_2$ | \( 1 + \)\(15\!\cdots\!54\)\( T + \)\(31\!\cdots\!55\)\( p T^{2} + \)\(40\!\cdots\!20\)\( p^{2} T^{3} + \)\(31\!\cdots\!55\)\( p^{70} T^{4} + \)\(15\!\cdots\!54\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 43 | $S_4\times C_2$ | \( 1 - \)\(35\!\cdots\!52\)\( T + \)\(42\!\cdots\!79\)\( p T^{2} - \)\(19\!\cdots\!24\)\( p^{2} T^{3} + \)\(42\!\cdots\!79\)\( p^{70} T^{4} - \)\(35\!\cdots\!52\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 47 | $S_4\times C_2$ | \( 1 + \)\(16\!\cdots\!56\)\( T + \)\(13\!\cdots\!79\)\( p T^{2} + \)\(33\!\cdots\!68\)\( p^{2} T^{3} + \)\(13\!\cdots\!79\)\( p^{70} T^{4} + \)\(16\!\cdots\!56\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 53 | $S_4\times C_2$ | \( 1 + \)\(37\!\cdots\!98\)\( T + \)\(31\!\cdots\!39\)\( p T^{2} + \)\(39\!\cdots\!96\)\( p^{2} T^{3} + \)\(31\!\cdots\!39\)\( p^{70} T^{4} + \)\(37\!\cdots\!98\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 59 | $S_4\times C_2$ | \( 1 + \)\(50\!\cdots\!40\)\( p^{2} T + \)\(93\!\cdots\!57\)\( p^{2} T^{2} + \)\(27\!\cdots\!80\)\( p^{3} T^{3} + \)\(93\!\cdots\!57\)\( p^{71} T^{4} + \)\(50\!\cdots\!40\)\( p^{140} T^{5} + p^{207} T^{6} \) | |
| 61 | $S_4\times C_2$ | \( 1 + \)\(53\!\cdots\!94\)\( T + \)\(58\!\cdots\!35\)\( p T^{2} - \)\(18\!\cdots\!00\)\( p^{2} T^{3} + \)\(58\!\cdots\!35\)\( p^{70} T^{4} + \)\(53\!\cdots\!94\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 67 | $S_4\times C_2$ | \( 1 + \)\(24\!\cdots\!28\)\( p T + \)\(81\!\cdots\!97\)\( p^{2} T^{2} + \)\(11\!\cdots\!64\)\( p^{3} T^{3} + \)\(81\!\cdots\!97\)\( p^{71} T^{4} + \)\(24\!\cdots\!28\)\( p^{139} T^{5} + p^{207} T^{6} \) | |
| 71 | $S_4\times C_2$ | \( 1 + \)\(40\!\cdots\!24\)\( T + \)\(13\!\cdots\!85\)\( T^{2} + \)\(46\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!85\)\( p^{69} T^{4} + \)\(40\!\cdots\!24\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 73 | $S_4\times C_2$ | \( 1 + \)\(45\!\cdots\!18\)\( T + \)\(83\!\cdots\!47\)\( T^{2} + \)\(11\!\cdots\!84\)\( T^{3} + \)\(83\!\cdots\!47\)\( p^{69} T^{4} + \)\(45\!\cdots\!18\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 79 | $S_4\times C_2$ | \( 1 - \)\(56\!\cdots\!60\)\( T + \)\(30\!\cdots\!57\)\( T^{2} - \)\(89\!\cdots\!80\)\( T^{3} + \)\(30\!\cdots\!57\)\( p^{69} T^{4} - \)\(56\!\cdots\!60\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 83 | $S_4\times C_2$ | \( 1 - \)\(64\!\cdots\!92\)\( T + \)\(19\!\cdots\!59\)\( p T^{2} - \)\(12\!\cdots\!96\)\( T^{3} + \)\(19\!\cdots\!59\)\( p^{70} T^{4} - \)\(64\!\cdots\!92\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 89 | $S_4\times C_2$ | \( 1 + \)\(33\!\cdots\!30\)\( T + \)\(59\!\cdots\!27\)\( T^{2} - \)\(10\!\cdots\!60\)\( T^{3} + \)\(59\!\cdots\!27\)\( p^{69} T^{4} + \)\(33\!\cdots\!30\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
| 97 | $S_4\times C_2$ | \( 1 + \)\(45\!\cdots\!86\)\( T + \)\(37\!\cdots\!83\)\( T^{2} + \)\(11\!\cdots\!52\)\( T^{3} + \)\(37\!\cdots\!83\)\( p^{69} T^{4} + \)\(45\!\cdots\!86\)\( p^{138} T^{5} + p^{207} T^{6} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.13999031237188797654755140252, −12.31060433406153966367367710774, −11.99473387537745184712562510096, −11.87287685996414072521265580791, −10.42086452214232670997396253116, −10.13323317834017042948061891026, −9.969922536532931450111677181010, −9.862203183326930478625494974780, −9.198389182929337230498148399067, −9.059589997102762532559549895894, −7.961002680026113336258418024764, −7.85554169959480868090558237827, −7.27009757837899972003552630646, −6.85281810963275674325746423913, −6.24261181046014926882506028474, −6.02494895818035198921634816151, −5.27818019958938766618962504307, −4.74312734936639500371990134339, −3.84639846529468383706830643085, −3.05292134010637277801594866804, −2.74365597658141933836770012229, −2.39884731750681231822141180102, −1.81020731469456865505901381563, −1.74984323485674083109362096588, −0.996830538164865634063952724421, 0, 0, 0, 0.996830538164865634063952724421, 1.74984323485674083109362096588, 1.81020731469456865505901381563, 2.39884731750681231822141180102, 2.74365597658141933836770012229, 3.05292134010637277801594866804, 3.84639846529468383706830643085, 4.74312734936639500371990134339, 5.27818019958938766618962504307, 6.02494895818035198921634816151, 6.24261181046014926882506028474, 6.85281810963275674325746423913, 7.27009757837899972003552630646, 7.85554169959480868090558237827, 7.961002680026113336258418024764, 9.059589997102762532559549895894, 9.198389182929337230498148399067, 9.862203183326930478625494974780, 9.969922536532931450111677181010, 10.13323317834017042948061891026, 10.42086452214232670997396253116, 11.87287685996414072521265580791, 11.99473387537745184712562510096, 12.31060433406153966367367710774, 13.13999031237188797654755140252