Dirichlet series
| L(s) = 1 | − 2.44e10·3-s + 5.35e14·5-s − 3.01e17·7-s + 4.40e19·9-s − 2.66e22·11-s + 2.67e24·13-s − 1.30e25·15-s − 4.08e26·17-s − 1.57e27·19-s + 7.36e27·21-s + 1.21e27·23-s − 2.71e30·25-s + 2.22e30·27-s + 5.71e31·29-s + 2.55e32·31-s + 6.50e32·33-s − 1.61e32·35-s − 2.32e33·37-s − 6.52e34·39-s + 2.57e34·41-s − 2.40e35·43-s + 2.35e34·45-s − 3.08e35·47-s − 1.51e36·49-s + 9.97e36·51-s + 1.50e37·53-s − 1.42e37·55-s + ⋯ |
| L(s) = 1 | − 1.34·3-s + 0.501·5-s − 0.204·7-s + 0.134·9-s − 1.08·11-s + 3.00·13-s − 0.676·15-s − 1.43·17-s − 0.506·19-s + 0.275·21-s + 0.00642·23-s − 2.39·25-s + 0.374·27-s + 2.06·29-s + 2.20·31-s + 1.46·33-s − 0.102·35-s − 0.445·37-s − 4.04·39-s + 0.544·41-s − 1.82·43-s + 0.0673·45-s − 0.346·47-s − 0.692·49-s + 1.93·51-s + 1.27·53-s − 0.545·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+43/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(6\) |
| Conductor: | \(4096\) = \(2^{12}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(6.57879\times 10^{6}\) |
| Root analytic conductor: | \(13.6885\) |
| Motivic weight: | \(43\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(3\) |
| Selberg data: | \((6,\ 4096,\ (\ :43/2, 43/2, 43/2),\ -1)\) |
Particular Values
| \(L(22)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{45}{2})\) | 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 | \( 1 \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + 903756956 p^{3} T + 9337321074916153 p^{10} T^{2} + \)\(87\!\cdots\!20\)\( p^{19} T^{3} + 9337321074916153 p^{53} T^{4} + 903756956 p^{89} T^{5} + p^{129} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 107041076154834 p T + \)\(48\!\cdots\!99\)\( p^{4} T^{2} - \)\(51\!\cdots\!56\)\( p^{9} T^{3} + \)\(48\!\cdots\!99\)\( p^{47} T^{4} - 107041076154834 p^{87} T^{5} + p^{129} T^{6} \) | |
| 7 | $S_4\times C_2$ | \( 1 + 6162682156438344 p^{2} T + \)\(95\!\cdots\!99\)\( p^{5} T^{2} + \)\(94\!\cdots\!00\)\( p^{9} T^{3} + \)\(95\!\cdots\!99\)\( p^{48} T^{4} + 6162682156438344 p^{88} T^{5} + p^{129} T^{6} \) | |
| 11 | $S_4\times C_2$ | \( 1 + \)\(24\!\cdots\!36\)\( p T + \)\(13\!\cdots\!65\)\( p^{2} T^{2} + \)\(17\!\cdots\!20\)\( p^{5} T^{3} + \)\(13\!\cdots\!65\)\( p^{45} T^{4} + \)\(24\!\cdots\!36\)\( p^{87} T^{5} + p^{129} T^{6} \) | |
| 13 | $S_4\times C_2$ | \( 1 - \)\(20\!\cdots\!14\)\( p T + \)\(20\!\cdots\!71\)\( p^{3} T^{2} - \)\(98\!\cdots\!20\)\( p^{6} T^{3} + \)\(20\!\cdots\!71\)\( p^{46} T^{4} - \)\(20\!\cdots\!14\)\( p^{87} T^{5} + p^{129} T^{6} \) | |
| 17 | $S_4\times C_2$ | \( 1 + \)\(40\!\cdots\!94\)\( T + \)\(98\!\cdots\!79\)\( p T^{2} + \)\(14\!\cdots\!40\)\( p^{3} T^{3} + \)\(98\!\cdots\!79\)\( p^{44} T^{4} + \)\(40\!\cdots\!94\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 19 | $S_4\times C_2$ | \( 1 + \)\(83\!\cdots\!00\)\( p T + \)\(15\!\cdots\!03\)\( p^{3} T^{2} + \)\(21\!\cdots\!00\)\( p^{5} T^{3} + \)\(15\!\cdots\!03\)\( p^{46} T^{4} + \)\(83\!\cdots\!00\)\( p^{87} T^{5} + p^{129} T^{6} \) | |
| 23 | $S_4\times C_2$ | \( 1 - \)\(12\!\cdots\!48\)\( T + \)\(23\!\cdots\!99\)\( p T^{2} - \)\(70\!\cdots\!20\)\( p^{2} T^{3} + \)\(23\!\cdots\!99\)\( p^{44} T^{4} - \)\(12\!\cdots\!48\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 29 | $S_4\times C_2$ | \( 1 - \)\(57\!\cdots\!50\)\( T + \)\(94\!\cdots\!23\)\( p T^{2} - \)\(95\!\cdots\!00\)\( p^{2} T^{3} + \)\(94\!\cdots\!23\)\( p^{44} T^{4} - \)\(57\!\cdots\!50\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 31 | $S_4\times C_2$ | \( 1 - \)\(82\!\cdots\!04\)\( p T + \)\(55\!\cdots\!65\)\( p^{2} T^{2} - \)\(22\!\cdots\!80\)\( p^{3} T^{3} + \)\(55\!\cdots\!65\)\( p^{45} T^{4} - \)\(82\!\cdots\!04\)\( p^{87} T^{5} + p^{129} T^{6} \) | |
| 37 | $S_4\times C_2$ | \( 1 + \)\(62\!\cdots\!62\)\( p T + \)\(57\!\cdots\!47\)\( p^{2} T^{2} + \)\(24\!\cdots\!20\)\( p^{3} T^{3} + \)\(57\!\cdots\!47\)\( p^{45} T^{4} + \)\(62\!\cdots\!62\)\( p^{87} T^{5} + p^{129} T^{6} \) | |
| 41 | $S_4\times C_2$ | \( 1 - \)\(62\!\cdots\!26\)\( p T + \)\(15\!\cdots\!15\)\( p^{2} T^{2} - \)\(12\!\cdots\!20\)\( p^{3} T^{3} + \)\(15\!\cdots\!15\)\( p^{45} T^{4} - \)\(62\!\cdots\!26\)\( p^{87} T^{5} + p^{129} T^{6} \) | |
| 43 | $S_4\times C_2$ | \( 1 + \)\(24\!\cdots\!92\)\( T + \)\(66\!\cdots\!57\)\( T^{2} + \)\(83\!\cdots\!00\)\( T^{3} + \)\(66\!\cdots\!57\)\( p^{43} T^{4} + \)\(24\!\cdots\!92\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 47 | $S_4\times C_2$ | \( 1 + \)\(30\!\cdots\!56\)\( T + \)\(20\!\cdots\!93\)\( T^{2} + \)\(40\!\cdots\!20\)\( T^{3} + \)\(20\!\cdots\!93\)\( p^{43} T^{4} + \)\(30\!\cdots\!56\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 53 | $S_4\times C_2$ | \( 1 - \)\(15\!\cdots\!62\)\( T + \)\(48\!\cdots\!47\)\( T^{2} - \)\(42\!\cdots\!40\)\( T^{3} + \)\(48\!\cdots\!47\)\( p^{43} T^{4} - \)\(15\!\cdots\!62\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 59 | $S_4\times C_2$ | \( 1 + \)\(22\!\cdots\!00\)\( T + \)\(50\!\cdots\!37\)\( T^{2} + \)\(62\!\cdots\!00\)\( T^{3} + \)\(50\!\cdots\!37\)\( p^{43} T^{4} + \)\(22\!\cdots\!00\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 61 | $S_4\times C_2$ | \( 1 + \)\(93\!\cdots\!54\)\( T + \)\(89\!\cdots\!15\)\( T^{2} - \)\(75\!\cdots\!20\)\( T^{3} + \)\(89\!\cdots\!15\)\( p^{43} T^{4} + \)\(93\!\cdots\!54\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 67 | $S_4\times C_2$ | \( 1 - \)\(73\!\cdots\!44\)\( T + \)\(79\!\cdots\!93\)\( T^{2} - \)\(52\!\cdots\!20\)\( T^{3} + \)\(79\!\cdots\!93\)\( p^{43} T^{4} - \)\(73\!\cdots\!44\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 71 | $S_4\times C_2$ | \( 1 - \)\(18\!\cdots\!64\)\( T + \)\(22\!\cdots\!65\)\( T^{2} - \)\(16\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!65\)\( p^{43} T^{4} - \)\(18\!\cdots\!64\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 73 | $S_4\times C_2$ | \( 1 + \)\(20\!\cdots\!98\)\( T + \)\(46\!\cdots\!27\)\( T^{2} + \)\(50\!\cdots\!80\)\( T^{3} + \)\(46\!\cdots\!27\)\( p^{43} T^{4} + \)\(20\!\cdots\!98\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 79 | $S_4\times C_2$ | \( 1 + \)\(15\!\cdots\!00\)\( T + \)\(11\!\cdots\!17\)\( T^{2} + \)\(15\!\cdots\!00\)\( p T^{3} + \)\(11\!\cdots\!17\)\( p^{43} T^{4} + \)\(15\!\cdots\!00\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 83 | $S_4\times C_2$ | \( 1 - \)\(89\!\cdots\!28\)\( T + \)\(68\!\cdots\!17\)\( T^{2} - \)\(57\!\cdots\!40\)\( T^{3} + \)\(68\!\cdots\!17\)\( p^{43} T^{4} - \)\(89\!\cdots\!28\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 89 | $S_4\times C_2$ | \( 1 - \)\(20\!\cdots\!50\)\( T + \)\(14\!\cdots\!07\)\( T^{2} - \)\(37\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!07\)\( p^{43} T^{4} - \)\(20\!\cdots\!50\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
| 97 | $S_4\times C_2$ | \( 1 + \)\(38\!\cdots\!94\)\( T + \)\(63\!\cdots\!43\)\( T^{2} + \)\(18\!\cdots\!80\)\( T^{3} + \)\(63\!\cdots\!43\)\( p^{43} T^{4} + \)\(38\!\cdots\!94\)\( p^{86} T^{5} + p^{129} T^{6} \) | |
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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
−10.49512312760056659698817132279, −9.920022282412437513849268312572, −9.699387355000175627031550736889, −9.067143802035980803245081270462, −8.433769324198956889791718053261, −8.341915159894626928468467216339, −8.258145285186619947531062502293, −7.56834966210593859296726582525, −6.79113116879165612274287575969, −6.57186821723813724061768589912, −6.10614595687118437343557638714, −5.99962449566981373697633648621, −5.96856472454495374472248989708, −5.11058865712505620351615083938, −4.96175234453999793558738453284, −4.50118757908438172541015628582, −4.02079311618412851041077451945, −3.61885933117728017996191418989, −3.31527973296784594901915008109, −2.65999247952819467966859925671, −2.39485455357327008323246321416, −2.00147464839728397103899223670, −1.43530222990690563340429859231, −1.05165243470454079668918967295, −0.934894964646956320275881702013, 0, 0, 0, 0.934894964646956320275881702013, 1.05165243470454079668918967295, 1.43530222990690563340429859231, 2.00147464839728397103899223670, 2.39485455357327008323246321416, 2.65999247952819467966859925671, 3.31527973296784594901915008109, 3.61885933117728017996191418989, 4.02079311618412851041077451945, 4.50118757908438172541015628582, 4.96175234453999793558738453284, 5.11058865712505620351615083938, 5.96856472454495374472248989708, 5.99962449566981373697633648621, 6.10614595687118437343557638714, 6.57186821723813724061768589912, 6.79113116879165612274287575969, 7.56834966210593859296726582525, 8.258145285186619947531062502293, 8.341915159894626928468467216339, 8.433769324198956889791718053261, 9.067143802035980803245081270462, 9.699387355000175627031550736889, 9.920022282412437513849268312572, 10.49512312760056659698817132279