Dirichlet series
| L(s) = 1 | − 6.59e12·2-s − 1.01e20·3-s + 2.90e25·4-s − 3.01e27·5-s + 6.70e32·6-s − 5.68e34·7-s − 1.06e38·8-s − 1.46e39·9-s + 1.99e40·10-s − 1.49e43·11-s − 2.95e45·12-s − 1.24e45·13-s + 3.75e47·14-s + 3.06e47·15-s + 3.50e50·16-s + 3.08e50·17-s + 9.65e51·18-s − 1.15e53·19-s − 8.75e52·20-s + 5.78e54·21-s + 9.86e55·22-s + 1.94e56·23-s + 1.08e58·24-s − 9.05e57·25-s + 8.23e57·26-s + 5.37e59·27-s − 1.64e60·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.60·3-s + 3·4-s − 0.0296·5-s + 3.41·6-s − 0.482·7-s − 3.53·8-s − 0.366·9-s + 0.0629·10-s − 0.905·11-s − 4.82·12-s − 0.0737·13-s + 1.02·14-s + 0.0477·15-s + 15/4·16-s + 0.266·17-s + 0.777·18-s − 0.987·19-s − 0.0890·20-s + 0.776·21-s + 1.92·22-s + 0.599·23-s + 5.69·24-s − 0.875·25-s + 0.156·26-s + 2.13·27-s − 1.44·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(84-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+83/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(6\) |
| Conductor: | \(8\) = \(2^{3}\) |
| Sign: | $1$ |
| Analytic conductor: | \(664297.\) |
| Root analytic conductor: | \(9.34100\) |
| Motivic weight: | \(83\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((6,\ 8,\ (\ :83/2, 83/2, 83/2),\ 1)\) |
Particular Values
| \(L(42)\) | \(\approx\) | \(0.01857966202\) |
| \(L(\frac12)\) | \(\approx\) | \(0.01857966202\) |
| \(L(\frac{85}{2})\) | 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 | $C_1$ | \( ( 1 + p^{41} T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 3766768380958173188 p^{3} T + \)\(66\!\cdots\!59\)\( p^{11} T^{2} + \)\(95\!\cdots\!44\)\( p^{25} T^{3} + \)\(66\!\cdots\!59\)\( p^{94} T^{4} + 3766768380958173188 p^{169} T^{5} + p^{249} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + \)\(48\!\cdots\!06\)\( p^{4} T + \)\(18\!\cdots\!43\)\( p^{11} T^{2} - \)\(33\!\cdots\!28\)\( p^{22} T^{3} + \)\(18\!\cdots\!43\)\( p^{94} T^{4} + \)\(48\!\cdots\!06\)\( p^{170} T^{5} + p^{249} T^{6} \) | |
| 7 | $S_4\times C_2$ | \( 1 + \)\(11\!\cdots\!12\)\( p^{2} T + \)\(59\!\cdots\!77\)\( p^{8} T^{2} + \)\(45\!\cdots\!04\)\( p^{16} T^{3} + \)\(59\!\cdots\!77\)\( p^{91} T^{4} + \)\(11\!\cdots\!12\)\( p^{168} T^{5} + p^{249} T^{6} \) | |
| 11 | $S_4\times C_2$ | \( 1 + \)\(13\!\cdots\!24\)\( p T + \)\(46\!\cdots\!25\)\( p^{2} T^{2} + \)\(34\!\cdots\!40\)\( p^{7} T^{3} + \)\(46\!\cdots\!25\)\( p^{85} T^{4} + \)\(13\!\cdots\!24\)\( p^{167} T^{5} + p^{249} T^{6} \) | |
| 13 | $S_4\times C_2$ | \( 1 + \)\(73\!\cdots\!54\)\( p^{2} T + \)\(79\!\cdots\!87\)\( p^{6} T^{2} + \)\(12\!\cdots\!36\)\( p^{11} T^{3} + \)\(79\!\cdots\!87\)\( p^{89} T^{4} + \)\(73\!\cdots\!54\)\( p^{168} T^{5} + p^{249} T^{6} \) | |
| 17 | $S_4\times C_2$ | \( 1 - \)\(18\!\cdots\!06\)\( p T + \)\(13\!\cdots\!63\)\( p^{2} T^{2} + \)\(12\!\cdots\!92\)\( p^{5} T^{3} + \)\(13\!\cdots\!63\)\( p^{85} T^{4} - \)\(18\!\cdots\!06\)\( p^{167} T^{5} + p^{249} T^{6} \) | |
| 19 | $S_4\times C_2$ | \( 1 + \)\(60\!\cdots\!00\)\( p T + \)\(20\!\cdots\!03\)\( p^{3} T^{2} - \)\(23\!\cdots\!00\)\( p^{6} T^{3} + \)\(20\!\cdots\!03\)\( p^{86} T^{4} + \)\(60\!\cdots\!00\)\( p^{167} T^{5} + p^{249} T^{6} \) | |
| 23 | $S_4\times C_2$ | \( 1 - \)\(84\!\cdots\!48\)\( p T + \)\(41\!\cdots\!37\)\( p^{2} T^{2} - \)\(76\!\cdots\!48\)\( p^{4} T^{3} + \)\(41\!\cdots\!37\)\( p^{85} T^{4} - \)\(84\!\cdots\!48\)\( p^{167} T^{5} + p^{249} T^{6} \) | |
| 29 | $S_4\times C_2$ | \( 1 + \)\(13\!\cdots\!70\)\( T + \)\(41\!\cdots\!23\)\( p T^{2} + \)\(27\!\cdots\!40\)\( p^{3} T^{3} + \)\(41\!\cdots\!23\)\( p^{84} T^{4} + \)\(13\!\cdots\!70\)\( p^{166} T^{5} + p^{249} T^{6} \) | |
| 31 | $S_4\times C_2$ | \( 1 + \)\(15\!\cdots\!44\)\( T + \)\(49\!\cdots\!35\)\( p T^{2} - \)\(32\!\cdots\!00\)\( p^{3} T^{3} + \)\(49\!\cdots\!35\)\( p^{84} T^{4} + \)\(15\!\cdots\!44\)\( p^{166} T^{5} + p^{249} T^{6} \) | |
| 37 | $S_4\times C_2$ | \( 1 + \)\(39\!\cdots\!54\)\( p T + \)\(35\!\cdots\!59\)\( p^{3} T^{2} - \)\(18\!\cdots\!88\)\( p^{5} T^{3} + \)\(35\!\cdots\!59\)\( p^{86} T^{4} + \)\(39\!\cdots\!54\)\( p^{167} T^{5} + p^{249} T^{6} \) | |
| 41 | $S_4\times C_2$ | \( 1 + \)\(36\!\cdots\!54\)\( p T + \)\(14\!\cdots\!95\)\( p^{2} T^{2} + \)\(27\!\cdots\!60\)\( p^{3} T^{3} + \)\(14\!\cdots\!95\)\( p^{85} T^{4} + \)\(36\!\cdots\!54\)\( p^{167} T^{5} + p^{249} T^{6} \) | |
| 43 | $S_4\times C_2$ | \( 1 + \)\(18\!\cdots\!96\)\( T + \)\(21\!\cdots\!93\)\( T^{2} + \)\(36\!\cdots\!84\)\( p T^{3} + \)\(21\!\cdots\!93\)\( p^{83} T^{4} + \)\(18\!\cdots\!96\)\( p^{166} T^{5} + p^{249} T^{6} \) | |
| 47 | $S_4\times C_2$ | \( 1 + \)\(34\!\cdots\!88\)\( T + \)\(15\!\cdots\!17\)\( T^{2} + \)\(74\!\cdots\!72\)\( p T^{3} + \)\(15\!\cdots\!17\)\( p^{83} T^{4} + \)\(34\!\cdots\!88\)\( p^{166} T^{5} + p^{249} T^{6} \) | |
| 53 | $S_4\times C_2$ | \( 1 + \)\(38\!\cdots\!66\)\( T + \)\(37\!\cdots\!11\)\( p T^{2} + \)\(80\!\cdots\!68\)\( p^{2} T^{3} + \)\(37\!\cdots\!11\)\( p^{84} T^{4} + \)\(38\!\cdots\!66\)\( p^{166} T^{5} + p^{249} T^{6} \) | |
| 59 | $S_4\times C_2$ | \( 1 - \)\(31\!\cdots\!60\)\( T + \)\(25\!\cdots\!37\)\( T^{2} - \)\(83\!\cdots\!20\)\( p T^{3} + \)\(25\!\cdots\!37\)\( p^{83} T^{4} - \)\(31\!\cdots\!60\)\( p^{166} T^{5} + p^{249} T^{6} \) | |
| 61 | $S_4\times C_2$ | \( 1 + \)\(49\!\cdots\!54\)\( p T + \)\(16\!\cdots\!55\)\( p^{2} T^{2} + \)\(41\!\cdots\!20\)\( p^{3} T^{3} + \)\(16\!\cdots\!55\)\( p^{85} T^{4} + \)\(49\!\cdots\!54\)\( p^{167} T^{5} + p^{249} T^{6} \) | |
| 67 | $S_4\times C_2$ | \( 1 + \)\(98\!\cdots\!28\)\( T + \)\(49\!\cdots\!51\)\( p T^{2} - \)\(47\!\cdots\!64\)\( p^{2} T^{3} + \)\(49\!\cdots\!51\)\( p^{84} T^{4} + \)\(98\!\cdots\!28\)\( p^{166} T^{5} + p^{249} T^{6} \) | |
| 71 | $S_4\times C_2$ | \( 1 - \)\(81\!\cdots\!96\)\( T + \)\(18\!\cdots\!55\)\( p T^{2} - \)\(13\!\cdots\!80\)\( p^{2} T^{3} + \)\(18\!\cdots\!55\)\( p^{84} T^{4} - \)\(81\!\cdots\!96\)\( p^{166} T^{5} + p^{249} T^{6} \) | |
| 73 | $S_4\times C_2$ | \( 1 - \)\(75\!\cdots\!18\)\( p T + \)\(43\!\cdots\!27\)\( p^{2} T^{2} - \)\(14\!\cdots\!44\)\( p^{3} T^{3} + \)\(43\!\cdots\!27\)\( p^{85} T^{4} - \)\(75\!\cdots\!18\)\( p^{167} T^{5} + p^{249} T^{6} \) | |
| 79 | $S_4\times C_2$ | \( 1 + \)\(65\!\cdots\!80\)\( p^{2} T + \)\(10\!\cdots\!37\)\( p^{2} T^{2} + \)\(60\!\cdots\!60\)\( p^{3} T^{3} + \)\(10\!\cdots\!37\)\( p^{85} T^{4} + \)\(65\!\cdots\!80\)\( p^{168} T^{5} + p^{249} T^{6} \) | |
| 83 | $S_4\times C_2$ | \( 1 - \)\(52\!\cdots\!84\)\( T + \)\(37\!\cdots\!13\)\( T^{2} - \)\(94\!\cdots\!68\)\( T^{3} + \)\(37\!\cdots\!13\)\( p^{83} T^{4} - \)\(52\!\cdots\!84\)\( p^{166} T^{5} + p^{249} T^{6} \) | |
| 89 | $S_4\times C_2$ | \( 1 - \)\(73\!\cdots\!30\)\( T - \)\(73\!\cdots\!93\)\( T^{2} + \)\(70\!\cdots\!60\)\( T^{3} - \)\(73\!\cdots\!93\)\( p^{83} T^{4} - \)\(73\!\cdots\!30\)\( p^{166} T^{5} + p^{249} T^{6} \) | |
| 97 | $S_4\times C_2$ | \( 1 - \)\(11\!\cdots\!82\)\( T + \)\(67\!\cdots\!27\)\( T^{2} - \)\(23\!\cdots\!56\)\( T^{3} + \)\(67\!\cdots\!27\)\( p^{83} T^{4} - \)\(11\!\cdots\!82\)\( p^{166} T^{5} + p^{249} T^{6} \) | |
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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
−11.31778640454824613380333939006, −10.88167920773412221064390714451, −10.51144590641007809546441433925, −10.10692298276767702176028971077, −9.446593554235572662888117061235, −9.341850961182419423369906951027, −8.593948661947357553317514111734, −8.157539316703607536004586605043, −7.894911232942774342096213026871, −7.39198553362586102701435136574, −6.58905180427988984388776399401, −6.42883167997121748803007677234, −6.20535689093710138932044460470, −5.38990677386099909510799920598, −5.24326065145192439281204632879, −4.89231643953424577113814838626, −3.59039794397489261291743093431, −3.36477498500808209903509353717, −3.02793283270186350080828190482, −2.01471965848111635953369991381, −2.01410371192593249767918229774, −1.60325493786589936209995264031, −0.791014437359105890905061053263, −0.27143987742286290018096475355, −0.11179335629051929744315893201, 0.11179335629051929744315893201, 0.27143987742286290018096475355, 0.791014437359105890905061053263, 1.60325493786589936209995264031, 2.01410371192593249767918229774, 2.01471965848111635953369991381, 3.02793283270186350080828190482, 3.36477498500808209903509353717, 3.59039794397489261291743093431, 4.89231643953424577113814838626, 5.24326065145192439281204632879, 5.38990677386099909510799920598, 6.20535689093710138932044460470, 6.42883167997121748803007677234, 6.58905180427988984388776399401, 7.39198553362586102701435136574, 7.894911232942774342096213026871, 8.157539316703607536004586605043, 8.593948661947357553317514111734, 9.341850961182419423369906951027, 9.446593554235572662888117061235, 10.10692298276767702176028971077, 10.51144590641007809546441433925, 10.88167920773412221064390714451, 11.31778640454824613380333939006