Dirichlet series
| L(s) = 1 | + 1.08e10·3-s − 2.12e14·5-s − 5.78e16·7-s + 1.04e19·9-s + 3.06e21·11-s − 9.85e22·13-s − 2.29e24·15-s + 3.55e25·17-s + 2.33e26·19-s − 6.26e26·21-s − 2.81e27·23-s − 5.19e28·25-s − 4.54e29·27-s − 1.27e28·29-s + 5.59e28·31-s + 3.31e31·33-s + 1.22e31·35-s + 4.94e30·37-s − 1.06e33·39-s − 3.12e33·41-s − 1.49e33·43-s − 2.22e33·45-s + 6.30e34·47-s − 1.13e35·49-s + 3.85e35·51-s + 7.98e34·53-s − 6.50e35·55-s + ⋯ |
| L(s) = 1 | + 1.79·3-s − 0.995·5-s − 0.274·7-s + 0.287·9-s + 1.37·11-s − 1.43·13-s − 1.78·15-s + 2.12·17-s + 1.42·19-s − 0.491·21-s − 0.341·23-s − 1.14·25-s − 2.06·27-s − 0.0133·29-s + 0.0149·31-s + 2.46·33-s + 0.272·35-s + 0.0351·37-s − 2.57·39-s − 2.70·41-s − 0.487·43-s − 0.285·45-s + 3.32·47-s − 2.55·49-s + 3.80·51-s + 0.358·53-s − 1.36·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(42-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+41/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(6\) |
| Conductor: | \(4096\) = \(2^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.94381\times 10^{6}\) |
| Root analytic conductor: | \(13.0519\) |
| Motivic weight: | \(41\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((6,\ 4096,\ (\ :41/2, 41/2, 41/2),\ 1)\) |
Particular Values
| \(L(21)\) | \(\approx\) | \(9.179225684\) |
| \(L(\frac12)\) | \(\approx\) | \(9.179225684\) |
| \(L(\frac{43}{2})\) | 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 | $S_4\times C_2$ | \( 1 - 1202328116 p^{2} T + 48750573799961899 p^{7} T^{2} - \)\(13\!\cdots\!40\)\( p^{16} T^{3} + 48750573799961899 p^{48} T^{4} - 1202328116 p^{84} T^{5} + p^{123} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 8492094011262 p^{2} T + \)\(12\!\cdots\!71\)\( p^{7} T^{2} + \)\(16\!\cdots\!68\)\( p^{13} T^{3} + \)\(12\!\cdots\!71\)\( p^{48} T^{4} + 8492094011262 p^{84} T^{5} + p^{123} T^{6} \) | |
| 7 | $S_4\times C_2$ | \( 1 + 8268345179748456 p T + \)\(34\!\cdots\!99\)\( p^{3} T^{2} + \)\(46\!\cdots\!00\)\( p^{6} T^{3} + \)\(34\!\cdots\!99\)\( p^{44} T^{4} + 8268345179748456 p^{83} T^{5} + p^{123} T^{6} \) | |
| 11 | $S_4\times C_2$ | \( 1 - \)\(30\!\cdots\!64\)\( T + \)\(80\!\cdots\!65\)\( p^{2} T^{2} - \)\(10\!\cdots\!80\)\( p^{5} T^{3} + \)\(80\!\cdots\!65\)\( p^{43} T^{4} - \)\(30\!\cdots\!64\)\( p^{82} T^{5} + p^{123} T^{6} \) | |
| 13 | $S_4\times C_2$ | \( 1 + \)\(75\!\cdots\!38\)\( p T + \)\(25\!\cdots\!39\)\( p^{3} T^{2} + \)\(26\!\cdots\!60\)\( p^{5} T^{3} + \)\(25\!\cdots\!39\)\( p^{44} T^{4} + \)\(75\!\cdots\!38\)\( p^{83} T^{5} + p^{123} T^{6} \) | |
| 17 | $S_4\times C_2$ | \( 1 - \)\(20\!\cdots\!06\)\( p T + \)\(24\!\cdots\!39\)\( p^{3} T^{2} - \)\(14\!\cdots\!40\)\( p^{5} T^{3} + \)\(24\!\cdots\!39\)\( p^{44} T^{4} - \)\(20\!\cdots\!06\)\( p^{83} T^{5} + p^{123} T^{6} \) | |
| 19 | $S_4\times C_2$ | \( 1 - \)\(12\!\cdots\!20\)\( p T + \)\(24\!\cdots\!37\)\( p^{2} T^{2} - \)\(90\!\cdots\!40\)\( p^{4} T^{3} + \)\(24\!\cdots\!37\)\( p^{43} T^{4} - \)\(12\!\cdots\!20\)\( p^{83} T^{5} + p^{123} T^{6} \) | |
| 23 | $S_4\times C_2$ | \( 1 + \)\(28\!\cdots\!56\)\( T + \)\(37\!\cdots\!11\)\( p T^{2} + \)\(13\!\cdots\!20\)\( p^{2} T^{3} + \)\(37\!\cdots\!11\)\( p^{42} T^{4} + \)\(28\!\cdots\!56\)\( p^{82} T^{5} + p^{123} T^{6} \) | |
| 29 | $S_4\times C_2$ | \( 1 + \)\(43\!\cdots\!30\)\( p T + \)\(18\!\cdots\!07\)\( p^{2} T^{2} - \)\(16\!\cdots\!60\)\( p^{3} T^{3} + \)\(18\!\cdots\!07\)\( p^{43} T^{4} + \)\(43\!\cdots\!30\)\( p^{83} T^{5} + p^{123} T^{6} \) | |
| 31 | $S_4\times C_2$ | \( 1 - \)\(55\!\cdots\!04\)\( T + \)\(11\!\cdots\!15\)\( p T^{2} - \)\(48\!\cdots\!80\)\( p^{2} T^{3} + \)\(11\!\cdots\!15\)\( p^{42} T^{4} - \)\(55\!\cdots\!04\)\( p^{82} T^{5} + p^{123} T^{6} \) | |
| 37 | $S_4\times C_2$ | \( 1 - \)\(13\!\cdots\!06\)\( p T + \)\(34\!\cdots\!03\)\( p^{2} T^{2} - \)\(46\!\cdots\!80\)\( p^{3} T^{3} + \)\(34\!\cdots\!03\)\( p^{43} T^{4} - \)\(13\!\cdots\!06\)\( p^{83} T^{5} + p^{123} T^{6} \) | |
| 41 | $S_4\times C_2$ | \( 1 + \)\(31\!\cdots\!74\)\( T + \)\(70\!\cdots\!15\)\( T^{2} + \)\(92\!\cdots\!80\)\( T^{3} + \)\(70\!\cdots\!15\)\( p^{41} T^{4} + \)\(31\!\cdots\!74\)\( p^{82} T^{5} + p^{123} T^{6} \) | |
| 43 | $S_4\times C_2$ | \( 1 + \)\(14\!\cdots\!56\)\( T + \)\(27\!\cdots\!93\)\( T^{2} + \)\(28\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!93\)\( p^{41} T^{4} + \)\(14\!\cdots\!56\)\( p^{82} T^{5} + p^{123} T^{6} \) | |
| 47 | $S_4\times C_2$ | \( 1 - \)\(63\!\cdots\!68\)\( T + \)\(22\!\cdots\!57\)\( T^{2} - \)\(51\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!57\)\( p^{41} T^{4} - \)\(63\!\cdots\!68\)\( p^{82} T^{5} + p^{123} T^{6} \) | |
| 53 | $S_4\times C_2$ | \( 1 - \)\(79\!\cdots\!06\)\( T + \)\(59\!\cdots\!63\)\( T^{2} - \)\(16\!\cdots\!60\)\( T^{3} + \)\(59\!\cdots\!63\)\( p^{41} T^{4} - \)\(79\!\cdots\!06\)\( p^{82} T^{5} + p^{123} T^{6} \) | |
| 59 | $S_4\times C_2$ | \( 1 + \)\(19\!\cdots\!60\)\( T + \)\(77\!\cdots\!77\)\( T^{2} + \)\(18\!\cdots\!80\)\( T^{3} + \)\(77\!\cdots\!77\)\( p^{41} T^{4} + \)\(19\!\cdots\!60\)\( p^{82} T^{5} + p^{123} T^{6} \) | |
| 61 | $S_4\times C_2$ | \( 1 - \)\(87\!\cdots\!86\)\( T + \)\(62\!\cdots\!15\)\( T^{2} - \)\(27\!\cdots\!20\)\( T^{3} + \)\(62\!\cdots\!15\)\( p^{41} T^{4} - \)\(87\!\cdots\!86\)\( p^{82} T^{5} + p^{123} T^{6} \) | |
| 67 | $S_4\times C_2$ | \( 1 + \)\(11\!\cdots\!52\)\( T + \)\(78\!\cdots\!57\)\( T^{2} - \)\(10\!\cdots\!20\)\( T^{3} + \)\(78\!\cdots\!57\)\( p^{41} T^{4} + \)\(11\!\cdots\!52\)\( p^{82} T^{5} + p^{123} T^{6} \) | |
| 71 | $S_4\times C_2$ | \( 1 - \)\(14\!\cdots\!84\)\( T + \)\(22\!\cdots\!65\)\( T^{2} - \)\(18\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!65\)\( p^{41} T^{4} - \)\(14\!\cdots\!84\)\( p^{82} T^{5} + p^{123} T^{6} \) | |
| 73 | $S_4\times C_2$ | \( 1 - \)\(45\!\cdots\!06\)\( T + \)\(18\!\cdots\!03\)\( T^{2} - \)\(58\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!03\)\( p^{41} T^{4} - \)\(45\!\cdots\!06\)\( p^{82} T^{5} + p^{123} T^{6} \) | |
| 79 | $S_4\times C_2$ | \( 1 - \)\(52\!\cdots\!20\)\( T + \)\(83\!\cdots\!37\)\( T^{2} - \)\(82\!\cdots\!60\)\( T^{3} + \)\(83\!\cdots\!37\)\( p^{41} T^{4} - \)\(52\!\cdots\!20\)\( p^{82} T^{5} + p^{123} T^{6} \) | |
| 83 | $S_4\times C_2$ | \( 1 - \)\(61\!\cdots\!44\)\( T + \)\(61\!\cdots\!73\)\( T^{2} - \)\(12\!\cdots\!60\)\( T^{3} + \)\(61\!\cdots\!73\)\( p^{41} T^{4} - \)\(61\!\cdots\!44\)\( p^{82} T^{5} + p^{123} T^{6} \) | |
| 89 | $S_4\times C_2$ | \( 1 + \)\(14\!\cdots\!10\)\( T + \)\(20\!\cdots\!67\)\( T^{2} + \)\(53\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!67\)\( p^{41} T^{4} + \)\(14\!\cdots\!10\)\( p^{82} T^{5} + p^{123} T^{6} \) | |
| 97 | $S_4\times C_2$ | \( 1 - \)\(11\!\cdots\!82\)\( T + \)\(11\!\cdots\!07\)\( T^{2} - \)\(67\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!07\)\( p^{41} T^{4} - \)\(11\!\cdots\!82\)\( p^{82} T^{5} + p^{123} T^{6} \) | |
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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
−9.817921663789346546097485622011, −9.279728580754437846351829576526, −9.091247011890349075209546213335, −8.687367171862925053047101129188, −8.057936390433703256711730331283, −7.911976984756685258733198548847, −7.77410492738579636953347541986, −7.27018824652830763187193781947, −6.82148914356472567587336569401, −6.46776280793846319977020270228, −5.58643302181201304482138635808, −5.56260570622222425296239885835, −5.16929710847027113436461262077, −4.43689960583359305733121689718, −4.06691434910330895685061342543, −3.60617127183675748070494871891, −3.37848724479128325608436836594, −3.07229194482454871747433298777, −3.01875490172319456978546780114, −2.15123604750240725049266562929, −1.93707998264149476183416925964, −1.72163277966508061869142567163, −0.885635994623396927961061896561, −0.58054078729883841220651756939, −0.43421455628799899663502840943, 0.43421455628799899663502840943, 0.58054078729883841220651756939, 0.885635994623396927961061896561, 1.72163277966508061869142567163, 1.93707998264149476183416925964, 2.15123604750240725049266562929, 3.01875490172319456978546780114, 3.07229194482454871747433298777, 3.37848724479128325608436836594, 3.60617127183675748070494871891, 4.06691434910330895685061342543, 4.43689960583359305733121689718, 5.16929710847027113436461262077, 5.56260570622222425296239885835, 5.58643302181201304482138635808, 6.46776280793846319977020270228, 6.82148914356472567587336569401, 7.27018824652830763187193781947, 7.77410492738579636953347541986, 7.911976984756685258733198548847, 8.057936390433703256711730331283, 8.687367171862925053047101129188, 9.091247011890349075209546213335, 9.279728580754437846351829576526, 9.817921663789346546097485622011