Dirichlet series
| L(s) = 1 | − 5.33e5·2-s − 2.03e11·4-s + 5.33e13·5-s − 1.57e15·7-s + 1.07e17·8-s − 2.84e19·10-s + 5.31e20·11-s + 5.47e21·13-s + 8.41e20·14-s − 8.72e22·16-s − 7.23e23·17-s − 1.06e25·19-s − 1.08e25·20-s − 2.83e26·22-s − 4.16e26·23-s − 1.90e27·25-s − 2.91e27·26-s + 3.21e26·28-s − 7.29e28·29-s − 3.83e27·31-s + 2.89e28·32-s + 3.86e29·34-s − 8.42e28·35-s + 2.90e30·37-s + 5.70e30·38-s + 5.76e30·40-s + 1.38e31·41-s + ⋯ |
| L(s) = 1 | − 0.719·2-s − 0.370·4-s + 1.25·5-s − 0.0523·7-s + 0.264·8-s − 0.900·10-s + 2.61·11-s + 1.03·13-s + 0.0376·14-s − 0.288·16-s − 0.734·17-s − 1.23·19-s − 0.463·20-s − 1.88·22-s − 1.16·23-s − 1.04·25-s − 0.747·26-s + 0.0193·28-s − 2.21·29-s − 0.0317·31-s + 0.129·32-s + 0.528·34-s − 0.0654·35-s + 0.764·37-s + 0.891·38-s + 0.331·40-s + 0.491·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+39/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(6\) |
| Conductor: | \(729\) = \(3^{6}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(651840.\) |
| Root analytic conductor: | \(9.31158\) |
| Motivic weight: | \(39\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(3\) |
| Selberg data: | \((6,\ 729,\ (\ :39/2, 39/2, 39/2),\ -1)\) |
Particular Values
| \(L(20)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{41}{2})\) | 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 | 3 | \( 1 \) | |
| good | 2 | $S_4\times C_2$ | \( 1 + 266787 p T + 3814554693 p^{7} T^{2} + 3985936932681 p^{16} T^{3} + 3814554693 p^{46} T^{4} + 266787 p^{79} T^{5} + p^{117} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 10676394788886 p T + \)\(15\!\cdots\!23\)\( p^{5} T^{2} - \)\(38\!\cdots\!44\)\( p^{11} T^{3} + \)\(15\!\cdots\!23\)\( p^{44} T^{4} - 10676394788886 p^{79} T^{5} + p^{117} T^{6} \) | |
| 7 | $S_4\times C_2$ | \( 1 + 225330382317504 p T + \)\(18\!\cdots\!65\)\( p^{2} T^{2} + \)\(10\!\cdots\!52\)\( p^{6} T^{3} + \)\(18\!\cdots\!65\)\( p^{41} T^{4} + 225330382317504 p^{79} T^{5} + p^{117} T^{6} \) | |
| 11 | $S_4\times C_2$ | \( 1 - \)\(53\!\cdots\!40\)\( T + \)\(19\!\cdots\!11\)\( p T^{2} - \)\(33\!\cdots\!76\)\( p^{4} T^{3} + \)\(19\!\cdots\!11\)\( p^{40} T^{4} - \)\(53\!\cdots\!40\)\( p^{78} T^{5} + p^{117} T^{6} \) | |
| 13 | $S_4\times C_2$ | \( 1 - \)\(42\!\cdots\!30\)\( p T + \)\(45\!\cdots\!31\)\( p^{2} T^{2} - \)\(13\!\cdots\!92\)\( p^{3} T^{3} + \)\(45\!\cdots\!31\)\( p^{41} T^{4} - \)\(42\!\cdots\!30\)\( p^{79} T^{5} + p^{117} T^{6} \) | |
| 17 | $S_4\times C_2$ | \( 1 + \)\(72\!\cdots\!58\)\( T + \)\(17\!\cdots\!23\)\( p^{2} T^{2} + \)\(75\!\cdots\!24\)\( p^{4} T^{3} + \)\(17\!\cdots\!23\)\( p^{41} T^{4} + \)\(72\!\cdots\!58\)\( p^{78} T^{5} + p^{117} T^{6} \) | |
| 19 | $S_4\times C_2$ | \( 1 + \)\(56\!\cdots\!96\)\( p T + \)\(40\!\cdots\!77\)\( p^{2} T^{2} + \)\(19\!\cdots\!88\)\( p^{3} T^{3} + \)\(40\!\cdots\!77\)\( p^{41} T^{4} + \)\(56\!\cdots\!96\)\( p^{79} T^{5} + p^{117} T^{6} \) | |
| 23 | $S_4\times C_2$ | \( 1 + \)\(41\!\cdots\!64\)\( T + \)\(25\!\cdots\!55\)\( p T^{2} - \)\(64\!\cdots\!76\)\( p^{2} T^{3} + \)\(25\!\cdots\!55\)\( p^{40} T^{4} + \)\(41\!\cdots\!64\)\( p^{78} T^{5} + p^{117} T^{6} \) | |
| 29 | $S_4\times C_2$ | \( 1 + \)\(25\!\cdots\!78\)\( p T + \)\(58\!\cdots\!07\)\( p^{2} T^{2} + \)\(69\!\cdots\!04\)\( p^{3} T^{3} + \)\(58\!\cdots\!07\)\( p^{41} T^{4} + \)\(25\!\cdots\!78\)\( p^{79} T^{5} + p^{117} T^{6} \) | |
| 31 | $S_4\times C_2$ | \( 1 + \)\(38\!\cdots\!56\)\( T + \)\(65\!\cdots\!87\)\( p T^{2} + \)\(79\!\cdots\!32\)\( p^{2} T^{3} + \)\(65\!\cdots\!87\)\( p^{40} T^{4} + \)\(38\!\cdots\!56\)\( p^{78} T^{5} + p^{117} T^{6} \) | |
| 37 | $S_4\times C_2$ | \( 1 - \)\(78\!\cdots\!06\)\( p T + \)\(17\!\cdots\!15\)\( p^{2} T^{2} - \)\(59\!\cdots\!44\)\( p^{3} T^{3} + \)\(17\!\cdots\!15\)\( p^{41} T^{4} - \)\(78\!\cdots\!06\)\( p^{79} T^{5} + p^{117} T^{6} \) | |
| 41 | $S_4\times C_2$ | \( 1 - \)\(13\!\cdots\!26\)\( T + \)\(11\!\cdots\!27\)\( T^{2} - \)\(14\!\cdots\!72\)\( T^{3} + \)\(11\!\cdots\!27\)\( p^{39} T^{4} - \)\(13\!\cdots\!26\)\( p^{78} T^{5} + p^{117} T^{6} \) | |
| 43 | $S_4\times C_2$ | \( 1 + \)\(24\!\cdots\!28\)\( T + \)\(33\!\cdots\!77\)\( T^{2} + \)\(28\!\cdots\!60\)\( T^{3} + \)\(33\!\cdots\!77\)\( p^{39} T^{4} + \)\(24\!\cdots\!28\)\( p^{78} T^{5} + p^{117} T^{6} \) | |
| 47 | $S_4\times C_2$ | \( 1 - \)\(10\!\cdots\!04\)\( T + \)\(86\!\cdots\!53\)\( T^{2} - \)\(39\!\cdots\!80\)\( T^{3} + \)\(86\!\cdots\!53\)\( p^{39} T^{4} - \)\(10\!\cdots\!04\)\( p^{78} T^{5} + p^{117} T^{6} \) | |
| 53 | $S_4\times C_2$ | \( 1 + \)\(62\!\cdots\!34\)\( T + \)\(47\!\cdots\!55\)\( T^{2} + \)\(22\!\cdots\!36\)\( T^{3} + \)\(47\!\cdots\!55\)\( p^{39} T^{4} + \)\(62\!\cdots\!34\)\( p^{78} T^{5} + p^{117} T^{6} \) | |
| 59 | $S_4\times C_2$ | \( 1 - \)\(75\!\cdots\!56\)\( T + \)\(29\!\cdots\!37\)\( T^{2} - \)\(16\!\cdots\!68\)\( T^{3} + \)\(29\!\cdots\!37\)\( p^{39} T^{4} - \)\(75\!\cdots\!56\)\( p^{78} T^{5} + p^{117} T^{6} \) | |
| 61 | $S_4\times C_2$ | \( 1 + \)\(71\!\cdots\!98\)\( T + \)\(73\!\cdots\!99\)\( T^{2} + \)\(33\!\cdots\!84\)\( T^{3} + \)\(73\!\cdots\!99\)\( p^{39} T^{4} + \)\(71\!\cdots\!98\)\( p^{78} T^{5} + p^{117} T^{6} \) | |
| 67 | $S_4\times C_2$ | \( 1 + \)\(51\!\cdots\!72\)\( T + \)\(32\!\cdots\!37\)\( T^{2} + \)\(82\!\cdots\!56\)\( T^{3} + \)\(32\!\cdots\!37\)\( p^{39} T^{4} + \)\(51\!\cdots\!72\)\( p^{78} T^{5} + p^{117} T^{6} \) | |
| 71 | $S_4\times C_2$ | \( 1 + \)\(84\!\cdots\!76\)\( T + \)\(40\!\cdots\!85\)\( T^{2} + \)\(20\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!85\)\( p^{39} T^{4} + \)\(84\!\cdots\!76\)\( p^{78} T^{5} + p^{117} T^{6} \) | |
| 73 | $S_4\times C_2$ | \( 1 - \)\(63\!\cdots\!14\)\( T + \)\(59\!\cdots\!15\)\( T^{2} - \)\(10\!\cdots\!96\)\( T^{3} + \)\(59\!\cdots\!15\)\( p^{39} T^{4} - \)\(63\!\cdots\!14\)\( p^{78} T^{5} + p^{117} T^{6} \) | |
| 79 | $S_4\times C_2$ | \( 1 + \)\(16\!\cdots\!00\)\( T + \)\(16\!\cdots\!57\)\( T^{2} + \)\(94\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!57\)\( p^{39} T^{4} + \)\(16\!\cdots\!00\)\( p^{78} T^{5} + p^{117} T^{6} \) | |
| 83 | $S_4\times C_2$ | \( 1 + \)\(59\!\cdots\!48\)\( T + \)\(32\!\cdots\!21\)\( T^{2} + \)\(91\!\cdots\!88\)\( T^{3} + \)\(32\!\cdots\!21\)\( p^{39} T^{4} + \)\(59\!\cdots\!48\)\( p^{78} T^{5} + p^{117} T^{6} \) | |
| 89 | $S_4\times C_2$ | \( 1 + \)\(18\!\cdots\!86\)\( T + \)\(36\!\cdots\!47\)\( T^{2} + \)\(36\!\cdots\!48\)\( T^{3} + \)\(36\!\cdots\!47\)\( p^{39} T^{4} + \)\(18\!\cdots\!86\)\( p^{78} T^{5} + p^{117} T^{6} \) | |
| 97 | $S_4\times C_2$ | \( 1 + \)\(99\!\cdots\!42\)\( T + \)\(88\!\cdots\!87\)\( T^{2} + \)\(45\!\cdots\!16\)\( T^{3} + \)\(88\!\cdots\!87\)\( p^{39} T^{4} + \)\(99\!\cdots\!42\)\( p^{78} T^{5} + p^{117} T^{6} \) | |
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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
−11.65588016595643027438521711004, −11.53756235325437882652120538500, −10.96497324176937979658613156977, −10.51101092305724839111716653334, −9.804579862624790285268774516852, −9.498459857877660906230830032454, −9.389169074085909005568441814049, −8.897311611735753744308193834141, −8.529629600426950667437549187344, −8.137496527383678097590627286021, −7.44280810163340957356229705808, −6.73968795267376810786170792682, −6.55859582362164202901510335783, −6.06278578195609420933438771131, −5.78795892004392739128573622884, −5.40120466672508845341138317873, −4.22365549455644628886410488112, −4.14942220072611658706091386665, −4.10178354518153850819283123140, −3.31428431660204359774275833422, −2.70773939954142032934275492517, −1.97132818802240748313811782670, −1.72936132001786070772022821403, −1.39982225316087691733363175614, −1.26651104107005287960870089381, 0, 0, 0, 1.26651104107005287960870089381, 1.39982225316087691733363175614, 1.72936132001786070772022821403, 1.97132818802240748313811782670, 2.70773939954142032934275492517, 3.31428431660204359774275833422, 4.10178354518153850819283123140, 4.14942220072611658706091386665, 4.22365549455644628886410488112, 5.40120466672508845341138317873, 5.78795892004392739128573622884, 6.06278578195609420933438771131, 6.55859582362164202901510335783, 6.73968795267376810786170792682, 7.44280810163340957356229705808, 8.137496527383678097590627286021, 8.529629600426950667437549187344, 8.897311611735753744308193834141, 9.389169074085909005568441814049, 9.498459857877660906230830032454, 9.804579862624790285268774516852, 10.51101092305724839111716653334, 10.96497324176937979658613156977, 11.53756235325437882652120538500, 11.65588016595643027438521711004