Dirichlet series
| L(s) = 1 | + 3.81e6·2-s + 5.35e9·3-s − 4.64e13·4-s − 9.12e14·5-s + 2.04e16·6-s − 7.61e18·7-s − 2.76e20·8-s − 3.83e21·9-s − 3.48e21·10-s − 2.92e23·11-s − 2.49e23·12-s − 2.43e25·13-s − 2.90e25·14-s − 4.89e24·15-s + 1.40e26·16-s + 1.16e27·17-s − 1.46e28·18-s + 1.39e29·19-s + 4.23e28·20-s − 4.08e28·21-s − 1.11e30·22-s − 1.06e31·23-s − 1.48e30·24-s − 2.02e31·25-s − 9.29e31·26-s + 6.96e31·27-s + 3.54e32·28-s + ⋯ |
| L(s) = 1 | + 0.643·2-s + 0.0986·3-s − 1.32·4-s − 0.171·5-s + 0.0634·6-s − 0.736·7-s − 1.32·8-s − 1.29·9-s − 0.110·10-s − 1.08·11-s − 0.130·12-s − 2.10·13-s − 0.473·14-s − 0.0168·15-s + 0.113·16-s + 0.241·17-s − 0.835·18-s + 2.36·19-s + 0.226·20-s − 0.0726·21-s − 0.696·22-s − 2.45·23-s − 0.130·24-s − 0.711·25-s − 1.35·26-s + 0.433·27-s + 0.972·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(46-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+45/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(6\) |
| Conductor: | \(1\) |
| Sign: | $-1$ |
| Analytic conductor: | \(2109.74\) |
| Root analytic conductor: | \(3.58128\) |
| Motivic weight: | \(45\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(3\) |
| Selberg data: | \((6,\ 1,\ (\ :45/2, 45/2, 45/2),\ -1)\) |
Particular Values
| \(L(23)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{47}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| good | 2 | $S_4\times C_2$ | \( 1 - 29799 p^{7} T + 14894647467 p^{12} T^{2} - 3968335633725 p^{25} T^{3} + 14894647467 p^{57} T^{4} - 29799 p^{97} T^{5} + p^{135} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 - 66171196 p^{4} T + 589312962345019913 p^{8} T^{2} - \)\(85\!\cdots\!00\)\( p^{17} T^{3} + 589312962345019913 p^{53} T^{4} - 66171196 p^{94} T^{5} + p^{135} T^{6} \) | |
| 5 | $S_4\times C_2$ | \( 1 + 36497938338414 p^{2} T + \)\(26\!\cdots\!19\)\( p^{7} T^{2} + \)\(15\!\cdots\!88\)\( p^{13} T^{3} + \)\(26\!\cdots\!19\)\( p^{52} T^{4} + 36497938338414 p^{92} T^{5} + p^{135} T^{6} \) | |
| 7 | $S_4\times C_2$ | \( 1 + 1088530132376865144 p T + \)\(62\!\cdots\!57\)\( p^{4} T^{2} + \)\(71\!\cdots\!00\)\( p^{8} T^{3} + \)\(62\!\cdots\!57\)\( p^{49} T^{4} + 1088530132376865144 p^{91} T^{5} + p^{135} T^{6} \) | |
| 11 | $S_4\times C_2$ | \( 1 + \)\(26\!\cdots\!04\)\( p T + \)\(11\!\cdots\!15\)\( p^{3} T^{2} + \)\(28\!\cdots\!80\)\( p^{5} T^{3} + \)\(11\!\cdots\!15\)\( p^{48} T^{4} + \)\(26\!\cdots\!04\)\( p^{91} T^{5} + p^{135} T^{6} \) | |
| 13 | $S_4\times C_2$ | \( 1 + \)\(18\!\cdots\!58\)\( p T + \)\(22\!\cdots\!19\)\( p^{3} T^{2} + \)\(12\!\cdots\!00\)\( p^{6} T^{3} + \)\(22\!\cdots\!19\)\( p^{48} T^{4} + \)\(18\!\cdots\!58\)\( p^{91} T^{5} + p^{135} T^{6} \) | |
| 17 | $S_4\times C_2$ | \( 1 - \)\(68\!\cdots\!46\)\( p T + \)\(17\!\cdots\!63\)\( p^{2} T^{2} - \)\(91\!\cdots\!00\)\( p^{4} T^{3} + \)\(17\!\cdots\!63\)\( p^{47} T^{4} - \)\(68\!\cdots\!46\)\( p^{91} T^{5} + p^{135} T^{6} \) | |
| 19 | $S_4\times C_2$ | \( 1 - \)\(73\!\cdots\!20\)\( p T + \)\(43\!\cdots\!77\)\( p^{2} T^{2} - \)\(79\!\cdots\!40\)\( p^{4} T^{3} + \)\(43\!\cdots\!77\)\( p^{47} T^{4} - \)\(73\!\cdots\!20\)\( p^{91} T^{5} + p^{135} T^{6} \) | |
| 23 | $S_4\times C_2$ | \( 1 + \)\(10\!\cdots\!84\)\( T + \)\(93\!\cdots\!93\)\( T^{2} + \)\(19\!\cdots\!00\)\( p T^{3} + \)\(93\!\cdots\!93\)\( p^{45} T^{4} + \)\(10\!\cdots\!84\)\( p^{90} T^{5} + p^{135} T^{6} \) | |
| 29 | $S_4\times C_2$ | \( 1 + \)\(67\!\cdots\!30\)\( T + \)\(35\!\cdots\!43\)\( p T^{2} + \)\(99\!\cdots\!40\)\( p^{2} T^{3} + \)\(35\!\cdots\!43\)\( p^{46} T^{4} + \)\(67\!\cdots\!30\)\( p^{90} T^{5} + p^{135} T^{6} \) | |
| 31 | $S_4\times C_2$ | \( 1 + \)\(43\!\cdots\!44\)\( T + \)\(13\!\cdots\!15\)\( p T^{2} + \)\(11\!\cdots\!80\)\( p^{2} T^{3} + \)\(13\!\cdots\!15\)\( p^{46} T^{4} + \)\(43\!\cdots\!44\)\( p^{90} T^{5} + p^{135} T^{6} \) | |
| 37 | $S_4\times C_2$ | \( 1 + \)\(10\!\cdots\!74\)\( p T + \)\(11\!\cdots\!03\)\( p^{2} T^{2} + \)\(60\!\cdots\!00\)\( p^{3} T^{3} + \)\(11\!\cdots\!03\)\( p^{47} T^{4} + \)\(10\!\cdots\!74\)\( p^{91} T^{5} + p^{135} T^{6} \) | |
| 41 | $S_4\times C_2$ | \( 1 - \)\(66\!\cdots\!66\)\( p T + \)\(74\!\cdots\!15\)\( p^{2} T^{2} - \)\(29\!\cdots\!20\)\( p^{3} T^{3} + \)\(74\!\cdots\!15\)\( p^{47} T^{4} - \)\(66\!\cdots\!66\)\( p^{91} T^{5} + p^{135} T^{6} \) | |
| 43 | $S_4\times C_2$ | \( 1 - \)\(28\!\cdots\!92\)\( p T + \)\(16\!\cdots\!57\)\( p^{2} T^{2} - \)\(26\!\cdots\!00\)\( p^{3} T^{3} + \)\(16\!\cdots\!57\)\( p^{47} T^{4} - \)\(28\!\cdots\!92\)\( p^{91} T^{5} + p^{135} T^{6} \) | |
| 47 | $S_4\times C_2$ | \( 1 + \)\(53\!\cdots\!48\)\( T + \)\(27\!\cdots\!57\)\( T^{2} - \)\(60\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!57\)\( p^{45} T^{4} + \)\(53\!\cdots\!48\)\( p^{90} T^{5} + p^{135} T^{6} \) | |
| 53 | $S_4\times C_2$ | \( 1 + \)\(18\!\cdots\!74\)\( T + \)\(22\!\cdots\!43\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!43\)\( p^{45} T^{4} + \)\(18\!\cdots\!74\)\( p^{90} T^{5} + p^{135} T^{6} \) | |
| 59 | $S_4\times C_2$ | \( 1 + \)\(35\!\cdots\!60\)\( T + \)\(46\!\cdots\!97\)\( T^{2} - \)\(10\!\cdots\!80\)\( p T^{3} + \)\(46\!\cdots\!97\)\( p^{45} T^{4} + \)\(35\!\cdots\!60\)\( p^{90} T^{5} + p^{135} T^{6} \) | |
| 61 | $S_4\times C_2$ | \( 1 + \)\(51\!\cdots\!94\)\( T + \)\(13\!\cdots\!15\)\( T^{2} + \)\(23\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!15\)\( p^{45} T^{4} + \)\(51\!\cdots\!94\)\( p^{90} T^{5} + p^{135} T^{6} \) | |
| 67 | $S_4\times C_2$ | \( 1 - \)\(14\!\cdots\!32\)\( T - \)\(71\!\cdots\!43\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} - \)\(71\!\cdots\!43\)\( p^{45} T^{4} - \)\(14\!\cdots\!32\)\( p^{90} T^{5} + p^{135} T^{6} \) | |
| 71 | $S_4\times C_2$ | \( 1 + \)\(12\!\cdots\!44\)\( T + \)\(39\!\cdots\!65\)\( T^{2} + \)\(60\!\cdots\!80\)\( T^{3} + \)\(39\!\cdots\!65\)\( p^{45} T^{4} + \)\(12\!\cdots\!44\)\( p^{90} T^{5} + p^{135} T^{6} \) | |
| 73 | $S_4\times C_2$ | \( 1 - \)\(54\!\cdots\!66\)\( T + \)\(10\!\cdots\!43\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!43\)\( p^{45} T^{4} - \)\(54\!\cdots\!66\)\( p^{90} T^{5} + p^{135} T^{6} \) | |
| 79 | $S_4\times C_2$ | \( 1 - \)\(90\!\cdots\!20\)\( T + \)\(66\!\cdots\!97\)\( T^{2} - \)\(49\!\cdots\!60\)\( T^{3} + \)\(66\!\cdots\!97\)\( p^{45} T^{4} - \)\(90\!\cdots\!20\)\( p^{90} T^{5} + p^{135} T^{6} \) | |
| 83 | $S_4\times C_2$ | \( 1 - \)\(14\!\cdots\!36\)\( T + \)\(64\!\cdots\!93\)\( T^{2} - \)\(63\!\cdots\!00\)\( T^{3} + \)\(64\!\cdots\!93\)\( p^{45} T^{4} - \)\(14\!\cdots\!36\)\( p^{90} T^{5} + p^{135} T^{6} \) | |
| 89 | $S_4\times C_2$ | \( 1 + \)\(15\!\cdots\!90\)\( T + \)\(23\!\cdots\!47\)\( T^{2} + \)\(17\!\cdots\!20\)\( T^{3} + \)\(23\!\cdots\!47\)\( p^{45} T^{4} + \)\(15\!\cdots\!90\)\( p^{90} T^{5} + p^{135} T^{6} \) | |
| 97 | $S_4\times C_2$ | \( 1 - \)\(37\!\cdots\!02\)\( T + \)\(66\!\cdots\!07\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(66\!\cdots\!07\)\( p^{45} T^{4} - \)\(37\!\cdots\!02\)\( p^{90} T^{5} + p^{135} T^{6} \) | |
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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
−20.00141330000586300776155390382, −19.02943616266343248258861587188, −18.06574641877894839779638823471, −17.93672447684804605224613643041, −17.05939314902054771368099854329, −16.10130476472056037967319118127, −15.56062319188556808650697312767, −14.23348000659179508911246174538, −14.21371923317822788965977870210, −13.66256273110436947434727107367, −12.43077072814230613599510364838, −12.40426487154665503106833194686, −11.38744322909410240225914390179, −10.10487225466447691656417923583, −9.497692399288722992339187258041, −9.146910085028181413545895132657, −7.75364568294789525381115787121, −7.71136879901769074032724678054, −6.13161181216014691666479320551, −5.33002238531523506864245254531, −5.09196327013335612925130720207, −4.15376264120983200749644933658, −3.18153651066159805009838408765, −2.91839662490823476782717262862, −1.82170644350664873402382721900, 0, 0, 0, 1.82170644350664873402382721900, 2.91839662490823476782717262862, 3.18153651066159805009838408765, 4.15376264120983200749644933658, 5.09196327013335612925130720207, 5.33002238531523506864245254531, 6.13161181216014691666479320551, 7.71136879901769074032724678054, 7.75364568294789525381115787121, 9.146910085028181413545895132657, 9.497692399288722992339187258041, 10.10487225466447691656417923583, 11.38744322909410240225914390179, 12.40426487154665503106833194686, 12.43077072814230613599510364838, 13.66256273110436947434727107367, 14.21371923317822788965977870210, 14.23348000659179508911246174538, 15.56062319188556808650697312767, 16.10130476472056037967319118127, 17.05939314902054771368099854329, 17.93672447684804605224613643041, 18.06574641877894839779638823471, 19.02943616266343248258861587188, 20.00141330000586300776155390382