Dirichlet series
| L(s) = 1 | − 1.94e8·2-s − 1.04e18·4-s − 1.73e20·5-s − 7.53e24·7-s + 4.38e26·8-s + 3.36e28·10-s + 7.38e30·11-s − 2.42e32·13-s + 1.46e33·14-s + 4.08e35·16-s − 3.93e35·17-s − 8.13e37·19-s + 1.80e38·20-s − 1.43e39·22-s − 2.19e39·23-s − 4.72e41·25-s + 4.70e40·26-s + 7.85e42·28-s − 8.34e42·29-s + 1.27e44·31-s − 4.42e44·32-s + 7.64e43·34-s + 1.30e45·35-s − 1.52e46·37-s + 1.58e46·38-s − 7.59e46·40-s − 8.42e47·41-s + ⋯ |
| L(s) = 1 | − 0.255·2-s − 1.80·4-s − 0.415·5-s − 0.884·7-s + 1.00·8-s + 0.106·10-s + 1.40·11-s − 0.333·13-s + 0.226·14-s + 1.23·16-s − 0.197·17-s − 1.53·19-s + 0.752·20-s − 0.359·22-s − 0.148·23-s − 2.72·25-s + 0.0852·26-s + 1.59·28-s − 0.603·29-s + 1.28·31-s − 1.75·32-s + 0.0506·34-s + 0.367·35-s − 0.832·37-s + 0.394·38-s − 0.416·40-s − 2.22·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(60-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+59/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(6561\) = \(3^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.54979\times 10^{9}\) |
| Root analytic conductor: | \(14.0858\) |
| Motivic weight: | \(59\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 6561,\ (\ :59/2, 59/2, 59/2, 59/2),\ 1)\) |
Particular Values
| \(L(30)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{61}{2})\) | 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 | $C_2 \wr S_4$ | \( 1 + 6073353 p^{5} T + 1055487122120105 p^{10} T^{2} - 3038581438255650573 p^{23} T^{3} + \)\(91\!\cdots\!21\)\( p^{36} T^{4} - 3038581438255650573 p^{82} T^{5} + 1055487122120105 p^{128} T^{6} + 6073353 p^{182} T^{7} + p^{236} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 34651637541172603224 p T + \)\(16\!\cdots\!56\)\( p^{5} T^{2} + \)\(22\!\cdots\!24\)\( p^{11} T^{3} + \)\(60\!\cdots\!78\)\( p^{19} T^{4} + \)\(22\!\cdots\!24\)\( p^{70} T^{5} + \)\(16\!\cdots\!56\)\( p^{123} T^{6} + 34651637541172603224 p^{178} T^{7} + p^{236} T^{8} \) | |
| 7 | $C_2 \wr S_4$ | \( 1 + \)\(15\!\cdots\!12\)\( p^{2} T + \)\(90\!\cdots\!44\)\( p^{5} T^{2} + \)\(31\!\cdots\!52\)\( p^{12} T^{3} + \)\(46\!\cdots\!70\)\( p^{17} T^{4} + \)\(31\!\cdots\!52\)\( p^{71} T^{5} + \)\(90\!\cdots\!44\)\( p^{123} T^{6} + \)\(15\!\cdots\!12\)\( p^{179} T^{7} + p^{236} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - \)\(73\!\cdots\!44\)\( T + \)\(41\!\cdots\!32\)\( p^{2} T^{2} - \)\(96\!\cdots\!40\)\( p^{4} T^{3} + \)\(48\!\cdots\!06\)\( p^{8} T^{4} - \)\(96\!\cdots\!40\)\( p^{63} T^{5} + \)\(41\!\cdots\!32\)\( p^{120} T^{6} - \)\(73\!\cdots\!44\)\( p^{177} T^{7} + p^{236} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + \)\(18\!\cdots\!28\)\( p T + \)\(42\!\cdots\!48\)\( p^{3} T^{2} + \)\(10\!\cdots\!28\)\( p^{6} T^{3} + \)\(38\!\cdots\!90\)\( p^{10} T^{4} + \)\(10\!\cdots\!28\)\( p^{65} T^{5} + \)\(42\!\cdots\!48\)\( p^{121} T^{6} + \)\(18\!\cdots\!28\)\( p^{178} T^{7} + p^{236} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + \)\(39\!\cdots\!04\)\( T + \)\(72\!\cdots\!04\)\( p T^{2} + \)\(12\!\cdots\!80\)\( p^{3} T^{3} + \)\(46\!\cdots\!38\)\( p^{5} T^{4} + \)\(12\!\cdots\!80\)\( p^{62} T^{5} + \)\(72\!\cdots\!04\)\( p^{119} T^{6} + \)\(39\!\cdots\!04\)\( p^{177} T^{7} + p^{236} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + \)\(42\!\cdots\!32\)\( p T + \)\(86\!\cdots\!88\)\( p^{3} T^{2} + \)\(25\!\cdots\!64\)\( p^{5} T^{3} + \)\(42\!\cdots\!86\)\( p^{7} T^{4} + \)\(25\!\cdots\!64\)\( p^{64} T^{5} + \)\(86\!\cdots\!88\)\( p^{121} T^{6} + \)\(42\!\cdots\!32\)\( p^{178} T^{7} + p^{236} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + \)\(21\!\cdots\!24\)\( T + \)\(13\!\cdots\!04\)\( p T^{2} + \)\(12\!\cdots\!92\)\( p^{3} T^{3} + \)\(89\!\cdots\!10\)\( p^{3} T^{4} + \)\(12\!\cdots\!92\)\( p^{62} T^{5} + \)\(13\!\cdots\!04\)\( p^{119} T^{6} + \)\(21\!\cdots\!24\)\( p^{177} T^{7} + p^{236} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + \)\(28\!\cdots\!16\)\( p T + \)\(68\!\cdots\!80\)\( p^{2} T^{2} + \)\(14\!\cdots\!72\)\( p^{3} T^{3} + \)\(20\!\cdots\!78\)\( p^{4} T^{4} + \)\(14\!\cdots\!72\)\( p^{62} T^{5} + \)\(68\!\cdots\!80\)\( p^{120} T^{6} + \)\(28\!\cdots\!16\)\( p^{178} T^{7} + p^{236} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - \)\(12\!\cdots\!44\)\( T + \)\(31\!\cdots\!68\)\( T^{2} - \)\(71\!\cdots\!12\)\( p T^{3} + \)\(38\!\cdots\!34\)\( p^{2} T^{4} - \)\(71\!\cdots\!12\)\( p^{60} T^{5} + \)\(31\!\cdots\!68\)\( p^{118} T^{6} - \)\(12\!\cdots\!44\)\( p^{177} T^{7} + p^{236} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + \)\(15\!\cdots\!08\)\( T + \)\(14\!\cdots\!04\)\( p T^{2} + \)\(39\!\cdots\!48\)\( p^{2} T^{3} + \)\(27\!\cdots\!50\)\( p^{3} T^{4} + \)\(39\!\cdots\!48\)\( p^{61} T^{5} + \)\(14\!\cdots\!04\)\( p^{119} T^{6} + \)\(15\!\cdots\!08\)\( p^{177} T^{7} + p^{236} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + \)\(84\!\cdots\!44\)\( T + \)\(63\!\cdots\!68\)\( T^{2} + \)\(71\!\cdots\!52\)\( p T^{3} + \)\(78\!\cdots\!34\)\( p^{2} T^{4} + \)\(71\!\cdots\!52\)\( p^{60} T^{5} + \)\(63\!\cdots\!68\)\( p^{118} T^{6} + \)\(84\!\cdots\!44\)\( p^{177} T^{7} + p^{236} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - \)\(40\!\cdots\!40\)\( T + \)\(15\!\cdots\!60\)\( p T^{2} - \)\(22\!\cdots\!20\)\( p^{2} T^{3} + \)\(25\!\cdots\!14\)\( p^{3} T^{4} - \)\(22\!\cdots\!20\)\( p^{61} T^{5} + \)\(15\!\cdots\!60\)\( p^{119} T^{6} - \)\(40\!\cdots\!40\)\( p^{177} T^{7} + p^{236} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - \)\(17\!\cdots\!40\)\( T + \)\(16\!\cdots\!80\)\( T^{2} - \)\(46\!\cdots\!60\)\( p T^{3} + \)\(48\!\cdots\!42\)\( p^{2} T^{4} - \)\(46\!\cdots\!60\)\( p^{60} T^{5} + \)\(16\!\cdots\!80\)\( p^{118} T^{6} - \)\(17\!\cdots\!40\)\( p^{177} T^{7} + p^{236} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - \)\(49\!\cdots\!76\)\( T + \)\(29\!\cdots\!48\)\( p^{2} T^{2} + \)\(56\!\cdots\!96\)\( p^{2} T^{3} + \)\(15\!\cdots\!30\)\( p^{3} T^{4} + \)\(56\!\cdots\!96\)\( p^{61} T^{5} + \)\(29\!\cdots\!48\)\( p^{120} T^{6} - \)\(49\!\cdots\!76\)\( p^{177} T^{7} + p^{236} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + \)\(20\!\cdots\!52\)\( p T + \)\(57\!\cdots\!32\)\( T^{2} - \)\(17\!\cdots\!96\)\( p T^{3} + \)\(11\!\cdots\!54\)\( T^{4} - \)\(17\!\cdots\!96\)\( p^{60} T^{5} + \)\(57\!\cdots\!32\)\( p^{118} T^{6} + \)\(20\!\cdots\!52\)\( p^{178} T^{7} + p^{236} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + \)\(51\!\cdots\!80\)\( T + \)\(72\!\cdots\!76\)\( T^{2} + \)\(28\!\cdots\!20\)\( T^{3} + \)\(22\!\cdots\!06\)\( T^{4} + \)\(28\!\cdots\!20\)\( p^{59} T^{5} + \)\(72\!\cdots\!76\)\( p^{118} T^{6} + \)\(51\!\cdots\!80\)\( p^{177} T^{7} + p^{236} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + \)\(13\!\cdots\!36\)\( T + \)\(15\!\cdots\!48\)\( T^{2} + \)\(14\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!06\)\( T^{4} + \)\(14\!\cdots\!40\)\( p^{59} T^{5} + \)\(15\!\cdots\!48\)\( p^{118} T^{6} + \)\(13\!\cdots\!36\)\( p^{177} T^{7} + p^{236} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - \)\(14\!\cdots\!32\)\( T + \)\(12\!\cdots\!08\)\( T^{2} - \)\(70\!\cdots\!24\)\( T^{3} + \)\(31\!\cdots\!70\)\( T^{4} - \)\(70\!\cdots\!24\)\( p^{59} T^{5} + \)\(12\!\cdots\!08\)\( p^{118} T^{6} - \)\(14\!\cdots\!32\)\( p^{177} T^{7} + p^{236} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + \)\(36\!\cdots\!36\)\( T + \)\(79\!\cdots\!52\)\( T^{2} + \)\(11\!\cdots\!36\)\( T^{3} + \)\(12\!\cdots\!10\)\( T^{4} + \)\(11\!\cdots\!36\)\( p^{59} T^{5} + \)\(79\!\cdots\!52\)\( p^{118} T^{6} + \)\(36\!\cdots\!36\)\( p^{177} T^{7} + p^{236} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + \)\(19\!\cdots\!60\)\( T + \)\(40\!\cdots\!76\)\( T^{2} + \)\(46\!\cdots\!20\)\( T^{3} + \)\(55\!\cdots\!66\)\( T^{4} + \)\(46\!\cdots\!20\)\( p^{59} T^{5} + \)\(40\!\cdots\!76\)\( p^{118} T^{6} + \)\(19\!\cdots\!60\)\( p^{177} T^{7} + p^{236} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - \)\(96\!\cdots\!68\)\( T + \)\(68\!\cdots\!00\)\( T^{2} - \)\(36\!\cdots\!92\)\( T^{3} + \)\(17\!\cdots\!66\)\( T^{4} - \)\(36\!\cdots\!92\)\( p^{59} T^{5} + \)\(68\!\cdots\!00\)\( p^{118} T^{6} - \)\(96\!\cdots\!68\)\( p^{177} T^{7} + p^{236} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + \)\(40\!\cdots\!12\)\( T + \)\(14\!\cdots\!92\)\( T^{2} + \)\(91\!\cdots\!44\)\( T^{3} + \)\(30\!\cdots\!94\)\( T^{4} + \)\(91\!\cdots\!44\)\( p^{59} T^{5} + \)\(14\!\cdots\!92\)\( p^{118} T^{6} + \)\(40\!\cdots\!12\)\( p^{177} T^{7} + p^{236} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + \)\(81\!\cdots\!36\)\( T + \)\(50\!\cdots\!68\)\( T^{2} + \)\(34\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!46\)\( T^{4} + \)\(34\!\cdots\!80\)\( p^{59} T^{5} + \)\(50\!\cdots\!68\)\( p^{118} T^{6} + \)\(81\!\cdots\!36\)\( p^{177} T^{7} + p^{236} T^{8} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.100569426951641559493683151530, −7.36280358186993730459740495171, −7.29221134424690727251316771398, −7.15885844618831772371904734683, −6.49644384397453643139096457354, −6.45128326552963783120386482706, −5.89616121187691884983358348649, −5.88004854788727849473296175490, −5.65456930880082188757769319561, −4.86796104821008869148661735150, −4.66163915304171081458985674696, −4.63972424512685395590184402803, −4.45638098377376295915060026338, −3.81567933676924138953772406762, −3.78219211485244573395232320814, −3.56075346294941952520287051071, −3.47457716352865795400794762233, −2.70056164814103534016635843761, −2.65631545225083206546200443152, −2.09805814368029801992931597892, −1.73864388350278649528488535092, −1.62684114407945317541911381863, −1.43007194554989660744902271973, −0.866478211427395917195347818681, −0.67084282598141009863611001273, 0, 0, 0, 0, 0.67084282598141009863611001273, 0.866478211427395917195347818681, 1.43007194554989660744902271973, 1.62684114407945317541911381863, 1.73864388350278649528488535092, 2.09805814368029801992931597892, 2.65631545225083206546200443152, 2.70056164814103534016635843761, 3.47457716352865795400794762233, 3.56075346294941952520287051071, 3.78219211485244573395232320814, 3.81567933676924138953772406762, 4.45638098377376295915060026338, 4.63972424512685395590184402803, 4.66163915304171081458985674696, 4.86796104821008869148661735150, 5.65456930880082188757769319561, 5.88004854788727849473296175490, 5.89616121187691884983358348649, 6.45128326552963783120386482706, 6.49644384397453643139096457354, 7.15885844618831772371904734683, 7.29221134424690727251316771398, 7.36280358186993730459740495171, 8.100569426951641559493683151530