Dirichlet series
| L(s) = 1 | + 9·2-s − 1.00e3·4-s + 1.20e4·5-s − 1.20e4·7-s + 8.83e4·8-s + 1.08e5·10-s − 5.26e5·11-s − 2.22e6·13-s − 1.08e5·14-s + 5.89e5·16-s + 8.88e6·17-s + 6.17e6·19-s − 1.20e7·20-s − 4.74e6·22-s + 7.69e7·23-s + 2.53e7·25-s − 2.00e7·26-s + 1.20e7·28-s + 1.80e8·29-s − 3.56e7·31-s + 1.55e6·32-s + 7.99e7·34-s − 1.45e8·35-s + 5.96e7·37-s + 5.55e7·38-s + 1.06e9·40-s − 5.40e8·41-s + ⋯ |
| L(s) = 1 | + 0.198·2-s − 0.488·4-s + 1.72·5-s − 0.271·7-s + 0.953·8-s + 0.343·10-s − 0.986·11-s − 1.66·13-s − 0.0540·14-s + 0.140·16-s + 1.51·17-s + 0.571·19-s − 0.843·20-s − 0.196·22-s + 2.49·23-s + 0.518·25-s − 0.330·26-s + 0.132·28-s + 1.63·29-s − 0.223·31-s + 0.00818·32-s + 0.301·34-s − 0.468·35-s + 0.141·37-s + 0.113·38-s + 1.64·40-s − 0.728·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(531441\) = \(3^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(185214.\) |
| Root analytic conductor: | \(4.55469\) |
| Motivic weight: | \(11\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 531441,\ (\ :11/2, 11/2, 11/2, 11/2),\ 1)\) |
Particular Values
| \(L(6)\) | \(\approx\) | \(1.946701981\) |
| \(L(\frac12)\) | \(\approx\) | \(1.946701981\) |
| \(L(\frac{13}{2})\) | 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 | $C_2 \wr S_4$ | \( 1 - 9 T + 541 p T^{2} - 837 p^{7} T^{3} + 17607 p^{7} T^{4} - 837 p^{18} T^{5} + 541 p^{23} T^{6} - 9 p^{33} T^{7} + p^{44} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 2412 p T + 24028126 p T^{2} - 38909247912 p^{2} T^{3} + 62688308007891 p^{3} T^{4} - 38909247912 p^{13} T^{5} + 24028126 p^{23} T^{6} - 2412 p^{34} T^{7} + p^{44} T^{8} \) | |
| 7 | $C_2 \wr S_4$ | \( 1 + 12076 T + 3523686214 T^{2} + 14469482528272 p T^{3} + 151417928656476919 p^{2} T^{4} + 14469482528272 p^{12} T^{5} + 3523686214 p^{22} T^{6} + 12076 p^{33} T^{7} + p^{44} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 526788 T + 39624253102 p T^{2} + 242807357995546104 T^{3} + \)\(21\!\cdots\!67\)\( T^{4} + 242807357995546104 p^{11} T^{5} + 39624253102 p^{23} T^{6} + 526788 p^{33} T^{7} + p^{44} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 2227432 T + 4340556730132 T^{2} + 461256626445491704 p T^{3} + \)\(89\!\cdots\!30\)\( T^{4} + 461256626445491704 p^{12} T^{5} + 4340556730132 p^{22} T^{6} + 2227432 p^{33} T^{7} + p^{44} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 8884296 T + 152188590444548 T^{2} - \)\(90\!\cdots\!08\)\( T^{3} + \)\(80\!\cdots\!06\)\( T^{4} - \)\(90\!\cdots\!08\)\( p^{11} T^{5} + 152188590444548 p^{22} T^{6} - 8884296 p^{33} T^{7} + p^{44} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 6172784 T + 385477479434692 T^{2} - \)\(10\!\cdots\!24\)\( p T^{3} + \)\(62\!\cdots\!82\)\( T^{4} - \)\(10\!\cdots\!24\)\( p^{12} T^{5} + 385477479434692 p^{22} T^{6} - 6172784 p^{33} T^{7} + p^{44} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 76901040 T + 5412772069781156 T^{2} - \)\(22\!\cdots\!40\)\( T^{3} + \)\(83\!\cdots\!82\)\( T^{4} - \)\(22\!\cdots\!40\)\( p^{11} T^{5} + 5412772069781156 p^{22} T^{6} - 76901040 p^{33} T^{7} + p^{44} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 180046080 T + 30796015347898508 T^{2} - \)\(38\!\cdots\!60\)\( T^{3} + \)\(51\!\cdots\!82\)\( T^{4} - \)\(38\!\cdots\!60\)\( p^{11} T^{5} + 30796015347898508 p^{22} T^{6} - 180046080 p^{33} T^{7} + p^{44} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 35694652 T + 67125076158615838 T^{2} + \)\(23\!\cdots\!88\)\( T^{3} + \)\(21\!\cdots\!95\)\( T^{4} + \)\(23\!\cdots\!88\)\( p^{11} T^{5} + 67125076158615838 p^{22} T^{6} + 35694652 p^{33} T^{7} + p^{44} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 59658848 T + 518629292153510428 T^{2} + \)\(16\!\cdots\!28\)\( T^{3} + \)\(11\!\cdots\!90\)\( T^{4} + \)\(16\!\cdots\!28\)\( p^{11} T^{5} + 518629292153510428 p^{22} T^{6} - 59658848 p^{33} T^{7} + p^{44} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 540559152 T + 1077564717734609852 T^{2} + \)\(66\!\cdots\!96\)\( T^{3} + \)\(84\!\cdots\!02\)\( T^{4} + \)\(66\!\cdots\!96\)\( p^{11} T^{5} + 1077564717734609852 p^{22} T^{6} + 540559152 p^{33} T^{7} + p^{44} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 2521421368 T + 5093790666347846452 T^{2} + \)\(69\!\cdots\!28\)\( T^{3} + \)\(76\!\cdots\!30\)\( T^{4} + \)\(69\!\cdots\!28\)\( p^{11} T^{5} + 5093790666347846452 p^{22} T^{6} + 2521421368 p^{33} T^{7} + p^{44} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 2691337608 T + 11877920084136312020 T^{2} + \)\(20\!\cdots\!12\)\( T^{3} + \)\(46\!\cdots\!42\)\( T^{4} + \)\(20\!\cdots\!12\)\( p^{11} T^{5} + 11877920084136312020 p^{22} T^{6} + 2691337608 p^{33} T^{7} + p^{44} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 132122340 T + 14408869704130488974 T^{2} + \)\(46\!\cdots\!04\)\( T^{3} + \)\(78\!\cdots\!07\)\( T^{4} + \)\(46\!\cdots\!04\)\( p^{11} T^{5} + 14408869704130488974 p^{22} T^{6} + 132122340 p^{33} T^{7} + p^{44} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 8139028320 T + \)\(11\!\cdots\!08\)\( T^{2} + \)\(72\!\cdots\!40\)\( T^{3} + \)\(50\!\cdots\!82\)\( T^{4} + \)\(72\!\cdots\!40\)\( p^{11} T^{5} + \)\(11\!\cdots\!08\)\( p^{22} T^{6} + 8139028320 p^{33} T^{7} + p^{44} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 3516171856 T + 26722632168180254740 T^{2} - \)\(17\!\cdots\!84\)\( T^{3} + \)\(49\!\cdots\!70\)\( T^{4} - \)\(17\!\cdots\!84\)\( p^{11} T^{5} + 26722632168180254740 p^{22} T^{6} + 3516171856 p^{33} T^{7} + p^{44} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 711923512 T + \)\(41\!\cdots\!80\)\( T^{2} + \)\(31\!\cdots\!88\)\( T^{3} + \)\(71\!\cdots\!02\)\( T^{4} + \)\(31\!\cdots\!88\)\( p^{11} T^{5} + \)\(41\!\cdots\!80\)\( p^{22} T^{6} + 711923512 p^{33} T^{7} + p^{44} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 36309366072 T + \)\(10\!\cdots\!72\)\( T^{2} - \)\(18\!\cdots\!36\)\( T^{3} + \)\(31\!\cdots\!42\)\( T^{4} - \)\(18\!\cdots\!36\)\( p^{11} T^{5} + \)\(10\!\cdots\!72\)\( p^{22} T^{6} - 36309366072 p^{33} T^{7} + p^{44} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 28769206364 T + \)\(83\!\cdots\!50\)\( T^{2} - \)\(17\!\cdots\!84\)\( T^{3} + \)\(40\!\cdots\!67\)\( T^{4} - \)\(17\!\cdots\!84\)\( p^{11} T^{5} + \)\(83\!\cdots\!50\)\( p^{22} T^{6} - 28769206364 p^{33} T^{7} + p^{44} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 108210339448 T + \)\(68\!\cdots\!12\)\( T^{2} + \)\(28\!\cdots\!76\)\( T^{3} + \)\(91\!\cdots\!58\)\( T^{4} + \)\(28\!\cdots\!76\)\( p^{11} T^{5} + \)\(68\!\cdots\!12\)\( p^{22} T^{6} + 108210339448 p^{33} T^{7} + p^{44} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 68197592556 T + \)\(34\!\cdots\!74\)\( T^{2} - \)\(81\!\cdots\!72\)\( T^{3} + \)\(30\!\cdots\!59\)\( T^{4} - \)\(81\!\cdots\!72\)\( p^{11} T^{5} + \)\(34\!\cdots\!74\)\( p^{22} T^{6} - 68197592556 p^{33} T^{7} + p^{44} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 69699290616 T + \)\(10\!\cdots\!28\)\( T^{2} - \)\(55\!\cdots\!72\)\( T^{3} + \)\(40\!\cdots\!42\)\( T^{4} - \)\(55\!\cdots\!72\)\( p^{11} T^{5} + \)\(10\!\cdots\!28\)\( p^{22} T^{6} - 69699290616 p^{33} T^{7} + p^{44} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 153951848548 T + \)\(14\!\cdots\!38\)\( T^{2} + \)\(10\!\cdots\!32\)\( T^{3} - \)\(18\!\cdots\!65\)\( T^{4} + \)\(10\!\cdots\!32\)\( p^{11} T^{5} + \)\(14\!\cdots\!38\)\( p^{22} T^{6} + 153951848548 p^{33} T^{7} + p^{44} T^{8} \) | |
| show more | |||
| show less | |||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.30239645430955453257749604037, −10.01627647697177524654281796270, −9.593536417743516560176970795399, −9.564468152949570720802804840721, −9.385554897440297217522560631154, −8.409165090900624692890498239840, −8.172574898294954937541417782508, −8.109088934463181747584606257365, −7.34645902140491210012070296630, −7.02255548179160255285904831535, −6.84664809143298575614917854346, −6.19257776195965212497866739334, −5.86695111872713815517293993061, −5.15695002497515956476512671477, −5.08642410397676405462296509948, −4.71697582821639801301478442572, −4.70544964460530256181273854185, −3.41117966574473761668158788171, −3.16500970332420060391178702395, −2.94288744046508713114437722897, −2.06227084566454383669354225961, −1.98648101391524390234348389150, −1.19153806326204571706981111112, −1.04851411336423615628253681273, −0.20208713358249857514827274253, 0.20208713358249857514827274253, 1.04851411336423615628253681273, 1.19153806326204571706981111112, 1.98648101391524390234348389150, 2.06227084566454383669354225961, 2.94288744046508713114437722897, 3.16500970332420060391178702395, 3.41117966574473761668158788171, 4.70544964460530256181273854185, 4.71697582821639801301478442572, 5.08642410397676405462296509948, 5.15695002497515956476512671477, 5.86695111872713815517293993061, 6.19257776195965212497866739334, 6.84664809143298575614917854346, 7.02255548179160255285904831535, 7.34645902140491210012070296630, 8.109088934463181747584606257365, 8.172574898294954937541417782508, 8.409165090900624692890498239840, 9.385554897440297217522560631154, 9.564468152949570720802804840721, 9.593536417743516560176970795399, 10.01627647697177524654281796270, 10.30239645430955453257749604037