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Properties

Label 1-6e3-216.139-r1-0-0
Degree $1$
Conductor $216$
Sign $-0.957 + 0.286i$
Analytic cond. $23.2124$
Root an. cond. $23.2124$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.939 + 0.342i)5-s + (−0.173 + 0.984i)7-s + (−0.939 + 0.342i)11-s + (−0.766 + 0.642i)13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)23-s + (0.766 + 0.642i)25-s + (−0.766 − 0.642i)29-s + (−0.173 − 0.984i)31-s + (−0.5 + 0.866i)35-s + (0.5 + 0.866i)37-s + (0.766 − 0.642i)41-s + (−0.939 + 0.342i)43-s + (−0.173 + 0.984i)47-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)5-s + (−0.173 + 0.984i)7-s + (−0.939 + 0.342i)11-s + (−0.766 + 0.642i)13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)23-s + (0.766 + 0.642i)25-s + (−0.766 − 0.642i)29-s + (−0.173 − 0.984i)31-s + (−0.5 + 0.866i)35-s + (0.5 + 0.866i)37-s + (0.766 − 0.642i)41-s + (−0.939 + 0.342i)43-s + (−0.173 + 0.984i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.957 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.957 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.957 + 0.286i$
Analytic conductor: \(23.2124\)
Root analytic conductor: \(23.2124\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 216,\ (1:\ ),\ -0.957 + 0.286i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1162753175 + 0.7938050426i\)
\(L(\frac12)\) \(\approx\) \(0.1162753175 + 0.7938050426i\)
\(L(1)\) \(\approx\) \(0.8799950881 + 0.2756544801i\)
\(L(1)\) \(\approx\) \(0.8799950881 + 0.2756544801i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (-0.173 + 0.984i)T \)
11 \( 1 + (-0.939 + 0.342i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (-0.766 - 0.642i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.766 + 0.642i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.04818686483636514602286112209, −25.040271244128277785982792770774, −24.008547693693500672064867661802, −23.32680490319742792637177380086, −21.94934861558792470513289247364, −21.39105968417182545270823589546, −20.20296445263460585999769609985, −19.577692165774976265780178788940, −18.09262324755295377911653611471, −17.38753339684611671840742607500, −16.579188580626206800470989357812, −15.41399333649500487703046811378, −14.24155966068730830222590576582, −13.19689709779944865276473044865, −12.7797667178302382589621938897, −10.97807400947896686227556357981, −10.29232896042147104334446018317, −9.265009016139163944446111302032, −8.015925653187256718656769875206, −6.89867016089143232589443440419, −5.66571773551089599337760288377, −4.68045482487456250767816472812, −3.16588569374049946092487526030, −1.79455181156939101922079322594, −0.23960734309534655226228880051, 2.064674091096163875254194798400, 2.70005449238056537059847005097, 4.612259796022863378179161381506, 5.69467734046118976747503494030, 6.63275629716157040232480040892, 7.94693574720505792208664414204, 9.29097166962987991394689138266, 9.92968188765765056874471802543, 11.16385231579941527707244978448, 12.36045866302804207461705503239, 13.22138412030784845190793911911, 14.37584569909007013156356607430, 15.17600629875851816973853687425, 16.332855670941466536314030190596, 17.36060019565541124325929518532, 18.47611105762137891037572567492, 18.82581853659900457417922095083, 20.43629622561586578680544092638, 21.20365324042076683990479418649, 22.11762609537799832679147684575, 22.80055740477756365150895077508, 24.23067182265711540994399606326, 24.93645801166074085710237932967, 25.8618192144198389497523665776, 26.57161216933253813477451187693

Graph of the $Z$-function along the critical line