Properties

Label 1-6e3-216.67-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  + (−0.173 − 0.984i)5-s + (−0.766 − 0.642i)7-s + (0.173 − 0.984i)11-s + (0.939 + 0.342i)13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.766 + 0.642i)23-s + (−0.939 + 0.342i)25-s + (0.939 − 0.342i)29-s + (−0.766 + 0.642i)31-s + (−0.5 + 0.866i)35-s + (0.5 + 0.866i)37-s + (−0.939 − 0.342i)41-s + (0.173 − 0.984i)43-s + (−0.766 − 0.642i)47-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)5-s + (−0.766 − 0.642i)7-s + (0.173 − 0.984i)11-s + (0.939 + 0.342i)13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.766 + 0.642i)23-s + (−0.939 + 0.342i)25-s + (0.939 − 0.342i)29-s + (−0.766 + 0.642i)31-s + (−0.5 + 0.866i)35-s + (0.5 + 0.866i)37-s + (−0.939 − 0.342i)41-s + (0.173 − 0.984i)43-s + (−0.766 − 0.642i)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: $\chi_{216} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 216,\ (1:\ ),\ -0.957 + 0.286i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07004705688 - 0.4782073111i\)
\(L(\frac12)\) \(\approx\) \(-0.07004705688 - 0.4782073111i\)
\(L(1)\) \(\approx\) \(0.7465720621 - 0.2765173363i\)
\(L(1)\) \(\approx\) \(0.7465720621 - 0.2765173363i\)

Euler product

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

−26.70635526889639978775738308292, −25.757539150478666847149768438977, −25.42673353909107859052772778588, −23.9374992152144743893507209182, −22.98065909541499595159344373795, −22.2745962225191093512708522532, −21.484822874382752969604359281900, −20.07748338377867282766158978622, −19.34662714293110232343630777600, −18.286782852594538606092820159616, −17.669829737096196193715027513123, −16.17022691546259839200091650835, −15.32180201556651882325417794360, −14.623370310111875306558095153843, −13.25385924478450510406546458733, −12.41155656571975870881757806471, −11.18550973118990043804747835554, −10.29979251014140508053194806657, −9.21302156396212231590137934439, −8.01741960728673629887646676586, −6.669183250947065955624032863498, −6.09970007991104068093281576559, −4.36889445460158350840908218688, −3.1703349103764712855587768422, −2.04784975930870646815621763848, 0.16101123703023360935636934913, 1.42759026339693592833394236412, 3.36145683970511958173781044825, 4.2756603822905977160633707999, 5.68431904909687912650776669259, 6.70613152321371145203324873293, 8.12450710545130253031201874711, 8.96106097521543574456023963974, 10.05070477116647760868905009207, 11.27093120314574473119798078816, 12.27235701581957629180360148320, 13.43675893367577587591868136974, 13.9058541209321756648210601517, 15.67611096674893853875890645562, 16.305127667351752164313648507033, 17.00524987768890998603506625896, 18.34750665805556137511068302520, 19.382681526388762529878534499644, 20.182884709157325218783667867135, 21.043326572191961991780628921640, 22.0838215988834680001040954087, 23.30057072669894051958539371239, 23.82695115319305180232132497146, 24.97205274061113591722044984632, 25.748976307326861680633947163842

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