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Properties

Label 1-3e3-27.16-r0-0-0
Degree $1$
Conductor $27$
Sign $0.230 + 0.973i$
Analytic cond. $0.125387$
Root an. cond. $0.125387$
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)2-s + (−0.939 + 0.342i)4-s + (0.766 + 0.642i)5-s + (−0.939 − 0.342i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.766 − 0.642i)11-s + (0.173 − 0.984i)13-s + (0.173 − 0.984i)14-s + (0.766 − 0.642i)16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.939 − 0.342i)20-s + (0.766 + 0.642i)22-s + (−0.939 + 0.342i)23-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.766 + 0.642i)5-s + (−0.939 − 0.342i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.766 − 0.642i)11-s + (0.173 − 0.984i)13-s + (0.173 − 0.984i)14-s + (0.766 − 0.642i)16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.939 − 0.342i)20-s + (0.766 + 0.642i)22-s + (−0.939 + 0.342i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $0.230 + 0.973i$
Analytic conductor: \(0.125387\)
Root analytic conductor: \(0.125387\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 27,\ (0:\ ),\ 0.230 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5864441141 + 0.4636998845i\)
\(L(\frac12)\) \(\approx\) \(0.5864441141 + 0.4636998845i\)
\(L(1)\) \(\approx\) \(0.8427329235 + 0.4707643909i\)
\(L(1)\) \(\approx\) \(0.8427329235 + 0.4707643909i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (0.766 - 0.642i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.939 - 0.342i)T \)
53 \( 1 + T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (0.173 + 0.984i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.766 - 0.642i)T \)
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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

−37.8812762299544340837531002367, −36.24787188861703270843351123143, −35.69894159792211780149762288776, −33.42085692277663973916079217459, −32.30414487371379749627334556962, −31.28632447054244060364686848212, −29.71148696724070565443330919075, −28.80686862932710188919027788825, −27.848567299870614307616963908205, −26.11423057552191493523360971159, −24.68670713132286106213894065078, −22.93938639134165655027519123479, −21.8326294761811823189568089601, −20.617155414677183501939128501133, −19.395605517348846220190297732761, −17.98583564836765856291709864755, −16.46375089540683759948909763622, −14.29464831064455455971039064721, −13.00007918159710167639789411203, −11.87522622421689831252737256398, −9.87536732274703710343140532294, −9.03127531541962678034631241343, −6.11462586364526593506523367458, −4.2495330971972226603620081109, −2.06798021116922394942160955910, 3.51591294158896197287077315224, 5.845991025159952441041195618561, 6.9000338510018832653850192661, 8.890356909867259833440304182087, 10.407662944348542994807060476686, 12.9156156312229036216898190000, 13.993766917902001250790143652743, 15.39807568411316693714823259329, 16.86548122591468243909404774192, 17.98066168625784950975071409847, 19.52558615072679607799181230122, 21.80544909324418429039844859308, 22.514398106655226045878026890315, 24.020638462668331218531831734714, 25.48689432617047355463403256918, 26.14910938770239339712457551788, 27.59552286596135926477831038659, 29.43591271873541447139070754659, 30.56288175259935588474263260321, 32.429820614659926591330357523629, 32.848376770172237125898598545783, 34.388355675025192957880332392606, 35.29535114844820265659073307784, 36.6749197804418468982121865463, 37.90317748619161782170449268181

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