↑

Properties

Label 1-3e3-27.22-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

Related objects

Downloads

Learn more

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: $\chi_{27} (22, \cdot )$
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 \( 1 + (0.766 + 0.642i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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