↑

Properties

Label 1-3e3-27.13-r0-0-0
Degree $1$
Conductor $27$
Sign $0.727 + 0.686i$
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  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + (0.173 + 0.984i)5-s + (0.766 + 0.642i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.173 − 0.984i)11-s + (−0.939 − 0.342i)13-s + (−0.939 − 0.342i)14-s + (0.173 − 0.984i)16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.766 + 0.642i)20-s + (0.173 + 0.984i)22-s + (0.766 − 0.642i)23-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + (0.173 + 0.984i)5-s + (0.766 + 0.642i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.173 − 0.984i)11-s + (−0.939 − 0.342i)13-s + (−0.939 − 0.342i)14-s + (0.173 − 0.984i)16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.766 + 0.642i)20-s + (0.173 + 0.984i)22-s + (0.766 − 0.642i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $0.727 + 0.686i$
Analytic conductor: \(0.125387\)
Root analytic conductor: \(0.125387\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 27,\ (0:\ ),\ 0.727 + 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4904161974 + 0.1948298888i\)
\(L(\frac12)\) \(\approx\) \(0.4904161974 + 0.1948298888i\)
\(L(1)\) \(\approx\) \(0.6708527834 + 0.1803680129i\)
\(L(1)\) \(\approx\) \(0.6708527834 + 0.1803680129i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
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 \( 1 + (0.173 - 0.984i)T \)
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

−37.29804107595635392617108100640, −36.48437316178819957089312997132, −35.56886624754737798978069059418, −34.04395283299891854674509643110, −32.906079767968009297468755051336, −31.10397322491896607666635155264, −29.867208497144359344107460029628, −28.55384907944317333973465270055, −27.646524949936342287729738193344, −26.37212973297575410829294073475, −24.95012017012848931506399938462, −23.84061236803992608271763504131, −21.59841535091784604896837017347, −20.451087671551300633152709188393, −19.50458733471328116879835740164, −17.49383395912093936008620550323, −17.03910479009999802770534189648, −15.19147760380354340021567674139, −13.02402124402460833174722300178, −11.62750766701912580283362925472, −10.00752396427758668036330038709, −8.64039580907036702230177424168, −7.16060343511994922697492656148, −4.55007778091148975896620945674, −1.74993694859259126172085783628, 2.4647645825156502465750364336, 5.69563784338302977426148653043, 7.290203438280219908468402759274, 8.81169104513031416092090035974, 10.471836307563585828664309239382, 11.65396679535830822065831219641, 14.28854830331122535283861236746, 15.291734837449843918437260545636, 16.99631143531635371190370315198, 18.286083298559526939498895729894, 19.14296493059143658470160211392, 20.91543964275324839248893731515, 22.415363267806994461457555754775, 24.26169998724819837558081461783, 25.204475213023365565372891903, 26.73081007126011074245113306014, 27.393782537926762119804255960977, 29.06920731570446233616621258167, 30.064223929893360388995265648033, 31.755868599039456540972199212770, 33.53656496223680104973264036537, 34.29014437904429629398873340305, 35.27352116954896961096017102697, 37.01477588187186283406394188866, 37.58372954182998689031153394258

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