Properties

Degree 1
Conductor $ 3^{3} $
Sign $0.727 - 0.686i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.727 - 0.686i$
motivic weight  =  \(0\)
character  :  $\chi_{27} (25, \cdot )$
Sato-Tate  :  $\mu(9)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 27,\ (0:\ ),\ 0.727 - 0.686i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4904161974 - 0.1948298888i$
$L(\frac12,\chi)$  $\approx$  $0.4904161974 - 0.1948298888i$
$L(\chi,1)$  $\approx$  0.6708527834 - 0.1803680129i
$L(1,\chi)$  $\approx$  0.6708527834 - 0.1803680129i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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