Properties

Degree 1
Conductor $ 3^{4} $
Sign $0.0774 - 0.996i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.686 − 0.727i)2-s + (−0.0581 − 0.998i)4-s + (0.993 + 0.116i)5-s + (0.893 − 0.448i)7-s + (−0.766 − 0.642i)8-s + (0.766 − 0.642i)10-s + (−0.396 + 0.918i)11-s + (0.973 − 0.230i)13-s + (0.286 − 0.957i)14-s + (−0.993 + 0.116i)16-s + (−0.173 − 0.984i)17-s + (0.173 − 0.984i)19-s + (0.0581 − 0.998i)20-s + (0.396 + 0.918i)22-s + (−0.893 − 0.448i)23-s + ⋯
L(s,χ)  = 1  + (0.686 − 0.727i)2-s + (−0.0581 − 0.998i)4-s + (0.993 + 0.116i)5-s + (0.893 − 0.448i)7-s + (−0.766 − 0.642i)8-s + (0.766 − 0.642i)10-s + (−0.396 + 0.918i)11-s + (0.973 − 0.230i)13-s + (0.286 − 0.957i)14-s + (−0.993 + 0.116i)16-s + (−0.173 − 0.984i)17-s + (0.173 − 0.984i)19-s + (0.0581 − 0.998i)20-s + (0.396 + 0.918i)22-s + (−0.893 − 0.448i)23-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.0774 - 0.996i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.0774 - 0.996i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(81\)    =    \(3^{4}\)
\( \varepsilon \)  =  $0.0774 - 0.996i$
motivic weight  =  \(0\)
character  :  $\chi_{81} (50, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 81,\ (1:\ ),\ 0.0774 - 0.996i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.107106244 - 1.949684296i$
$L(\frac12,\chi)$  $\approx$  $2.107106244 - 1.949684296i$
$L(\chi,1)$  $\approx$  1.618427994 - 0.9056868716i
$L(1,\chi)$  $\approx$  1.618427994 - 0.9056868716i

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

−31.10516220359868132228959230959, −30.16010975498660174662513943913, −29.06305562607839197476821128007, −27.7252178152504595715873396119, −26.36170610024523076994022417828, −25.514448479569282810459371923331, −24.444520485745381868306379402464, −23.77213635188537395279032600420, −22.28562382980713243681153443712, −21.31834705108973595129495204887, −20.79258528578100210335787438524, −18.60436494297736017426025327176, −17.65867984131688942140664279913, −16.60678442544208042213153238469, −15.4207386499413830148193487753, −14.153261517630565279595445187603, −13.48535280733684881744985454359, −12.12821747048661924804010665683, −10.75047072932560272992738109965, −8.88633722057483845053778655391, −7.97154210078463746095180764839, −6.09773321236418388722299949259, −5.51545124185131524715381398673, −3.84046721260376598415495361839, −2.00078796799759549938461208327, 1.323282850427507990554958103325, 2.637960689681680118548723315457, 4.47143921374918441994420529926, 5.5262019200808615016911638096, 7.036338781840297695364750657682, 9.04197289101916887936745111176, 10.3199914924519273068502816932, 11.171591978196345202538129213205, 12.65556419790930653522462179869, 13.728384886403399584268737648928, 14.48477318180073022756282988572, 15.88114582972065334208771313046, 17.79816513339196796030092888669, 18.23374502131793768508457348818, 20.11469131127612700402930019846, 20.72653220080552851140016662160, 21.71762215473169495203786614091, 22.82340360804161002348031078821, 23.85572072699129567125402998304, 24.95458977886989231078564168612, 26.18211315912644746651723984044, 27.68537814799999714097686393658, 28.512342501887479683407637086174, 29.62714388633109677570489635641, 30.44301539641359780216486050780

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