Properties

Degree 1
Conductor $ 3 \cdot 19 $
Sign $0.290 + 0.956i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.766 + 0.642i)10-s + (0.5 − 0.866i)11-s + (0.939 + 0.342i)13-s + (0.173 − 0.984i)14-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s − 20-s + (0.939 − 0.342i)22-s + (−0.173 − 0.984i)23-s + (−0.939 − 0.342i)25-s + (0.5 + 0.866i)26-s + ⋯
L(s,χ)  = 1  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.766 + 0.642i)10-s + (0.5 − 0.866i)11-s + (0.939 + 0.342i)13-s + (0.173 − 0.984i)14-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s − 20-s + (0.939 − 0.342i)22-s + (−0.173 − 0.984i)23-s + (−0.939 − 0.342i)25-s + (0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.290 + 0.956i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.290 + 0.956i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(57\)    =    \(3 \cdot 19\)
\( \varepsilon \)  =  $0.290 + 0.956i$
motivic weight  =  \(0\)
character  :  $\chi_{57} (53, \cdot )$
Sato-Tate  :  $\mu(18)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 57,\ (0:\ ),\ 0.290 + 0.956i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.9767511401 + 0.7240934479i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.9767511401 + 0.7240934479i\)
\(L(\chi,1)\)  \(\approx\)  \(1.202616685 + 0.5966948384i\)
\(L(1,\chi)\)  \(\approx\)  \(1.202616685 + 0.5966948384i\)

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

−32.563371582925584937973189138047, −31.48275137764504562511800644629, −30.72166075342355603906820324253, −29.30042259031889790386309744726, −28.2217007642433886842942631201, −27.78304659129735242585904889748, −25.583302238983055944401736249774, −24.61700250965743683097795208460, −23.46222524327603173494161377350, −22.37654099363572910779026619177, −21.24586818531810739534993203841, −20.1630978287607629183099080210, −19.27196170744109415812590917122, −17.74950014902260358206666857757, −15.953687904634736807773296308304, −15.13195040821131067147850075480, −13.40881044129993141359404765825, −12.53618834464391696137203710792, −11.54452145669419879389219900533, −9.83945010535109530721733788189, −8.6648661256333105979411848080, −6.37963916873638088968906029163, −5.07560399290877310363568765270, −3.67571423190668730345682445197, −1.7658686899899328616425696788, 3.05264398190378724706401307006, 4.18971569348015419628187312811, 6.2588285258460393877005356274, 6.986347355265703604031475910026, 8.57656934094403387390761687380, 10.6108711247087023108230919278, 11.7725497887800773966645219789, 13.54992984075997721751113328659, 14.115053591798576692528383729876, 15.60520423219151143759517762256, 16.54257296197703713137949413541, 17.92550290524197437959346215209, 19.34722399455726448170423429061, 20.80121938811289109863826944851, 22.14434593921332734753186482834, 22.89763745434151662563110171146, 23.90117265168570621862572843080, 25.23082423092246179398228967895, 26.39901131709837712824127124542, 26.94518064586343169915945639911, 29.08330768633168966646313716740, 30.12711441603230070099260780331, 30.85952212499981240506074857458, 32.23811050457515754264183524501, 33.12251750024732692882093875921

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