Properties

Degree 1
Conductor $ 5 \cdot 19 $
Sign $-0.486 + 0.873i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.173 + 0.984i)2-s + (−0.766 − 0.642i)3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)6-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.766 + 0.642i)13-s + (−0.939 + 0.342i)14-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s − 18-s + (0.173 − 0.984i)21-s + (−0.766 − 0.642i)22-s + ⋯
L(s,χ)  = 1  + (−0.173 + 0.984i)2-s + (−0.766 − 0.642i)3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)6-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.766 + 0.642i)13-s + (−0.939 + 0.342i)14-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s − 18-s + (0.173 − 0.984i)21-s + (−0.766 − 0.642i)22-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.486 + 0.873i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.486 + 0.873i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(95\)    =    \(5 \cdot 19\)
\( \varepsilon \)  =  $-0.486 + 0.873i$
motivic weight  =  \(0\)
character  :  $\chi_{95} (44, \cdot )$
Sato-Tate  :  $\mu(18)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 95,\ (0:\ ),\ -0.486 + 0.873i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2889849750 + 0.4918920032i$
$L(\frac12,\chi)$  $\approx$  $0.2889849750 + 0.4918920032i$
$L(\chi,1)$  $\approx$  0.5712607687 + 0.3520494375i
$L(1,\chi)$  $\approx$  0.5712607687 + 0.3520494375i

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

−29.37030175171058875441614227092, −29.16730063606590436234174071161, −27.64810997870507466063725810964, −27.068313559407029138045022988210, −26.3367548921774362357736188656, −24.35724210473904135488017704306, −23.14029476890713997827917736935, −22.42397661831591506327381879477, −21.20122273275906246742395330275, −20.62072010004510285150972534033, −19.33186539661214580815328612633, −17.96892154417309495833182641252, −17.23513675972836087142021600612, −16.125059414909351747146022622641, −14.51171892997458936767333347085, −13.28397526172023901135217679034, −11.98580206652363561160016286628, −10.92522056111803779090221748709, −10.30037861370215781470781186142, −8.988717280386248030348546228166, −7.44581234608667745966945298625, −5.39509025107779972739445292015, −4.41209941781215047181457065867, −2.99082760385092158946060092064, −0.72177993422487286185860051996, 1.87740647536784338092634050845, 4.68125573538334042648971950858, 5.58016164358734039137170071963, 6.85672819430100289077290957776, 7.82920752897935063807843893933, 9.16244548123950335585367007225, 10.64955498492614543279382201327, 12.16042060369288096584766617984, 13.06161264587616843185824467831, 14.54133880145899064188023663080, 15.47915001017410849402160388316, 16.801151115103265659794186828, 17.63206016985073055640845262440, 18.48383722208035951503381822115, 19.43672250413615535431249409003, 21.46501134039593343283029555908, 22.40109401819842342588781886893, 23.509179197901480617429351607224, 24.25318694320516884053906855785, 25.12407242907385200242781438561, 26.1840959738156500326094856532, 27.60377891954124572948938275913, 28.24232568327549086491754093544, 29.27041479485375637103918637373, 30.93150659869267426805292676796

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