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Properties

Label 1-171-171.137-r1-0-0
Degree $1$
Conductor $171$
Sign $0.533 + 0.845i$
Analytic cond. $18.3765$
Root an. cond. $18.3765$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.533 + 0.845i$
Analytic conductor: \(18.3765\)
Root analytic conductor: \(18.3765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (1:\ ),\ 0.533 + 0.845i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8644177600 + 0.4768596062i\)
\(L(\frac12)\) \(\approx\) \(0.8644177600 + 0.4768596062i\)
\(L(1)\) \(\approx\) \(0.6932064531 + 0.2820558880i\)
\(L(1)\) \(\approx\) \(0.6932064531 + 0.2820558880i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 - T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.939 + 0.342i)T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (-0.173 - 0.984i)T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.88090355184544378687291540322, −26.62251469040476429597434074720, −25.68100534822897055025030854793, −23.67371842760417320069541224859, −23.27046112570711519239781187152, −22.29312010830226121040591479759, −21.18149173441324777200025315048, −20.258226331917758931241718463815, −19.309235385909246883404305850800, −18.67111171700748115961869856881, −17.57438211098288934829242058123, −16.373713715265331453831073568103, −15.24197133816946997999710296020, −13.83728935136448128346089247839, −13.12112429389892766352077678427, −11.88393373716326302169678645363, −10.79229622486095736029122502998, −10.332619300036329367731870395184, −8.81388656434196839695161191225, −7.753850598155606773366746025600, −6.52741468001504611461897278079, −4.57529227719355281760850313880, −3.657421141274518342355123065510, −2.52925387602958188901602445408, −0.67952730722515819481882634025, 0.70233495335438834372739332892, 3.06524015983573943067391005042, 4.645308953338248629745586842058, 5.53614049529563404107739206703, 6.79087377174817506332668764969, 8.09777632456375049820321964669, 8.71108953255124424340085279790, 9.8817854084178214433921085772, 11.36236457105206001058061038847, 12.79362653538280151141199594646, 13.34892108523252136006164158148, 15.010804219027077913143267331549, 15.76311682827330650740086611352, 16.2068350427442031276207363006, 17.63371349697103901772067770796, 18.51684164851025747045257324171, 19.403225814246282969189973236011, 20.602312053758644140776853829680, 21.8692784857561207915503415982, 23.05202071407046005109861790967, 23.536630534895023689883228400196, 24.80002376381124738933634293521, 25.277881772337446801953202054732, 26.491889996055349618424078112398, 27.30670253360846765389751720067

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