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Properties

Label 1-171-171.83-r1-0-0
Degree $1$
Conductor $171$
Sign $0.790 + 0.612i$
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  − 2-s + 4-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 13-s + (0.5 + 0.866i)14-s + 16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s − 23-s + (−0.5 + 0.866i)25-s − 26-s + ⋯
L(s)  = 1  − 2-s + 4-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 13-s + (0.5 + 0.866i)14-s + 16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s − 23-s + (−0.5 + 0.866i)25-s − 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.790 + 0.612i$
Analytic conductor: \(18.3765\)
Root analytic conductor: \(18.3765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (1:\ ),\ 0.790 + 0.612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.196151761 + 0.4095856929i\)
\(L(\frac12)\) \(\approx\) \(1.196151761 + 0.4095856929i\)
\(L(1)\) \(\approx\) \(0.8245196240 + 0.1130097660i\)
\(L(1)\) \(\approx\) \(0.8245196240 + 0.1130097660i\)

Euler product

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

−27.507973322915982434045101358330, −25.97866241044886089935984999021, −25.5188680492228971123008290898, −24.52180668004492389014113327878, −23.77188368137958700300409284882, −21.99818091417195968155635375722, −21.26772873979937269957792031030, −20.25639759184532002306308447977, −19.253809950120156828997716970321, −18.42050954773160938732805340853, −17.38473352527298894328035278148, −16.35174232318150422737555712765, −15.838766543342853768302779973503, −14.37979863456244346931986778133, −12.947598459281466526704154443828, −12.04916568539530108700784646735, −10.88336578718741095754596783404, −9.62317398568533853645004827231, −8.84226779681935215364975012375, −8.0870851889341545596436630806, −6.23496752721208428792972689987, −5.755727180777512695841988546224, −3.62146699152799137923240057304, −2.06845858081256758545414876213, −0.798240897417133618397988839551, 1.06827836500938434016175037304, 2.52591547138103308765165411715, 3.83934878447664089404187358998, 5.997385416805255755792902144856, 6.87035772354432560960095093733, 7.742118783116450617055170814, 9.32073034073056064938381551299, 10.06352128390321378072924656621, 10.9122216726776515244589860274, 12.06373846286434593750246405047, 13.56396738723424648637991477666, 14.56960935076615613778983629764, 15.78276847109270979006019376346, 16.71239183698979041131954009858, 17.78385198945493018316816404758, 18.388873954144026944863508164358, 19.5551822517819645960174294231, 20.3538860895302787112538453105, 21.378643892426904207355568576361, 22.69105609652665834045757535235, 23.46507790183185221878548281199, 25.01904910076642157125057737953, 25.64526881742607010850600228970, 26.39036170844739621487326485862, 27.268302793666786918659019587272

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