Properties

Label 1-171-171.157-r0-0-0
Degree $1$
Conductor $171$
Sign $0.845 + 0.533i$
Analytic cond. $0.794120$
Root an. cond. $0.794120$
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

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

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.845 + 0.533i$
Analytic conductor: \(0.794120\)
Root analytic conductor: \(0.794120\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (0:\ ),\ 0.845 + 0.533i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.212969464 + 0.3504839624i\)
\(L(\frac12)\) \(\approx\) \(1.212969464 + 0.3504839624i\)
\(L(1)\) \(\approx\) \(1.125676561 + 0.3342096573i\)
\(L(1)\) \(\approx\) \(1.125676561 + 0.3342096573i\)

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

−27.91325463855636768039673496075, −26.52410785767689862361332898370, −25.6239364083977178295954371006, −24.67252428010668028647507414779, −23.15324595526764281848118824562, −22.352309528887081199236434224461, −21.734812262949956555215547576440, −20.791877125355125217891435024677, −19.61342767353477348056734602841, −18.71014726500498542127466196948, −18.01516443762796281277071742717, −16.89435864458587163451109896117, −15.245040002690079762005454622105, −14.36371524137553058479633844372, −13.349236287267340888062923055374, −12.37147453082717029650531371186, −11.35548247301338288759720502602, −10.19876053559889431139112965277, −9.45482217835729878801020471236, −8.315187284157315699215999852368, −6.2915159662898636586537559455, −5.63014413302363946343334840296, −3.84818748414008452493530265389, −2.814720747504328235471660092068, −1.58329974624497510989451761762, 1.222416884352976629508714121133, 3.57648090381635927790122226649, 4.619789897793981948723269060774, 5.97262284152496779455272883301, 6.73745941919318549986869382544, 8.05438145203793219335353196531, 9.28398633182254685802365261393, 9.910297698269618321716089780539, 11.756000803435774778641485180736, 13.05457786728506713505315556115, 13.771053990910600352343830143141, 14.54008031280383523071143392266, 16.17925040168640285218723853641, 16.59415281982098547222924664718, 17.50018751096301901449919307899, 18.56352121378336030854606574740, 19.8764897082385542554183967896, 20.99977516462306814783495989759, 22.00089573645504885112827845474, 22.99134005479968637665155751300, 23.85233450007202133807896566927, 24.818768162601327807345076588235, 25.61468395063652327400491819942, 26.359502359608370025811906956, 27.51250861895078998641492168664

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