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Properties

Label 1-171-171.61-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: $\chi_{171} (61, \cdot )$
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

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

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