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Properties

Label 1-171-171.34-r1-0-0
Degree $1$
Conductor $171$
Sign $-0.973 - 0.226i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.973 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.973 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.973 - 0.226i$
Analytic conductor: \(18.3765\)
Root analytic conductor: \(18.3765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (1:\ ),\ -0.973 - 0.226i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1371105608 + 1.194028880i\)
\(L(\frac12)\) \(\approx\) \(-0.1371105608 + 1.194028880i\)
\(L(1)\) \(\approx\) \(0.6393102968 + 0.6728668112i\)
\(L(1)\) \(\approx\) \(0.6393102968 + 0.6728668112i\)

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

−27.10075223731750823620845282537, −25.89891118575572053889085961775, −25.02874898019382101776658368753, −23.74266591417688582741859071578, −22.58682633172710729057661456411, −21.90672676447094774431101780530, −20.74463159076052598521407406089, −20.04363140120461278816334895030, −19.27834838776472367333971747933, −17.853841556028871001989227698334, −17.16649454746183127562460481318, −16.3186916314003166934599686890, −14.35396349935185937360681969110, −13.65990887811379090837730849084, −12.61780136480796928956939122654, −11.800549356491895647885087879526, −10.23526579722915817014044514271, −9.78221911132118227425098595045, −8.662812555886292958371370752832, −7.27057256629005343927204746278, −5.63249590127930586112079259595, −4.409386480369194422490221394805, −3.19823354780954580346253649110, −1.66792103245550994046201798419, −0.46885708348924032847186094471, 1.80336550388124948931119736654, 3.52038051257271350732807072423, 5.1427475210217366939302510777, 6.24902403923661638923569643934, 6.86096600741273437978204344459, 8.39382689047880919684945377947, 9.47834932725644476958070912391, 10.13059329355618917628834903906, 11.88773494282077540145121856844, 13.03926123044760635655122350011, 14.43069930787484830783107057383, 14.64688297015437031761051430331, 16.07587738890389654539017175643, 16.965058353894686313068332294919, 17.917533928878881526899051398961, 18.825616266871955967219508559555, 19.613460748555073098092245818901, 21.63443751038426395509545729286, 21.99479616770331233470691375422, 22.981178795858531026411726157487, 24.231849525368238701568183230358, 25.09288804083943237516107883642, 25.74708936219432275912125211088, 26.58594201213155817707824241093, 27.67892712889635700679970043894

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