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Properties

Label 1-171-171.97-r1-0-0
Degree $1$
Conductor $171$
Sign $-0.320 + 0.947i$
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 + (0.173 + 0.984i)5-s + 7-s + (0.5 + 0.866i)8-s + (0.939 − 0.342i)10-s + (−0.5 + 0.866i)11-s + (−0.766 − 0.642i)13-s + (−0.173 − 0.984i)14-s + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + (−0.5 − 0.866i)20-s + (0.939 + 0.342i)22-s + (−0.939 + 0.342i)23-s + (−0.939 + 0.342i)25-s + (−0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.173 + 0.984i)5-s + 7-s + (0.5 + 0.866i)8-s + (0.939 − 0.342i)10-s + (−0.5 + 0.866i)11-s + (−0.766 − 0.642i)13-s + (−0.173 − 0.984i)14-s + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + (−0.5 − 0.866i)20-s + (0.939 + 0.342i)22-s + (−0.939 + 0.342i)23-s + (−0.939 + 0.342i)25-s + (−0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.320 + 0.947i$
Analytic conductor: \(18.3765\)
Root analytic conductor: \(18.3765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (1:\ ),\ -0.320 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3160844210 + 0.4404666805i\)
\(L(\frac12)\) \(\approx\) \(0.3160844210 + 0.4404666805i\)
\(L(1)\) \(\approx\) \(0.7600799813 - 0.1010703384i\)
\(L(1)\) \(\approx\) \(0.7600799813 - 0.1010703384i\)

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 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.766 - 0.642i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.766 - 0.642i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (0.766 + 0.642i)T \)
53 \( 1 + (-0.766 - 0.642i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.173 + 0.984i)T \)
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

−26.837693612584777282836465094922, −26.144251702015739534762802739532, −24.73294665022646316541107597622, −24.22323848340260712616353585482, −23.736481370367371758751306795563, −22.159151738370016682614294155047, −21.349045331691292427431751292817, −20.17872372248765903495997294558, −18.97724460824192548068534024408, −17.89173327614797319526667095774, −17.06208784970414070856633698534, −16.27348232795948103538241063621, −15.22329934864091673380744594796, −14.10616256298590279736736167952, −13.32435610495410370647409089689, −12.064090546617109533683745395054, −10.65610228839463681691788242347, −9.26744048084885731960876289389, −8.46271797451967902309660283123, −7.58938940542833805996444592880, −6.09249182163631282111124935155, −5.060015505886987185044741918679, −4.217156923959122817101107341573, −1.806188491079191197737817382281, −0.19992870315614496699004034417, 1.85929214646968815991163379751, 2.73344679040961569265973730377, 4.25760146706477809523007849419, 5.368631515898938324154264994999, 7.2252916911865177776510917910, 8.143921341780902972666306989702, 9.63379838785388960559312480832, 10.47207164158848009016681439816, 11.344715011877970877223543306642, 12.344884042351091023130624723802, 13.596925178860995799657055655477, 14.509684878146078317248354501353, 15.47419524240870069289408634506, 17.55974981239572161617296335364, 17.70826556736224984819244685208, 18.77420093624300387022914118261, 19.913760781460692484504462746483, 20.72467456476184926518592195885, 21.76739348327248596787726364881, 22.50331201539348251334050995418, 23.42827696052639393620063674861, 24.75126799658596363156282105220, 26.0370867713321965838925808335, 26.77766873703451666364396400551, 27.61308236656955799746024672476

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