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Properties

Label 1-171-171.40-r1-0-0
Degree $1$
Conductor $171$
Sign $-0.988 - 0.150i$
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  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.939 − 0.342i)5-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)8-s + (0.939 − 0.342i)10-s + 11-s + (0.939 − 0.342i)13-s + (0.939 + 0.342i)14-s + (−0.939 − 0.342i)16-s + (−0.939 − 0.342i)17-s + (−0.5 + 0.866i)20-s + (−0.766 + 0.642i)22-s + (0.173 − 0.984i)23-s + (0.766 + 0.642i)25-s + (−0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.939 − 0.342i)5-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)8-s + (0.939 − 0.342i)10-s + 11-s + (0.939 − 0.342i)13-s + (0.939 + 0.342i)14-s + (−0.939 − 0.342i)16-s + (−0.939 − 0.342i)17-s + (−0.5 + 0.866i)20-s + (−0.766 + 0.642i)22-s + (0.173 − 0.984i)23-s + (0.766 + 0.642i)25-s + (−0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.988 - 0.150i$
Analytic conductor: \(18.3765\)
Root analytic conductor: \(18.3765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (1:\ ),\ -0.988 - 0.150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009570453111 - 0.1262133161i\)
\(L(\frac12)\) \(\approx\) \(0.009570453111 - 0.1262133161i\)
\(L(1)\) \(\approx\) \(0.5304757797 + 0.003130501020i\)
\(L(1)\) \(\approx\) \(0.5304757797 + 0.003130501020i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + T \)
13 \( 1 + (0.939 - 0.342i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.173 + 0.984i)T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (-0.766 - 0.642i)T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.173 + 0.984i)T \)
97 \( 1 + (-0.766 + 0.642i)T \)
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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

−27.80706827457175981423078862260, −26.99056839440424449832541182091, −25.97284571595595899617623417807, −25.20374618925515464180280802722, −23.952167317817681981617740577368, −22.56488279915448895318154086588, −22.00733987794153887878120636876, −20.7905205552777912798260238981, −19.64533700480777445118430969059, −19.1229621667589027442651082660, −18.24138508545394303429287194514, −17.0833665078723463591534235846, −15.91827194561947361524054315761, −15.25519016385390230137984964668, −13.589820230870101933206076463150, −12.32209350549541295026803171682, −11.59537098769460685309623594435, −10.74335565342269303909033873686, −9.23634791635753844966469479302, −8.65437648430274515986315552258, −7.31643712641007337628649920257, −6.24290697089455449316439109652, −4.110270202242952045970192087133, −3.24631080271775855901329169750, −1.74094477094079613372762949303, 0.06552717816347794022094071257, 1.28189077062816647073695779762, 3.56337924587490513933582263243, 4.76621752419841007713263344813, 6.43421051044888712339082117791, 7.17459403366291929119062951582, 8.409559303888583954826967023214, 9.20068714996803369165392273609, 10.59330628061527590135849831023, 11.38974949403927565118446737232, 12.882589494520771789756343559117, 14.10475687704662471406616761819, 15.20148187901375679888248352065, 16.21671211996984346594719606327, 16.74260574485205626807730072856, 17.94973944366467420828054054689, 19.02114746087441393596220960583, 20.00064398166574343579639854935, 20.37969826312007825597309599618, 22.433342090451351757608395208309, 23.17945352071579644224412423516, 24.05391154774240566379568974109, 24.94565746524285592418034591365, 26.01828123065789131390118516466, 26.87967326145031444311087494556

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