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Properties

Label 1-189-189.32-r1-0-0
Degree $1$
Conductor $189$
Sign $0.941 + 0.337i$
Analytic cond. $20.3108$
Root an. cond. $20.3108$
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 + (−0.173 − 0.984i)5-s + (0.5 − 0.866i)8-s + (−0.5 + 0.866i)10-s + (−0.173 + 0.984i)11-s + (0.173 + 0.984i)13-s + (−0.939 + 0.342i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.939 − 0.342i)20-s + (0.766 − 0.642i)22-s + (−0.766 + 0.642i)23-s + (−0.939 + 0.342i)25-s + (0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.173 − 0.984i)5-s + (0.5 − 0.866i)8-s + (−0.5 + 0.866i)10-s + (−0.173 + 0.984i)11-s + (0.173 + 0.984i)13-s + (−0.939 + 0.342i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.939 − 0.342i)20-s + (0.766 − 0.642i)22-s + (−0.766 + 0.642i)23-s + (−0.939 + 0.342i)25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.941 + 0.337i$
Analytic conductor: \(20.3108\)
Root analytic conductor: \(20.3108\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (1:\ ),\ 0.941 + 0.337i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8984178710 + 0.1560090262i\)
\(L(\frac12)\) \(\approx\) \(0.8984178710 + 0.1560090262i\)
\(L(1)\) \(\approx\) \(0.6944002463 - 0.1410721785i\)
\(L(1)\) \(\approx\) \(0.6944002463 - 0.1410721785i\)

Euler product

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

−26.77139102606832574427660451435, −25.99164734642067606541773531027, −25.14425095744856113512391631897, −24.09214191478380556653004583559, −23.17841400420743697825579368419, −22.314935949961170775426638573253, −21.02438877993943225964501465852, −19.75622902553947252611052917503, −18.86551780846468738230222158724, −18.273357761224420884299697191642, −17.150962744482670338957566866077, −16.19424985132105634979002473970, −15.12181122161869857712702266741, −14.48399481684061156288127976706, −13.28655385901334937492401136611, −11.61397249859643203520069122458, −10.57788234190731170380586749836, −9.96720619780300325197742396541, −8.30292798216287588561582080900, −7.811627169888012504376889956207, −6.32327940818782456713463387224, −5.74843289844291875943965882563, −3.8031459895476981363273827620, −2.31121167886276885746072610205, −0.48758202934590137174230671827, 1.08253931817707624126714703112, 2.30313029489564890038498786624, 3.94588962840337290268393108316, 4.97642295716018278664693645653, 6.87845804156833683631580191565, 7.9002939337331786362810531271, 9.077076898550647970687924710144, 9.66590565024466965803960340606, 11.058308565914016242973451970776, 12.063947395071706050956084421154, 12.76501402819312192815379770892, 13.95437977651408838553779778518, 15.6495074883290643061391050001, 16.40139021856291234931059500474, 17.37895571338513447114698511625, 18.23574003079988860443662213402, 19.42325602526021970660363831382, 20.15322847231783137189653347836, 20.9851786996640069116129241420, 21.81248580459277094858779743879, 23.17700769169630708170085055126, 24.15912801286737221463700061721, 25.365005232928986489255956456285, 25.96817675350446348444636513995, 27.24188704202642638379313136555

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