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Properties

Label 1-189-189.167-r0-0-0
Degree $1$
Conductor $189$
Sign $0.993 - 0.116i$
Analytic cond. $0.877712$
Root an. cond. $0.877712$
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)8-s + (0.5 + 0.866i)10-s + (−0.766 − 0.642i)11-s + (−0.173 − 0.984i)13-s + (0.766 + 0.642i)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 + (0.939 + 0.342i)23-s + (0.173 − 0.984i)25-s + 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)8-s + (0.5 + 0.866i)10-s + (−0.766 − 0.642i)11-s + (−0.173 − 0.984i)13-s + (0.766 + 0.642i)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 + (0.939 + 0.342i)23-s + (0.173 − 0.984i)25-s + 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.993 - 0.116i$
Analytic conductor: \(0.877712\)
Root analytic conductor: \(0.877712\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (0:\ ),\ 0.993 - 0.116i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9873231926 - 0.05750502751i\)
\(L(\frac12)\) \(\approx\) \(0.9873231926 - 0.05750502751i\)
\(L(1)\) \(\approx\) \(0.9429080949 + 0.1386648350i\)
\(L(1)\) \(\approx\) \(0.9429080949 + 0.1386648350i\)

Euler product

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

−27.08223410001186230834996572362, −26.32175443080145247623571233807, −25.63253133886591233414072484435, −24.210978130620066939544961286602, −22.94166150854861940930409333696, −22.301409957602027427482240327331, −21.15125602711172400106060308380, −20.75807872961860934774487100063, −19.26530852579463934685999916307, −18.66049985604548825557438302426, −17.64791140321469033126875499708, −16.903928491613080986697142455308, −15.22741048633829458355106181910, −14.13015141173955162557331231616, −13.29998888638489685347968067537, −12.26230100129941744983206111782, −11.09808301792177567471104047628, −10.19193507089421224040678514784, −9.44641557116075281992880201594, −8.143262498620192400280920230114, −6.77959768595566269216772614876, −5.34541952379582365190757799569, −4.0671363106879611849435267383, −2.65037549756192176903769751827, −1.74324745517421815812791231326, 0.88367590088089953290643812499, 2.9333725354627882760901542618, 4.89241844601960945133531227381, 5.41394381712105064947255021838, 6.68734105678395328053039042710, 7.89595654174748832230792130557, 8.90999565702488941457029495320, 9.770186097202768315777067815387, 10.97456590244632919868908612596, 12.8209730720151400366139746900, 13.382040929359869044278711906431, 14.39337144907906387374584964836, 15.69993214874766766360571365626, 16.270765322055479045437842612038, 17.57500460440501881472897918549, 17.931736819195036532206394928594, 19.24182745432379049556596094601, 20.43660584921112655649840631789, 21.534366905889890053664302906427, 22.47432012937404560636009886726, 23.53390588690904062751336911817, 24.54799558245867320623143760363, 25.01223218576266418837140584419, 26.08654427941433517371734801934, 26.91537578736309977820669999761

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