Properties

Label 1-189-189.31-r1-0-0
Degree $1$
Conductor $189$
Sign $-0.860 - 0.508i$
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  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)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 + (−0.766 + 0.642i)13-s + (0.173 − 0.984i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.173 + 0.984i)20-s + (−0.939 − 0.342i)22-s + (−0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)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 + (−0.766 + 0.642i)13-s + (0.173 − 0.984i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.173 + 0.984i)20-s + (−0.939 − 0.342i)22-s + (−0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.860 - 0.508i$
Analytic conductor: \(20.3108\)
Root analytic conductor: \(20.3108\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (1:\ ),\ -0.860 - 0.508i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04887912241 + 0.1786989596i\)
\(L(\frac12)\) \(\approx\) \(-0.04887912241 + 0.1786989596i\)
\(L(1)\) \(\approx\) \(0.5025490787 + 0.1786257510i\)
\(L(1)\) \(\approx\) \(0.5025490787 + 0.1786257510i\)

Euler product

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

−26.633305615237466015049003974035, −25.4752947156718764121044963590, −24.50850416914300251191983689418, −23.81450383428887087752503705936, −22.26773768157884791886214917763, −21.43418269323006485368651839863, −20.10448413860108828085510688996, −19.73732830541227977963632244591, −18.79128354946554646902544866485, −17.51524984584292279108407671178, −16.780899361090291078469767844223, −15.85796225159599303318936954853, −14.876218174297095469600784118323, −13.21624255735914175564748883509, −12.10831537707010698702905901796, −11.4912903984839302249198697620, −10.223646336062722715052427676459, −9.147713158308206573358090876001, −8.19323562645635564330420577576, −7.36693597150752251352644831027, −5.893983926282675044037350512119, −4.20002552845265951183936879816, −3.03180014992084587196877684457, −1.32037124701252386156452954171, −0.09423503673404324240906175459, 1.68677313598448413514393032652, 3.18982649941045065771118628127, 4.78672577081319802849816852717, 6.433000878933211984294320128618, 7.2306282175124495475768396402, 8.133853957991186666164732691587, 9.4809621131391698550823016184, 10.25213738679886457213034859869, 11.60785536475296637410032202865, 12.113968641678216462807094399236, 14.33165254319881034827561437937, 14.70057486187444120559801182563, 16.03020212652908015157706753441, 16.663973193392728596196058881989, 17.990672729010107490087664435139, 18.62254093671790123428507107980, 19.724603739010986105214027344941, 20.23463851031172624692479779867, 21.77160079275836217789633847255, 22.900982032886032552821005503111, 23.73082635498220545306423967172, 24.83477191187470656735278618784, 25.60172388648029753427837535764, 26.7762513216129769238343329988, 27.174987108327431988835979218529

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