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Properties

Label 1-189-189.11-r1-0-0
Degree $1$
Conductor $189$
Sign $0.838 + 0.545i$
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 + (0.939 − 0.342i)5-s + (0.5 + 0.866i)8-s + 10-s + (0.939 + 0.342i)11-s + (0.173 − 0.984i)13-s + (0.173 + 0.984i)16-s − 17-s + 19-s + (0.939 + 0.342i)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  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (0.939 − 0.342i)5-s + (0.5 + 0.866i)8-s + 10-s + (0.939 + 0.342i)11-s + (0.173 − 0.984i)13-s + (0.173 + 0.984i)16-s − 17-s + 19-s + (0.939 + 0.342i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.838 + 0.545i$
Analytic conductor: \(20.3108\)
Root analytic conductor: \(20.3108\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (1:\ ),\ 0.838 + 0.545i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.073806264 + 1.208090659i\)
\(L(\frac12)\) \(\approx\) \(4.073806264 + 1.208090659i\)
\(L(1)\) \(\approx\) \(2.276430491 + 0.4713191014i\)
\(L(1)\) \(\approx\) \(2.276430491 + 0.4713191014i\)

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 \( 1 + (0.939 - 0.342i)T \)
11 \( 1 + (0.939 + 0.342i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (-0.173 - 0.984i)T \)
31 \( 1 + (0.766 + 0.642i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.766 + 0.642i)T \)
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 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (-0.173 - 0.984i)T \)
89 \( 1 - T \)
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

−26.66156545663691885295669594117, −25.70936207380047009716722088803, −24.56939466342966073107831732969, −24.1416890666940732296383843808, −22.54419021509195850824205928576, −22.20679620505055538370719953912, −21.21583775026906392213994590113, −20.339518082934219279510153716995, −19.2488519667780180003359459569, −18.26379351543756298163799225464, −16.97156677593912403957133877320, −15.98650909570459533533498570348, −14.65447744542322546540403853893, −14.00093962360047677390264642585, −13.21706469462480564705465775966, −11.92199042476939769332215267907, −11.07956373919889326761161355295, −9.94846340473634205909894335131, −8.9210450754882498097665463774, −6.8859549635921946851593208345, −6.30909852812901274471322665071, −5.04744437899843573622601305292, −3.82701171495580657059793288368, −2.50437300965475288267599012157, −1.3659538328348631479618204123, 1.49007315977817606743597906794, 2.85900798032414656368058624979, 4.24599390969715503494612762850, 5.41140909444608629898658925508, 6.2635820596941109013695369289, 7.42748287927659515550285162461, 8.78810643885416604166686383123, 9.98628062416778646594237863737, 11.326576321097451008273408041466, 12.38137053747960489139919505152, 13.376127994003567990241194966920, 14.03902599979355315672673819325, 15.19765968245316402585289768259, 16.09971967488167399380531854802, 17.3518662307220557551558328764, 17.77978711937919539561699795436, 19.70060407692062383066220719974, 20.44764907660217778546684308475, 21.42623416139393773789270361800, 22.29874070066875477224131252485, 22.97301302590036799321728630011, 24.40496231774486023479188505270, 24.85135854616174545840475269990, 25.67671363925628864334050269773, 26.74377689874451513146451483454

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