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Properties

Label 1-189-189.139-r1-0-0
Degree $1$
Conductor $189$
Sign $-0.802 + 0.597i$
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 + (0.939 + 0.342i)5-s + (−0.5 + 0.866i)8-s + (0.5 + 0.866i)10-s + (−0.939 + 0.342i)11-s + (−0.766 + 0.642i)13-s + (−0.939 + 0.342i)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 + (0.173 + 0.984i)23-s + (0.766 + 0.642i)25-s − 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)8-s + (0.5 + 0.866i)10-s + (−0.939 + 0.342i)11-s + (−0.766 + 0.642i)13-s + (−0.939 + 0.342i)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 + (0.173 + 0.984i)23-s + (0.766 + 0.642i)25-s − 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.802 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.802 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.802 + 0.597i$
Analytic conductor: \(20.3108\)
Root analytic conductor: \(20.3108\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (1:\ ),\ -0.802 + 0.597i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8678857429 + 2.619131747i\)
\(L(\frac12)\) \(\approx\) \(0.8678857429 + 2.619131747i\)
\(L(1)\) \(\approx\) \(1.326247619 + 1.062267302i\)
\(L(1)\) \(\approx\) \(1.326247619 + 1.062267302i\)

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 \( 1 + (0.939 + 0.342i)T \)
11 \( 1 + (-0.939 + 0.342i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.766 + 0.642i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + T \)
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 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
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

−26.63581596113214525048732096249, −25.12776241197959875152032640731, −24.68135124071201216177815612900, −23.502366020432597859235142235405, −22.56575733678765101096618119394, −21.66858417563327752077537311680, −20.80132510278491416291023667780, −20.20637363709215239386155952268, −18.81277997034423221128725416814, −18.06559816120822451316135066705, −16.71453004700555232461336221616, −15.634237841262707292996861816627, −14.4014933953686455331811332601, −13.655880141766017468656824759364, −12.71146096831359739260734383660, −11.8637744458841674849450967521, −10.304503394831624016062723415565, −9.981196668386768450205431899513, −8.45287572284655188139494354494, −6.82878604464609022700260950045, −5.47054487836303653805885032227, −4.98595888923830555252983728798, −3.209963103854021565961795604833, −2.21773943662014186643221246311, −0.71031859301773654746380843506, 2.05918375724898207231453494799, 3.19022822014912239213420512719, 4.78298553711151728982559387135, 5.632686644409760683378029927223, 6.792859440061526944360731160514, 7.702107858802127220447666924928, 9.13610630870793161822089556548, 10.28580019474807024514044529692, 11.60306898386671808097122946220, 12.81754565257487946475642396291, 13.56582998572688716322726209276, 14.5472072738594513688216308859, 15.36692334346429322808350310236, 16.56831471848171812461446924783, 17.45764722029150273337732282039, 18.25311264675069454237463443425, 19.71111147841718388588618245525, 21.08936894081735759338807340955, 21.574046683304702845365056058833, 22.47768349466612878072222999552, 23.5909719814103588781586019876, 24.29818077696062271370438397973, 25.46187196706735267918305224180, 26.01323684855597604701065241522, 26.84298997876825533990276843697

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