Properties

Label 2-189-189.47-c1-0-12
Degree $2$
Conductor $189$
Sign $0.665 - 0.746i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.87 + 0.330i)2-s + (0.583 + 1.63i)3-s + (1.52 + 0.553i)4-s + (−0.848 − 0.308i)5-s + (0.555 + 3.24i)6-s + (2.48 − 0.894i)7-s + (−0.629 − 0.363i)8-s + (−2.31 + 1.90i)9-s + (−1.48 − 0.858i)10-s + (−1.19 − 3.28i)11-s + (−0.0147 + 2.80i)12-s + (−1.35 + 3.72i)13-s + (4.95 − 0.853i)14-s + (0.00822 − 1.56i)15-s + (−3.53 − 2.96i)16-s + (−0.233 + 0.405i)17-s + ⋯
L(s)  = 1  + (1.32 + 0.233i)2-s + (0.337 + 0.941i)3-s + (0.760 + 0.276i)4-s + (−0.379 − 0.138i)5-s + (0.226 + 1.32i)6-s + (0.941 − 0.338i)7-s + (−0.222 − 0.128i)8-s + (−0.772 + 0.634i)9-s + (−0.470 − 0.271i)10-s + (−0.360 − 0.990i)11-s + (−0.00425 + 0.808i)12-s + (−0.375 + 1.03i)13-s + (1.32 − 0.228i)14-s + (0.00212 − 0.403i)15-s + (−0.884 − 0.742i)16-s + (−0.0567 + 0.0982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.665 - 0.746i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.665 - 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04254 + 0.914994i\)
\(L(\frac12)\) \(\approx\) \(2.04254 + 0.914994i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.583 - 1.63i)T \)
7 \( 1 + (-2.48 + 0.894i)T \)
good2 \( 1 + (-1.87 - 0.330i)T + (1.87 + 0.684i)T^{2} \)
5 \( 1 + (0.848 + 0.308i)T + (3.83 + 3.21i)T^{2} \)
11 \( 1 + (1.19 + 3.28i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (1.35 - 3.72i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.233 - 0.405i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.14 + 1.81i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.44 + 0.607i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (1.79 + 4.92i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (2.38 - 6.54i)T + (-23.7 - 19.9i)T^{2} \)
37 \( 1 - 9.85T + 37T^{2} \)
41 \( 1 + (-2.71 - 0.987i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.54 - 8.76i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (3.13 - 1.14i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (8.75 - 5.05i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.48 - 2.08i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.0209 - 0.0574i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.78 - 10.0i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (2.19 - 1.26i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.37iT - 73T^{2} \)
79 \( 1 + (-1.64 + 9.30i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (10.6 - 3.86i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (1.51 + 2.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.02 - 0.886i)T + (91.1 + 33.1i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.98377722528098393933359484660, −11.52216542018660756740909721675, −11.24324381381499326617283158355, −9.718379670183925820220762445982, −8.656385184019373709389890451378, −7.53423317503839545008139098489, −5.96511048829373325979799988787, −4.82724702482613196075096145106, −4.21830620834330837790633898637, −2.92999914974445558358520277675, 2.13405859673227140884382404607, 3.39103301457937071945087216692, 4.90978622787150760789711869180, 5.79170612284559118556525396294, 7.33211332509488762400074144965, 8.002334980662612255616928127362, 9.366786377171804740490226717998, 11.06423370733848590682802038159, 11.81530023803919797338574623561, 12.62681037422796846333179126269

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