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Properties

Label 2-189-189.101-c1-0-3
Degree $2$
Conductor $189$
Sign $-0.547 - 0.836i$
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  + (−0.720 + 0.858i)2-s + (0.864 − 1.50i)3-s + (0.129 + 0.731i)4-s + (−3.14 + 2.63i)5-s + (0.666 + 1.82i)6-s + (0.864 + 2.50i)7-s + (−2.66 − 1.53i)8-s + (−1.50 − 2.59i)9-s − 4.59i·10-s + (−0.647 + 0.771i)11-s + (1.21 + 0.438i)12-s + (−1.20 + 3.32i)13-s + (−2.77 − 1.05i)14-s + (1.24 + 6.99i)15-s + (1.84 − 0.670i)16-s + 6.15·17-s + ⋯
L(s)  = 1  + (−0.509 + 0.607i)2-s + (0.498 − 0.866i)3-s + (0.0645 + 0.365i)4-s + (−1.40 + 1.17i)5-s + (0.272 + 0.744i)6-s + (0.326 + 0.945i)7-s + (−0.941 − 0.543i)8-s + (−0.502 − 0.864i)9-s − 1.45i·10-s + (−0.195 + 0.232i)11-s + (0.349 + 0.126i)12-s + (−0.335 + 0.921i)13-s + (−0.740 − 0.283i)14-s + (0.320 + 1.80i)15-s + (0.460 − 0.167i)16-s + 1.49·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 - 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.547 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.547 - 0.836i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.547 - 0.836i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.352930 + 0.652927i\)
\(L(\frac12)\) \(\approx\) \(0.352930 + 0.652927i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.864 + 1.50i)T \)
7 \( 1 + (-0.864 - 2.50i)T \)
good2 \( 1 + (0.720 - 0.858i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (3.14 - 2.63i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (0.647 - 0.771i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (1.20 - 3.32i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 6.15T + 17T^{2} \)
19 \( 1 - 2.40iT - 19T^{2} \)
23 \( 1 + (0.696 - 1.91i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.707 - 1.94i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-0.819 + 0.144i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-1.98 + 3.43i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.37 - 3.41i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.78 + 10.1i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (1.26 - 7.16i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (3.60 + 2.08i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.60 - 0.949i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-3.70 - 0.652i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (1.49 - 1.25i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-5.21 + 3.00i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.29 - 3.63i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.62 - 4.72i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (7.39 - 2.69i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 + (5.98 + 1.05i)T + (91.1 + 33.1i)T^{2} \)
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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

−12.46501344320453596407943918486, −12.08791901465408189597889919239, −11.27638143950723792801105465026, −9.581583014504268988364229870760, −8.426525207880758305528346907851, −7.70179298773886130827067865202, −7.17938976440348254825512051840, −6.06469161878928060092221516325, −3.73786309860093471619970556083, −2.66389061221571072762093461123, 0.75754813847547919791296783365, 3.19112222626083430818546755688, 4.43493564768675815041512266577, 5.38488090844250184390472521428, 7.75605578084767173970132836303, 8.237598314195602470679778523696, 9.365009982141856418170167027511, 10.26903065850480388801883138225, 11.08630648309791453978892716405, 11.93039215200220171273052780115

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