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Properties

Label 2-189-63.11-c2-0-13
Degree $2$
Conductor $189$
Sign $-0.988 - 0.149i$
Analytic cond. $5.14987$
Root an. cond. $2.26933$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.28i·2-s + 2.34·4-s + (−6.82 − 3.93i)5-s + (−6.26 + 3.13i)7-s − 8.15i·8-s + (−5.06 + 8.76i)10-s + (−10.6 + 6.16i)11-s + (−3.39 − 5.87i)13-s + (4.02 + 8.04i)14-s − 1.09·16-s + (6.19 + 3.57i)17-s + (−15.4 − 26.8i)19-s + (−16.0 − 9.24i)20-s + (7.93 + 13.7i)22-s + (−1.87 − 1.08i)23-s + ⋯
L(s)  = 1  − 0.642i·2-s + 0.586·4-s + (−1.36 − 0.787i)5-s + (−0.894 + 0.447i)7-s − 1.01i·8-s + (−0.506 + 0.876i)10-s + (−0.971 + 0.560i)11-s + (−0.261 − 0.452i)13-s + (0.287 + 0.574i)14-s − 0.0685·16-s + (0.364 + 0.210i)17-s + (−0.815 − 1.41i)19-s + (−0.800 − 0.462i)20-s + (0.360 + 0.624i)22-s + (−0.0815 − 0.0470i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.988 - 0.149i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ -0.988 - 0.149i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0433248 + 0.576268i\)
\(L(\frac12)\) \(\approx\) \(0.0433248 + 0.576268i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (6.26 - 3.13i)T \)
good2 \( 1 + 1.28iT - 4T^{2} \)
5 \( 1 + (6.82 + 3.93i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (10.6 - 6.16i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (3.39 + 5.87i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-6.19 - 3.57i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (15.4 + 26.8i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (1.87 + 1.08i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-8.97 - 5.17i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 20.1T + 961T^{2} \)
37 \( 1 + (-2.46 - 4.26i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (28.0 - 16.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-25.5 + 44.1i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + 21.4iT - 2.20e3T^{2} \)
53 \( 1 + (-51.0 - 29.4i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + 103. iT - 3.48e3T^{2} \)
61 \( 1 - 8.93T + 3.72e3T^{2} \)
67 \( 1 - 90.6T + 4.48e3T^{2} \)
71 \( 1 + 2.12iT - 5.04e3T^{2} \)
73 \( 1 + (-35.2 + 61.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 144.T + 6.24e3T^{2} \)
83 \( 1 + (-13.6 - 7.85i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-4.08 + 2.35i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (47.0 - 81.4i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.96169690886177221083044988214, −10.96785414094637639206450757762, −10.03591816296672778415870744076, −8.834561421611391749892816308257, −7.72463630470258900232074020952, −6.76423124656190950738797446295, −5.14162412586675130224307090846, −3.77002294691974906816677026136, −2.54888134069343392462122483939, −0.31109557868860138579978870577, 2.80906815478554013887342065057, 3.94443399785466627369332667224, 5.77305107145623938779377972679, 6.84249215423539997977535389903, 7.57921826920889475158855120531, 8.361119384682838838992334533647, 10.14914194425554488732662345390, 10.89860475744378288342922781751, 11.78629040700939602518521902240, 12.72687871396406219879543199583

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