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Properties

Label 2-189-63.47-c1-0-2
Degree $2$
Conductor $189$
Sign $0.994 - 0.100i$
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  + (−1.81 + 1.04i)2-s + (1.19 − 2.07i)4-s + 2.08·5-s + (−0.879 − 2.49i)7-s + 0.819i·8-s + (−3.79 + 2.18i)10-s − 3.22i·11-s + (2.68 − 1.55i)13-s + (4.21 + 3.60i)14-s + (1.53 + 2.65i)16-s + (0.816 + 1.41i)17-s + (4.79 + 2.76i)19-s + (2.49 − 4.32i)20-s + (3.38 + 5.85i)22-s − 1.16i·23-s + ⋯
L(s)  = 1  + (−1.28 + 0.740i)2-s + (0.597 − 1.03i)4-s + 0.934·5-s + (−0.332 − 0.943i)7-s + 0.289i·8-s + (−1.19 + 0.692i)10-s − 0.973i·11-s + (0.745 − 0.430i)13-s + (1.12 + 0.964i)14-s + (0.383 + 0.663i)16-s + (0.197 + 0.342i)17-s + (1.09 + 0.634i)19-s + (0.558 − 0.967i)20-s + (0.721 + 1.24i)22-s − 0.242i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.994 - 0.100i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.994 - 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.714027 + 0.0359102i\)
\(L(\frac12)\) \(\approx\) \(0.714027 + 0.0359102i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.879 + 2.49i)T \)
good2 \( 1 + (1.81 - 1.04i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 2.08T + 5T^{2} \)
11 \( 1 + 3.22iT - 11T^{2} \)
13 \( 1 + (-2.68 + 1.55i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.816 - 1.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.79 - 2.76i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.16iT - 23T^{2} \)
29 \( 1 + (-7.05 - 4.07i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.16 + 2.98i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.82 + 4.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.35 - 2.34i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.974 - 1.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.06 + 7.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.27 - 3.04i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.98 + 3.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.15 - 2.39i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.336 + 0.583i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.01iT - 71T^{2} \)
73 \( 1 + (2.96 - 1.71i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.07 - 12.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.54 - 2.67i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.45 + 4.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.07 - 1.20i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.79577385998027653833223072361, −11.08877832986891244482295909236, −10.25864513105794485365177340733, −9.579077480861566462520030876349, −8.521298962299444881063489269920, −7.62846234820133264122015574550, −6.46998169340173996409497303694, −5.68819451932157532817349686165, −3.54475387748830543146082721901, −1.12281363211009344201097969591, 1.68298034256384422298002434118, 2.87875163801396666864982086374, 5.13613775594495937281790311806, 6.41042925293841625912823651350, 7.78307684289318666506586597424, 9.052086938808689185036216884268, 9.492248438363936639612991100250, 10.28129725581984707676929630400, 11.48303105888396197492852469791, 12.17180228565148129379330705557

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