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Properties

Label 2-189-189.25-c1-0-0
Degree $2$
Conductor $189$
Sign $-0.828 + 0.559i$
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  + (−2.58 + 0.939i)2-s + (−1.71 + 0.215i)3-s + (4.25 − 3.56i)4-s + (0.335 + 0.122i)5-s + (4.23 − 2.17i)6-s + (−0.921 + 2.48i)7-s + (−4.87 + 8.44i)8-s + (2.90 − 0.739i)9-s − 0.980·10-s + (−1.67 + 0.609i)11-s + (−6.53 + 7.04i)12-s + (−0.143 − 0.811i)13-s + (0.0492 − 7.26i)14-s + (−0.602 − 0.137i)15-s + (2.72 − 15.4i)16-s − 1.85·17-s + ⋯
L(s)  = 1  + (−1.82 + 0.664i)2-s + (−0.992 + 0.124i)3-s + (2.12 − 1.78i)4-s + (0.149 + 0.0545i)5-s + (1.72 − 0.886i)6-s + (−0.348 + 0.937i)7-s + (−1.72 + 2.98i)8-s + (0.969 − 0.246i)9-s − 0.310·10-s + (−0.504 + 0.183i)11-s + (−1.88 + 2.03i)12-s + (−0.0397 − 0.225i)13-s + (0.0131 − 1.94i)14-s + (−0.155 − 0.0355i)15-s + (0.680 − 3.86i)16-s − 0.448·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.828 + 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.828 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.828 + 0.559i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.828 + 0.559i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0173289 - 0.0565930i\)
\(L(\frac12)\) \(\approx\) \(0.0173289 - 0.0565930i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 - 0.215i)T \)
7 \( 1 + (0.921 - 2.48i)T \)
good2 \( 1 + (2.58 - 0.939i)T + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (-0.335 - 0.122i)T + (3.83 + 3.21i)T^{2} \)
11 \( 1 + (1.67 - 0.609i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (0.143 + 0.811i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + 1.85T + 17T^{2} \)
19 \( 1 + 4.90T + 19T^{2} \)
23 \( 1 + (1.13 + 6.43i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (0.600 - 3.40i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (7.06 - 5.93i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (2.49 - 4.32i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.733 + 4.16i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-3.83 - 3.21i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (7.02 + 5.89i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-2.16 + 3.74i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.27 - 7.21i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-3.62 - 3.04i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (4.10 + 1.49i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-2.19 - 3.80i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.34 + 4.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.7 + 3.91i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (1.28 - 7.28i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + 5.36T + 89T^{2} \)
97 \( 1 + (-1.12 - 0.939i)T + (16.8 + 95.5i)T^{2} \)
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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

−12.74182595926646072917242352283, −11.81202268773991712122172896704, −10.68304271595405209724009432713, −10.22888097693062394003423209330, −9.113776959068370421886089716564, −8.284026898928334097731181101838, −6.92969815778392383602250296541, −6.22869223126316873417019612535, −5.22917643723695043665744034440, −2.13170945207246699444707873485, 0.098846429448679656345367448931, 1.84926461014979153226847747070, 3.86044769113770150058336146941, 6.06572496513605693187817762922, 7.17168589166484947115762088230, 7.87345072406714310384627818842, 9.346635007905856847908724366242, 10.02760130948973252239318028622, 10.99254285132498301108511102262, 11.37125198972017445278756409361

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