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Properties

Label 2-189-189.5-c1-0-8
Degree $2$
Conductor $189$
Sign $0.301 - 0.953i$
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  + (0.284 − 0.0501i)2-s + (1.43 + 0.965i)3-s + (−1.80 + 0.655i)4-s + (−0.499 + 2.83i)5-s + (0.457 + 0.202i)6-s + (0.989 − 2.45i)7-s + (−0.979 + 0.565i)8-s + (1.13 + 2.77i)9-s + 0.830i·10-s + (2.61 − 0.460i)11-s + (−3.22 − 0.796i)12-s + (−4.11 + 4.90i)13-s + (0.158 − 0.747i)14-s + (−3.45 + 3.59i)15-s + (2.68 − 2.25i)16-s + 2.47·17-s + ⋯
L(s)  = 1  + (0.201 − 0.0354i)2-s + (0.830 + 0.557i)3-s + (−0.900 + 0.327i)4-s + (−0.223 + 1.26i)5-s + (0.186 + 0.0826i)6-s + (0.374 − 0.927i)7-s + (−0.346 + 0.199i)8-s + (0.378 + 0.925i)9-s + 0.262i·10-s + (0.787 − 0.138i)11-s + (−0.930 − 0.229i)12-s + (−1.14 + 1.36i)13-s + (0.0423 − 0.199i)14-s + (−0.891 + 0.927i)15-s + (0.671 − 0.563i)16-s + 0.600·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.301 - 0.953i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.301 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10574 + 0.809988i\)
\(L(\frac12)\) \(\approx\) \(1.10574 + 0.809988i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.43 - 0.965i)T \)
7 \( 1 + (-0.989 + 2.45i)T \)
good2 \( 1 + (-0.284 + 0.0501i)T + (1.87 - 0.684i)T^{2} \)
5 \( 1 + (0.499 - 2.83i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-2.61 + 0.460i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (4.11 - 4.90i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 - 2.47T + 17T^{2} \)
19 \( 1 + 4.65iT - 19T^{2} \)
23 \( 1 + (-5.08 + 6.05i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (0.285 + 0.340i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (2.04 + 5.60i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (1.80 + 3.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.96 - 5.00i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (2.51 + 0.915i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.418 - 0.152i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (2.28 - 1.32i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.11 - 3.45i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.27 - 3.50i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.361 + 2.04i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (12.0 + 6.97i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.49 - 0.863i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.786 - 4.46i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (11.7 - 9.85i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + (-2.49 + 6.85i)T + (-74.3 - 62.3i)T^{2} \)
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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

−13.05557488288397594363146821815, −11.65032036320476008751604537992, −10.73170708567127144824599006583, −9.682293145046689559837161585562, −8.926421932109087456972197365122, −7.60800564828121387752692868151, −6.88246865323250037382615669132, −4.71113595909824766135814752753, −3.98336699971698750264301825947, −2.76697913384372173875417674353, 1.31749936516393275974920267163, 3.40129650188163466895714432360, 4.88781045882087688700300374185, 5.69282907793278840573990782598, 7.57930415444015685338965962002, 8.484502577639717545323438174659, 9.137328121324319887059940342525, 9.956217148611612040758166264635, 12.00625750898210305741786660776, 12.52194349993525201531828199808

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