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Properties

Label 2-189-63.58-c3-0-17
Degree $2$
Conductor $189$
Sign $0.909 - 0.414i$
Analytic cond. $11.1513$
Root an. cond. $3.33936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.07·2-s + 17.8·4-s + (9.23 + 15.9i)5-s + (−8.82 − 16.2i)7-s + 49.8·8-s + (46.9 + 81.2i)10-s + (8.59 − 14.8i)11-s + (−27.2 + 47.2i)13-s + (−44.8 − 82.7i)14-s + 110.·16-s + (7.61 + 13.1i)17-s + (43.7 − 75.7i)19-s + (164. + 284. i)20-s + (43.6 − 75.5i)22-s + (−48.3 − 83.7i)23-s + ⋯
L(s)  = 1  + 1.79·2-s + 2.22·4-s + (0.826 + 1.43i)5-s + (−0.476 − 0.879i)7-s + 2.20·8-s + (1.48 + 2.56i)10-s + (0.235 − 0.407i)11-s + (−0.582 + 1.00i)13-s + (−0.855 − 1.57i)14-s + 1.72·16-s + (0.108 + 0.188i)17-s + (0.527 − 0.914i)19-s + (1.83 + 3.18i)20-s + (0.422 − 0.732i)22-s + (−0.438 − 0.759i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.414i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.909 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.909 - 0.414i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.909 - 0.414i)\)

Particular Values

\(L(2)\) \(\approx\) \(5.07965 + 1.10318i\)
\(L(\frac12)\) \(\approx\) \(5.07965 + 1.10318i\)
\(L(\frac{5}{2})\) 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 \( 1 + (8.82 + 16.2i)T \)
good2 \( 1 - 5.07T + 8T^{2} \)
5 \( 1 + (-9.23 - 15.9i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-8.59 + 14.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (27.2 - 47.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-7.61 - 13.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-43.7 + 75.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (48.3 + 83.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (57.3 + 99.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 50.0T + 2.97e4T^{2} \)
37 \( 1 + (-3.58 + 6.20i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-87.4 + 151. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (77.4 + 134. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 117.T + 1.03e5T^{2} \)
53 \( 1 + (-1.27 - 2.20i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 412.T + 2.05e5T^{2} \)
61 \( 1 + 711.T + 2.26e5T^{2} \)
67 \( 1 - 37.7T + 3.00e5T^{2} \)
71 \( 1 - 290.T + 3.57e5T^{2} \)
73 \( 1 + (-209. - 363. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 1.07e3T + 4.93e5T^{2} \)
83 \( 1 + (-634. - 1.09e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (5.74 - 9.95i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-935. - 1.61e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.36494811401693598755413615521, −11.30882137652696039384727777747, −10.61919340427351578438068007140, −9.585838090942757984200544819409, −7.28489881128715692159870875195, −6.67647360560838830477663661499, −5.89990435162317700896248326800, −4.42172146576045920627962238292, −3.29960079226718169664668538833, −2.26662005162999309404988465370, 1.75586654574212283353088874772, 3.16036673065929446755640746285, 4.69502261255952412178830827436, 5.49236949644965985642383342143, 6.10351015093998237322244254889, 7.70838986769296574016792432980, 9.173405492014961902863602940358, 10.10662015179264259277982511500, 11.75716124935973069922810589564, 12.43935485722716676974584860244

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