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Properties

Label 2-189-63.38-c3-0-2
Degree $2$
Conductor $189$
Sign $-0.236 + 0.971i$
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  + 3.46i·2-s − 4.01·4-s + (−9.02 + 15.6i)5-s + (−18.1 − 3.61i)7-s + 13.8i·8-s + (−54.1 − 31.2i)10-s + (10.2 − 5.89i)11-s + (45.9 − 26.5i)13-s + (12.5 − 62.9i)14-s − 79.9·16-s + (−5.93 + 10.2i)17-s + (−12.1 + 7.04i)19-s + (36.2 − 62.7i)20-s + (20.4 + 35.4i)22-s + (−139. − 80.6i)23-s + ⋯
L(s)  = 1  + 1.22i·2-s − 0.501·4-s + (−0.807 + 1.39i)5-s + (−0.980 − 0.194i)7-s + 0.610i·8-s + (−1.71 − 0.989i)10-s + (0.279 − 0.161i)11-s + (0.980 − 0.566i)13-s + (0.238 − 1.20i)14-s − 1.24·16-s + (−0.0847 + 0.146i)17-s + (−0.147 + 0.0850i)19-s + (0.405 − 0.701i)20-s + (0.198 + 0.343i)22-s + (−1.26 − 0.731i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.236 + 0.971i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.236 + 0.971i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.236 + 0.971i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.377921 - 0.481105i\)
\(L(\frac12)\) \(\approx\) \(0.377921 - 0.481105i\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.1 + 3.61i)T \)
good2 \( 1 - 3.46iT - 8T^{2} \)
5 \( 1 + (9.02 - 15.6i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-10.2 + 5.89i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-45.9 + 26.5i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (5.93 - 10.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (12.1 - 7.04i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (139. + 80.6i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (131. + 75.8i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 90.1iT - 2.97e4T^{2} \)
37 \( 1 + (-185. - 320. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-129. - 223. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (173. - 300. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 71.6T + 1.03e5T^{2} \)
53 \( 1 + (143. + 82.9i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 575.T + 2.05e5T^{2} \)
61 \( 1 + 496. iT - 2.26e5T^{2} \)
67 \( 1 + 700.T + 3.00e5T^{2} \)
71 \( 1 - 1.13e3iT - 3.57e5T^{2} \)
73 \( 1 + (-281. - 162. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 523.T + 4.93e5T^{2} \)
83 \( 1 + (410. - 710. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-181. - 313. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-273. - 157. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−13.07055902638439603022696754817, −11.63305373210944016807772229064, −10.89977312335887248516387465986, −9.819315914602125459015101647329, −8.291016861415644568595725239304, −7.59351427964684721389542287101, −6.42050693178947648048896782584, −6.15029089619839914329782801807, −4.07236099531453056049756642747, −2.88941094045856224435559985367, 0.26451489507726913676057001151, 1.69494762792305549124111126014, 3.53352552622825838168671338055, 4.27620909200782708794994247961, 5.93955922232399207498891654389, 7.38627083807167436068851028099, 8.894386179296082905858878922243, 9.319449385642389637375028371283, 10.59273002145142369789081262479, 11.66487386920292302304316842510

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