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Properties

Label 2-189-63.25-c3-0-8
Degree $2$
Conductor $189$
Sign $0.980 + 0.195i$
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  − 4.21·2-s + 9.76·4-s + (−3.91 + 6.77i)5-s + (−15.9 − 9.49i)7-s − 7.43·8-s + (16.4 − 28.5i)10-s + (−6.42 − 11.1i)11-s + (6.74 + 11.6i)13-s + (67.0 + 40.0i)14-s − 46.7·16-s + (−35.5 + 61.4i)17-s + (−46.2 − 80.0i)19-s + (−38.1 + 66.1i)20-s + (27.0 + 46.8i)22-s + (1.97 − 3.41i)23-s + ⋯
L(s)  = 1  − 1.49·2-s + 1.22·4-s + (−0.349 + 0.606i)5-s + (−0.858 − 0.512i)7-s − 0.328·8-s + (0.521 − 0.903i)10-s + (−0.176 − 0.304i)11-s + (0.143 + 0.249i)13-s + (1.27 + 0.764i)14-s − 0.730·16-s + (−0.506 + 0.877i)17-s + (−0.558 − 0.966i)19-s + (−0.427 + 0.739i)20-s + (0.262 + 0.454i)22-s + (0.0178 − 0.0309i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.980 + 0.195i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.980 + 0.195i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.546630 - 0.0540476i\)
\(L(\frac12)\) \(\approx\) \(0.546630 - 0.0540476i\)
\(L(\frac{5}{2})\) 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 \( 1 + (15.9 + 9.49i)T \)
good2 \( 1 + 4.21T + 8T^{2} \)
5 \( 1 + (3.91 - 6.77i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (6.42 + 11.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-6.74 - 11.6i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (35.5 - 61.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (46.2 + 80.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-1.97 + 3.41i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-90.3 + 156. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 270.T + 2.97e4T^{2} \)
37 \( 1 + (-110. - 190. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-33.6 - 58.2i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-237. + 411. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 512.T + 1.03e5T^{2} \)
53 \( 1 + (-238. + 413. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 717.T + 2.05e5T^{2} \)
61 \( 1 - 376.T + 2.26e5T^{2} \)
67 \( 1 - 694.T + 3.00e5T^{2} \)
71 \( 1 - 230.T + 3.57e5T^{2} \)
73 \( 1 + (258. - 446. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + 943.T + 4.93e5T^{2} \)
83 \( 1 + (-84.4 + 146. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-149. - 258. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-389. + 674. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.62399793064738780629699349393, −10.73041456725548331497856326254, −10.13742304606455604098645629035, −9.060630859963162609798165724831, −8.175552403169143959118385873717, −7.06715919729735247699392899328, −6.36580420188512833846956633352, −4.18808599673220062320430392235, −2.59458503834065716607451833788, −0.63253266076664205165872458152, 0.77854608856545783156332712215, 2.58002568529352659173871483176, 4.50366296755008649088923593409, 6.14684215836505736970917816146, 7.31930547804926342385024609838, 8.369094489311829240599833934187, 9.071407763122037535727802145259, 9.938947248080064493190650852641, 10.82658894591320243607353823382, 12.06634032708039260168422425807

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