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Properties

Label 2-189-63.25-c3-0-10
Degree $2$
Conductor $189$
Sign $0.330 + 0.943i$
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  − 0.983·2-s − 7.03·4-s + (−9.35 + 16.2i)5-s + (−18.4 − 0.890i)7-s + 14.7·8-s + (9.20 − 15.9i)10-s + (11.7 + 20.3i)11-s + (−23.8 − 41.3i)13-s + (18.1 + 0.875i)14-s + 41.7·16-s + (47.7 − 82.7i)17-s + (28.4 + 49.3i)19-s + (65.8 − 113. i)20-s + (−11.5 − 20.0i)22-s + (16.3 − 28.2i)23-s + ⋯
L(s)  = 1  − 0.347·2-s − 0.879·4-s + (−0.836 + 1.44i)5-s + (−0.998 − 0.0480i)7-s + 0.653·8-s + (0.291 − 0.504i)10-s + (0.322 + 0.557i)11-s + (−0.509 − 0.882i)13-s + (0.347 + 0.0167i)14-s + 0.651·16-s + (0.681 − 1.18i)17-s + (0.343 + 0.595i)19-s + (0.735 − 1.27i)20-s + (−0.111 − 0.193i)22-s + (0.147 − 0.256i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.330 + 0.943i$
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: \((2,\ 189,\ (\ :3/2),\ 0.330 + 0.943i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.313579 - 0.222562i\)
\(L(\frac12)\) \(\approx\) \(0.313579 - 0.222562i\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.4 + 0.890i)T \)
good2 \( 1 + 0.983T + 8T^{2} \)
5 \( 1 + (9.35 - 16.2i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-11.7 - 20.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (23.8 + 41.3i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-47.7 + 82.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-28.4 - 49.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-16.3 + 28.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (81.3 - 140. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 40.4T + 2.97e4T^{2} \)
37 \( 1 + (127. + 221. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-35.9 - 62.2i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-237. + 411. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 264.T + 1.03e5T^{2} \)
53 \( 1 + (13.1 - 22.8i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 39.1T + 2.05e5T^{2} \)
61 \( 1 - 421.T + 2.26e5T^{2} \)
67 \( 1 + 369.T + 3.00e5T^{2} \)
71 \( 1 + 685.T + 3.57e5T^{2} \)
73 \( 1 + (-113. + 196. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + 686.T + 4.93e5T^{2} \)
83 \( 1 + (-49.0 + 85.0i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (529. + 916. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (706. - 1.22e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.98202385536895857390978668691, −10.61435870792560048069836813734, −10.03237764318821043023395623053, −9.054645771410453241581805068087, −7.54033453102478370523059049962, −7.14827890346797508863235950512, −5.53474491992916270171967440516, −3.91098768491971191580788103650, −2.98198621022316000974977514011, −0.25303109415718840417573456438, 1.03420856945028638751143678701, 3.68563791769115297251868611527, 4.56437968751536420651339961517, 5.82359671682417707939041165917, 7.47586145932304362618642628562, 8.521452615655575819059404757613, 9.163246235698089976664562718718, 9.956398923626411714260599938678, 11.49920067619962546784916662708, 12.46420438103444861835824139080

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