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Properties

Label 2-189-63.38-c3-0-1
Degree $2$
Conductor $189$
Sign $-0.252 - 0.967i$
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  − 2.91i·2-s − 0.495·4-s + (−8.60 + 14.9i)5-s + (−4.20 − 18.0i)7-s − 21.8i·8-s + (43.4 + 25.0i)10-s + (−30.9 + 17.8i)11-s + (8.99 − 5.19i)13-s + (−52.5 + 12.2i)14-s − 67.7·16-s + (−64.1 + 111. i)17-s + (−84.7 + 48.9i)19-s + (4.26 − 7.38i)20-s + (52.0 + 90.0i)22-s + (−22.1 − 12.7i)23-s + ⋯
L(s)  = 1  − 1.03i·2-s − 0.0619·4-s + (−0.769 + 1.33i)5-s + (−0.227 − 0.973i)7-s − 0.966i·8-s + (1.37 + 0.793i)10-s + (−0.847 + 0.489i)11-s + (0.191 − 0.110i)13-s + (−1.00 + 0.234i)14-s − 1.05·16-s + (−0.915 + 1.58i)17-s + (−1.02 + 0.590i)19-s + (0.0476 − 0.0825i)20-s + (0.504 + 0.873i)22-s + (−0.200 − 0.115i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.252 - 0.967i$
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: \((2,\ 189,\ (\ :3/2),\ -0.252 - 0.967i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.161764 + 0.209490i\)
\(L(\frac12)\) \(\approx\) \(0.161764 + 0.209490i\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.20 + 18.0i)T \)
good2 \( 1 + 2.91iT - 8T^{2} \)
5 \( 1 + (8.60 - 14.9i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (30.9 - 17.8i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-8.99 + 5.19i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (64.1 - 111. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (84.7 - 48.9i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (22.1 + 12.7i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-14.1 - 8.16i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 185. iT - 2.97e4T^{2} \)
37 \( 1 + (0.462 + 0.801i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (231. + 401. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (43.2 - 74.9i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 287.T + 1.03e5T^{2} \)
53 \( 1 + (334. + 193. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 90.4T + 2.05e5T^{2} \)
61 \( 1 + 139. iT - 2.26e5T^{2} \)
67 \( 1 - 303.T + 3.00e5T^{2} \)
71 \( 1 + 351. iT - 3.57e5T^{2} \)
73 \( 1 + (-761. - 439. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 14.2T + 4.93e5T^{2} \)
83 \( 1 + (125. - 217. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (505. + 875. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (1.12e3 + 649. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.33781407089634730816621269387, −11.02610000651233056655471320919, −10.64558197377749659811573056662, −10.11707181884817206570030759484, −8.283885584237780983337810045121, −7.14697147812365566613385058003, −6.44914614967348934180767961403, −4.16012359466569325218102501555, −3.39384867080722723137323790391, −2.04255102076624162636456388341, 0.10669011777848808852239708978, 2.52660650548357319871967839304, 4.61627455193211776275682020624, 5.41091374303155843330544706956, 6.57661484918128912190246481060, 7.86315363167273866806136267731, 8.553893207816570903838963036084, 9.301780025125396918473249752905, 11.16495145953107666959184694485, 11.81357408456405462886773206195

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