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Properties

Label 2-189-63.38-c3-0-13
Degree $2$
Conductor $189$
Sign $0.868 - 0.495i$
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  + 1.83i·2-s + 4.62·4-s + (−0.207 + 0.359i)5-s + (5.26 − 17.7i)7-s + 23.1i·8-s + (−0.659 − 0.381i)10-s + (42.9 − 24.8i)11-s + (−1.43 + 0.828i)13-s + (32.6 + 9.67i)14-s − 5.67·16-s + (−20.6 + 35.7i)17-s + (130. − 75.5i)19-s + (−0.957 + 1.65i)20-s + (45.5 + 78.9i)22-s + (−114. − 65.8i)23-s + ⋯
L(s)  = 1  + 0.649i·2-s + 0.577·4-s + (−0.0185 + 0.0321i)5-s + (0.284 − 0.958i)7-s + 1.02i·8-s + (−0.0208 − 0.0120i)10-s + (1.17 − 0.679i)11-s + (−0.0306 + 0.0176i)13-s + (0.623 + 0.184i)14-s − 0.0886·16-s + (−0.294 + 0.510i)17-s + (1.57 − 0.911i)19-s + (−0.0107 + 0.0185i)20-s + (0.441 + 0.765i)22-s + (−1.03 − 0.596i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.868 - 0.495i$
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: \((2,\ 189,\ (\ :3/2),\ 0.868 - 0.495i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.19195 + 0.581594i\)
\(L(\frac12)\) \(\approx\) \(2.19195 + 0.581594i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.26 + 17.7i)T \)
good2 \( 1 - 1.83iT - 8T^{2} \)
5 \( 1 + (0.207 - 0.359i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-42.9 + 24.8i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (1.43 - 0.828i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (20.6 - 35.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-130. + 75.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (114. + 65.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-136. - 78.8i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 18.4iT - 2.97e4T^{2} \)
37 \( 1 + (-173. - 301. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (16.9 + 29.4i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-29.5 + 51.2i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 216.T + 1.03e5T^{2} \)
53 \( 1 + (-174. - 100. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 299.T + 2.05e5T^{2} \)
61 \( 1 - 80.4iT - 2.26e5T^{2} \)
67 \( 1 + 257.T + 3.00e5T^{2} \)
71 \( 1 + 1.03e3iT - 3.57e5T^{2} \)
73 \( 1 + (711. + 410. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 105.T + 4.93e5T^{2} \)
83 \( 1 + (-245. + 424. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (754. + 1.30e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (1.08e3 + 627. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.92437961373208983492790422488, −11.29859551263807167426001246264, −10.33777717230378360767232098400, −8.991677423096940543600913188102, −7.900673181382111113511364827875, −6.96258532259003566319665923825, −6.15201369051503401410222171849, −4.72445289721734442406425204837, −3.22711382722991109251497611675, −1.27647702334405943340166660944, 1.42076317640280417001891880855, 2.67338279269690300966168461708, 4.10569892364007279989660371938, 5.69588557278720827749564559704, 6.79349443016359326903076003740, 7.959230803915280082713846782269, 9.355313387567847740123824937931, 9.991406458068291900989344423979, 11.37949594706655937725006329836, 11.95261678282192955987200241774

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