Properties

Label 2-189-63.16-c3-0-21
Degree $2$
Conductor $189$
Sign $-0.961 + 0.276i$
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  + (0.667 + 1.15i)2-s + (3.10 − 5.38i)4-s − 9.00·5-s + (−14.2 + 11.8i)7-s + 18.9·8-s + (−6.01 − 10.4i)10-s − 29.3·11-s + (−21.1 − 36.6i)13-s + (−23.2 − 8.53i)14-s + (−12.1 − 21.1i)16-s + (2.56 + 4.45i)17-s + (−71.2 + 123. i)19-s + (−27.9 + 48.4i)20-s + (−19.6 − 33.9i)22-s − 178.·23-s + ⋯
L(s)  = 1  + (0.236 + 0.408i)2-s + (0.388 − 0.672i)4-s − 0.805·5-s + (−0.767 + 0.640i)7-s + 0.839·8-s + (−0.190 − 0.329i)10-s − 0.804·11-s + (−0.451 − 0.782i)13-s + (−0.443 − 0.162i)14-s + (−0.190 − 0.329i)16-s + (0.0366 + 0.0634i)17-s + (−0.860 + 1.49i)19-s + (−0.312 + 0.541i)20-s + (−0.189 − 0.329i)22-s − 1.61·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.276i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.961 + 0.276i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.961 + 0.276i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0107179 - 0.0761081i\)
\(L(\frac12)\) \(\approx\) \(0.0107179 - 0.0761081i\)
\(L(\frac{5}{2})\) 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 \( 1 + (14.2 - 11.8i)T \)
good2 \( 1 + (-0.667 - 1.15i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + 9.00T + 125T^{2} \)
11 \( 1 + 29.3T + 1.33e3T^{2} \)
13 \( 1 + (21.1 + 36.6i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-2.56 - 4.45i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (71.2 - 123. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 178.T + 1.21e4T^{2} \)
29 \( 1 + (-109. + 189. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (73.9 - 128. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (21.2 - 36.8i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (83.7 + 145. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-121. + 210. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (38.2 + 66.2i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-181. - 314. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (60.7 - 105. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-321. - 556. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-81.4 + 141. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 833.T + 3.57e5T^{2} \)
73 \( 1 + (62.4 + 108. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (421. + 729. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (566. - 982. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-248. + 429. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (128. - 223. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.87114219156747943442862776495, −10.43835449731535308354626971287, −9.998884808457759584854170368584, −8.334301357186418864819666545412, −7.53535935682650092351204963705, −6.20818249264990422507439671205, −5.46771499247057968135582983349, −3.94660828790602406150478756485, −2.31218545420804853089894996977, −0.02847357616155964156196676473, 2.40700787148082735624464145810, 3.68257210179443301734478044839, 4.60412988968369922670501921626, 6.58805503881453651667673814081, 7.41593833020518472277904843814, 8.341131622671281210649403399653, 9.767061897478896142587042142490, 10.84708026185082502868971804974, 11.59354986411049906921736015047, 12.56789879356542255303250203456

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