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Properties

Label 2-189-63.38-c3-0-9
Degree $2$
Conductor $189$
Sign $-0.941 - 0.337i$
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  + 3.06i·2-s − 1.37·4-s + (−2.75 + 4.76i)5-s + (8.35 + 16.5i)7-s + 20.2i·8-s + (−14.5 − 8.42i)10-s + (51.4 − 29.7i)11-s + (−12.7 + 7.35i)13-s + (−50.6 + 25.5i)14-s − 73.0·16-s + (−29.3 + 50.7i)17-s + (−66.1 + 38.2i)19-s + (3.77 − 6.53i)20-s + (90.9 + 157. i)22-s + (−22.5 − 13.0i)23-s + ⋯
L(s)  = 1  + 1.08i·2-s − 0.171·4-s + (−0.246 + 0.426i)5-s + (0.451 + 0.892i)7-s + 0.896i·8-s + (−0.461 − 0.266i)10-s + (1.41 − 0.814i)11-s + (−0.271 + 0.157i)13-s + (−0.965 + 0.488i)14-s − 1.14·16-s + (−0.418 + 0.724i)17-s + (−0.798 + 0.461i)19-s + (0.0421 − 0.0730i)20-s + (0.881 + 1.52i)22-s + (−0.204 − 0.118i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.337i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.941 - 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.941 - 0.337i$
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: \((2,\ 189,\ (\ :3/2),\ -0.941 - 0.337i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.299635 + 1.72433i\)
\(L(\frac12)\) \(\approx\) \(0.299635 + 1.72433i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-8.35 - 16.5i)T \)
good2 \( 1 - 3.06iT - 8T^{2} \)
5 \( 1 + (2.75 - 4.76i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-51.4 + 29.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (12.7 - 7.35i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (29.3 - 50.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (66.1 - 38.2i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (22.5 + 13.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (217. + 125. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 219. iT - 2.97e4T^{2} \)
37 \( 1 + (97.2 + 168. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-37.6 - 65.2i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-210. + 365. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 383.T + 1.03e5T^{2} \)
53 \( 1 + (-92.7 - 53.5i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 533.T + 2.05e5T^{2} \)
61 \( 1 + 41.2iT - 2.26e5T^{2} \)
67 \( 1 - 333.T + 3.00e5T^{2} \)
71 \( 1 - 500. iT - 3.57e5T^{2} \)
73 \( 1 + (-351. - 203. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 150.T + 4.93e5T^{2} \)
83 \( 1 + (-493. + 854. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-311. - 539. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-407. - 235. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.44487161974273805004145275534, −11.51670607349166548718699515389, −10.80889149547488999254367964133, −9.042273589530038197289125929546, −8.484276990794810290388605348035, −7.27183039782278412717841162980, −6.31673869625843755228140786008, −5.50852780329215417962274262510, −3.88973768108344882864290331219, −2.06920863705294074006470670969, 0.78532045183627094866412107617, 2.10378021225367334878868810756, 3.85513131233656086720751610671, 4.60789962050627175913921413291, 6.61570421446528301868691818374, 7.48218227635307055724771054348, 9.005137502525372286373287184535, 9.831720646953867892839129943015, 10.91056767816226693714973683638, 11.59920277195132148779367282010

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