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Properties

Label 2-189-7.2-c3-0-6
Degree $2$
Conductor $189$
Sign $-0.390 + 0.920i$
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  + (−1.64 + 2.85i)2-s + (−1.42 − 2.46i)4-s + (−10.8 + 18.8i)5-s + (1.08 + 18.4i)7-s − 16.9·8-s + (−35.7 − 61.9i)10-s + (20.9 + 36.3i)11-s + 46.0·13-s + (−54.5 − 27.3i)14-s + (39.3 − 68.1i)16-s + (−1.00 − 1.74i)17-s + (−36.6 + 63.4i)19-s + 61.7·20-s − 138.·22-s + (12.0 − 20.9i)23-s + ⋯
L(s)  = 1  + (−0.582 + 1.00i)2-s + (−0.177 − 0.307i)4-s + (−0.971 + 1.68i)5-s + (0.0587 + 0.998i)7-s − 0.750·8-s + (−1.13 − 1.95i)10-s + (0.575 + 0.996i)11-s + 0.982·13-s + (−1.04 − 0.521i)14-s + (0.614 − 1.06i)16-s + (−0.0143 − 0.0249i)17-s + (−0.442 + 0.765i)19-s + 0.690·20-s − 1.33·22-s + (0.109 − 0.189i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.390 + 0.920i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.390 + 0.920i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.473036 - 0.714706i\)
\(L(\frac12)\) \(\approx\) \(0.473036 - 0.714706i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.08 - 18.4i)T \)
good2 \( 1 + (1.64 - 2.85i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (10.8 - 18.8i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-20.9 - 36.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 46.0T + 2.19e3T^{2} \)
17 \( 1 + (1.00 + 1.74i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (36.6 - 63.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-12.0 + 20.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 90.7T + 2.43e4T^{2} \)
31 \( 1 + (26.6 + 46.0i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (33.9 - 58.7i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 341.T + 6.89e4T^{2} \)
43 \( 1 - 509.T + 7.95e4T^{2} \)
47 \( 1 + (19.2 - 33.3i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (194. + 337. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (225. + 390. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (112. - 195. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-386. - 668. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 962.T + 3.57e5T^{2} \)
73 \( 1 + (-526. - 912. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (16.7 - 29.0i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 446.T + 5.71e5T^{2} \)
89 \( 1 + (244. - 424. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 460.T + 9.12e5T^{2} \)
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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

−12.50156371498003426183814060290, −11.74118144847326328156664238712, −10.83670055441241586002439381129, −9.602070183351491963083685606801, −8.447638826174167158924358938957, −7.63394452691974640066024732026, −6.69510037953115278636857381763, −6.01707208465473661608137347334, −3.92897868005878531710629513532, −2.62565952314023300130763645709, 0.53717759926566335932561418132, 1.22421567805574216186788462594, 3.51180788564657472370923776728, 4.46007006849155139650222886666, 6.04672004892247760558777528801, 7.73791995455149475669257034367, 8.779424817691257467898941700150, 9.208407024993750800777556320342, 10.77825016796798550169995450953, 11.25898807366775336867792372209

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