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Properties

Label 2-189-63.4-c1-0-3
Degree $2$
Conductor $189$
Sign $0.990 + 0.137i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.335 − 0.580i)2-s + (0.775 + 1.34i)4-s − 1.42·5-s + (2.21 − 1.44i)7-s + 2.38·8-s + (−0.477 + 0.827i)10-s + 4.93·11-s + (−1.37 + 2.38i)13-s + (−0.0972 − 1.77i)14-s + (−0.752 + 1.30i)16-s + (−0.559 + 0.969i)17-s + (−2.00 − 3.47i)19-s + (−1.10 − 1.91i)20-s + (1.65 − 2.86i)22-s − 5.43·23-s + ⋯
L(s)  = 1  + (0.236 − 0.410i)2-s + (0.387 + 0.671i)4-s − 0.637·5-s + (0.837 − 0.546i)7-s + 0.841·8-s + (−0.151 + 0.261i)10-s + 1.48·11-s + (−0.381 + 0.661i)13-s + (−0.0259 − 0.473i)14-s + (−0.188 + 0.326i)16-s + (−0.135 + 0.235i)17-s + (−0.460 − 0.797i)19-s + (−0.247 − 0.427i)20-s + (0.352 − 0.610i)22-s − 1.13·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.990 + 0.137i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.990 + 0.137i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45302 - 0.100397i\)
\(L(\frac12)\) \(\approx\) \(1.45302 - 0.100397i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 + 1.44i)T \)
good2 \( 1 + (-0.335 + 0.580i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 1.42T + 5T^{2} \)
11 \( 1 - 4.93T + 11T^{2} \)
13 \( 1 + (1.37 - 2.38i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.559 - 0.969i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.00 + 3.47i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.43T + 23T^{2} \)
29 \( 1 + (3.40 + 5.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.25 + 2.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.709 - 1.22i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.124 - 0.215i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.498 + 0.863i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.73 - 8.20i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.410 + 0.710i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.29 + 5.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.0376 - 0.0651i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.29 - 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.0804T + 71T^{2} \)
73 \( 1 + (-5.34 + 9.25i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.922 + 1.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.23 - 12.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.76 + 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.70 - 4.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.24400796842095881608527071557, −11.55502058757626056835621082045, −11.08821418975611442722316294710, −9.620542536950171617667037722537, −8.323610017551720014268161959016, −7.49823361529581214137383593537, −6.48159196924410953952473878357, −4.40549554801095930869821352679, −3.87016066667740645455603453220, −1.94638829101051503003834538493, 1.77838196427970389813576563797, 3.93961028741123333841713152085, 5.20525700153184168522558104371, 6.24505654445241614328796997405, 7.39889691867851877220442134240, 8.380144807653560922742159585145, 9.621118011495299452086524648560, 10.78082555282579423191193940281, 11.66716237578248115515005765479, 12.34609032800038762376669053010

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