Properties

Label 2-189-189.79-c1-0-12
Degree $2$
Conductor $189$
Sign $0.952 + 0.304i$
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  + (0.757 + 0.635i)2-s + (0.532 − 1.64i)3-s + (−0.177 − 1.00i)4-s + (0.570 + 3.23i)5-s + (1.45 − 0.909i)6-s + (1.49 − 2.18i)7-s + (1.49 − 2.58i)8-s + (−2.43 − 1.75i)9-s + (−1.62 + 2.81i)10-s + (−0.598 + 3.39i)11-s + (−1.75 − 0.243i)12-s + (−0.176 − 1.00i)13-s + (2.51 − 0.707i)14-s + (5.63 + 0.783i)15-s + (0.853 − 0.310i)16-s + (0.335 − 0.581i)17-s + ⋯
L(s)  = 1  + (0.535 + 0.449i)2-s + (0.307 − 0.951i)3-s + (−0.0888 − 0.503i)4-s + (0.255 + 1.44i)5-s + (0.592 − 0.371i)6-s + (0.563 − 0.826i)7-s + (0.528 − 0.914i)8-s + (−0.810 − 0.585i)9-s + (−0.513 + 0.889i)10-s + (−0.180 + 1.02i)11-s + (−0.506 − 0.0704i)12-s + (−0.0490 − 0.278i)13-s + (0.672 − 0.189i)14-s + (1.45 + 0.202i)15-s + (0.213 − 0.0776i)16-s + (0.0813 − 0.140i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.952 + 0.304i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.952 + 0.304i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67854 - 0.262161i\)
\(L(\frac12)\) \(\approx\) \(1.67854 - 0.262161i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.532 + 1.64i)T \)
7 \( 1 + (-1.49 + 2.18i)T \)
good2 \( 1 + (-0.757 - 0.635i)T + (0.347 + 1.96i)T^{2} \)
5 \( 1 + (-0.570 - 3.23i)T + (-4.69 + 1.71i)T^{2} \)
11 \( 1 + (0.598 - 3.39i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (0.176 + 1.00i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.335 + 0.581i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.06 - 1.83i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.39 - 3.68i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-0.678 + 3.84i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-1.79 - 10.1i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + 9.11T + 37T^{2} \)
41 \( 1 + (-0.857 - 4.86i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (2.16 + 1.81i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (0.0859 - 0.487i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (4.04 + 7.01i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.08 - 3.30i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-1.52 + 8.62i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-7.46 + 6.26i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-4.72 - 8.18i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.1T + 73T^{2} \)
79 \( 1 + (0.158 + 0.132i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-2.37 + 13.4i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (5.99 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.577 - 0.484i)T + (16.8 + 95.5i)T^{2} \)
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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

−12.81080632204827638695518704332, −11.60756454140272258372643038908, −10.45375051868837702505072981419, −9.843397786582362657489294834972, −7.963735681191496773162046484450, −7.07941556721446574886961496117, −6.54234972022433218868598929823, −5.21333096628294111104356532396, −3.55271693185185964005413685077, −1.82166909864156719883604668659, 2.36807645355940692615897526647, 3.89294814796904618695827043631, 4.91800313920947179565892132386, 5.62563472960903298879469988496, 8.176750951109806197092689245304, 8.560353476804143290510554977080, 9.442449654638855494261285622125, 10.88916906067705279933991950211, 11.76962286386300999466938339900, 12.53400247914926072973438799009

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