Properties

Label 2-189-189.167-c1-0-8
Degree $2$
Conductor $189$
Sign $-0.303 - 0.952i$
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  + (1.22 + 1.46i)2-s + (1.33 + 1.10i)3-s + (−0.283 + 1.60i)4-s + (−3.99 + 1.45i)5-s + (0.0292 + 3.30i)6-s + (1.59 − 2.10i)7-s + (0.603 − 0.348i)8-s + (0.573 + 2.94i)9-s + (−7.02 − 4.05i)10-s + (−0.402 + 1.10i)11-s + (−2.15 + 1.83i)12-s + (1.01 − 1.21i)13-s + (5.03 − 0.249i)14-s + (−6.94 − 2.45i)15-s + (4.32 + 1.57i)16-s + (2.27 − 3.94i)17-s + ⋯
L(s)  = 1  + (0.866 + 1.03i)2-s + (0.771 + 0.635i)3-s + (−0.141 + 0.804i)4-s + (−1.78 + 0.650i)5-s + (0.0119 + 1.34i)6-s + (0.604 − 0.796i)7-s + (0.213 − 0.123i)8-s + (0.191 + 0.981i)9-s + (−2.22 − 1.28i)10-s + (−0.121 + 0.333i)11-s + (−0.621 + 0.530i)12-s + (0.282 − 0.336i)13-s + (1.34 − 0.0666i)14-s + (−1.79 − 0.635i)15-s + (1.08 + 0.393i)16-s + (0.552 − 0.957i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.303 - 0.952i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.303 - 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10971 + 1.51871i\)
\(L(\frac12)\) \(\approx\) \(1.10971 + 1.51871i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.33 - 1.10i)T \)
7 \( 1 + (-1.59 + 2.10i)T \)
good2 \( 1 + (-1.22 - 1.46i)T + (-0.347 + 1.96i)T^{2} \)
5 \( 1 + (3.99 - 1.45i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (0.402 - 1.10i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (-1.01 + 1.21i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-2.27 + 3.94i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.81 + 1.04i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.65 + 0.819i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (3.22 + 3.83i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (5.36 + 0.945i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (5.20 - 9.01i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.42 - 3.70i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (5.34 + 1.94i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.355 + 2.01i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 2.28iT - 53T^{2} \)
59 \( 1 + (-0.199 + 0.0727i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (2.42 - 0.427i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (0.424 + 0.355i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-7.56 - 4.36i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.58 - 0.914i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.929 + 0.780i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-2.49 + 2.09i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (1.62 + 2.80i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.36 - 12.0i)T + (-74.3 - 62.3i)T^{2} \)
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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

−13.31731816941132258462669298123, −11.90516284923056288870715931158, −10.94500714965075797590675860269, −10.00399305433135096522826393608, −8.184608334238660100410730632255, −7.68665926735633065654295037479, −6.96682911978912517777183283278, −5.06178725651527888163669512712, −4.15922312059482750427588690554, −3.38085991656531914480247730980, 1.67933287272620718217122924448, 3.38730558105379173962326403107, 4.06714654317615188278253248204, 5.48440023412257202741736312460, 7.52739011835361444310329007897, 8.161575001009586476758464944697, 9.031623154381269045948522441423, 10.88111290279710678044467532887, 11.68800551375083331157516811360, 12.38956903146412255144112224519

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