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Properties

Label 2-189-189.184-c1-0-7
Degree $2$
Conductor $189$
Sign $-0.927 - 0.372i$
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  + (0.475 + 2.69i)2-s + (1.62 + 0.612i)3-s + (−5.17 + 1.88i)4-s + (0.0114 − 0.0652i)5-s + (−0.881 + 4.66i)6-s + (2.53 + 0.763i)7-s + (−4.80 − 8.32i)8-s + (2.25 + 1.98i)9-s + 0.181·10-s + (−0.757 − 4.29i)11-s + (−9.54 − 0.116i)12-s + (−1.95 − 1.63i)13-s + (−0.854 + 7.19i)14-s + (0.0585 − 0.0986i)15-s + (11.7 − 9.85i)16-s − 4.47·17-s + ⋯
L(s)  = 1  + (0.336 + 1.90i)2-s + (0.935 + 0.353i)3-s + (−2.58 + 0.942i)4-s + (0.00514 − 0.0291i)5-s + (−0.359 + 1.90i)6-s + (0.957 + 0.288i)7-s + (−1.70 − 2.94i)8-s + (0.750 + 0.661i)9-s + 0.0573·10-s + (−0.228 − 1.29i)11-s + (−2.75 − 0.0336i)12-s + (−0.541 − 0.454i)13-s + (−0.228 + 1.92i)14-s + (0.0151 − 0.0254i)15-s + (2.93 − 2.46i)16-s − 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.927 - 0.372i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.927 - 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.306334 + 1.58341i\)
\(L(\frac12)\) \(\approx\) \(0.306334 + 1.58341i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.62 - 0.612i)T \)
7 \( 1 + (-2.53 - 0.763i)T \)
good2 \( 1 + (-0.475 - 2.69i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (-0.0114 + 0.0652i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (0.757 + 4.29i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (1.95 + 1.63i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 + (-0.757 - 0.635i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (1.49 - 1.25i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (3.91 - 1.42i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-0.415 - 0.720i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.97 + 3.33i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (4.96 + 1.80i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-4.03 - 1.47i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-0.866 - 1.50i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.05 + 7.60i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.27 + 0.464i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.766 + 4.34i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-1.92 + 3.33i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.05 + 10.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.16 - 6.60i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (5.61 - 4.71i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + 15.0T + 89T^{2} \)
97 \( 1 + (-3.32 - 1.21i)T + (74.3 + 62.3i)T^{2} \)
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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

−13.63093105477609863152593092321, −12.58908557997688084792245782321, −10.88427295588717190740524375968, −9.328498494486046158159044518783, −8.629383104345430707872177827313, −7.933087259522377725091494262837, −7.00043881315544649783018642278, −5.49743564110092981258553826268, −4.74660149831439921884070249211, −3.33304925137481186683400046281, 1.64323546089286183238035557796, 2.60909455978020382306027859992, 4.14975401036457192998709351298, 4.90428648032933728158599629409, 7.23566090353682914537375276661, 8.520933689332821272248503790865, 9.425367352168397060195019540181, 10.23325939694735811432531903672, 11.29907772494668238088361657210, 12.17007251149652595533168976896

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