Properties

Label 2-189-189.184-c1-0-15
Degree $2$
Conductor $189$
Sign $0.571 + 0.820i$
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  + (−0.199 − 1.13i)2-s + (1.60 + 0.647i)3-s + (0.640 − 0.233i)4-s + (−0.0295 + 0.167i)5-s + (0.411 − 1.94i)6-s + (−1.82 − 1.91i)7-s + (−1.53 − 2.66i)8-s + (2.16 + 2.07i)9-s + 0.195·10-s + (−0.0250 − 0.141i)11-s + (1.17 + 0.0400i)12-s + (4.82 + 4.04i)13-s + (−1.79 + 2.44i)14-s + (−0.155 + 0.250i)15-s + (−1.66 + 1.39i)16-s − 3.38·17-s + ⋯
L(s)  = 1  + (−0.140 − 0.799i)2-s + (0.927 + 0.373i)3-s + (0.320 − 0.116i)4-s + (−0.0132 + 0.0749i)5-s + (0.167 − 0.794i)6-s + (−0.691 − 0.722i)7-s + (−0.544 − 0.942i)8-s + (0.720 + 0.693i)9-s + 0.0618·10-s + (−0.00754 − 0.0427i)11-s + (0.340 + 0.0115i)12-s + (1.33 + 1.12i)13-s + (−0.480 + 0.654i)14-s + (−0.0402 + 0.0645i)15-s + (−0.415 + 0.349i)16-s − 0.819·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.571 + 0.820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.571 + 0.820i$
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: \((2,\ 189,\ (\ :1/2),\ 0.571 + 0.820i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35115 - 0.705433i\)
\(L(\frac12)\) \(\approx\) \(1.35115 - 0.705433i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.60 - 0.647i)T \)
7 \( 1 + (1.82 + 1.91i)T \)
good2 \( 1 + (0.199 + 1.13i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (0.0295 - 0.167i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (0.0250 + 0.141i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-4.82 - 4.04i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + 3.38T + 17T^{2} \)
19 \( 1 + 1.68T + 19T^{2} \)
23 \( 1 + (3.64 + 3.05i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (7.64 - 6.41i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (2.01 - 0.734i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (2.27 + 3.94i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.05 - 3.39i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (4.02 + 1.46i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-9.05 - 3.29i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (1.26 + 2.18i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.23 - 1.03i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (11.8 + 4.30i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.674 - 3.82i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-4.00 + 6.92i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.47 + 7.74i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.903 - 5.12i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-6.82 + 5.72i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 - 9.26T + 89T^{2} \)
97 \( 1 + (4.89 + 1.78i)T + (74.3 + 62.3i)T^{2} \)
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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

−12.50819872490332990588177672929, −10.93283405060788065014870791751, −10.73164785170819082380024612831, −9.424707955202133367501060424729, −8.837352160623161850467302472735, −7.23990237240271889116227609511, −6.35979188429364184707097544110, −4.17555583473143033511743861876, −3.32388070976051545658128248993, −1.82431363197097107970886603315, 2.34578830750910171949965465059, 3.60566630732752415474246201188, 5.76106110535459125283017551784, 6.53875775208109855910064564938, 7.71348135198688318959398676511, 8.521773386974970234314144745389, 9.269054569649459801206908541445, 10.69890673770652996920313889606, 11.94544345887463688189503446391, 12.93054341993508782612281866936

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