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Properties

Label 2-189-63.59-c1-0-4
Degree $2$
Conductor $189$
Sign $0.805 - 0.592i$
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  + (1.5 + 0.866i)2-s + (0.5 + 0.866i)4-s + 3·5-s + (−2.5 + 0.866i)7-s − 1.73i·8-s + (4.5 + 2.59i)10-s + 1.73i·11-s + (−1.5 − 0.866i)13-s + (−4.5 − 0.866i)14-s + (2.49 − 4.33i)16-s + (−1.5 + 2.59i)17-s + (−4.5 + 2.59i)19-s + (1.5 + 2.59i)20-s + (−1.49 + 2.59i)22-s − 5.19i·23-s + ⋯
L(s)  = 1  + (1.06 + 0.612i)2-s + (0.250 + 0.433i)4-s + 1.34·5-s + (−0.944 + 0.327i)7-s − 0.612i·8-s + (1.42 + 0.821i)10-s + 0.522i·11-s + (−0.416 − 0.240i)13-s + (−1.20 − 0.231i)14-s + (0.624 − 1.08i)16-s + (−0.363 + 0.630i)17-s + (−1.03 + 0.596i)19-s + (0.335 + 0.580i)20-s + (−0.319 + 0.553i)22-s − 1.08i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.805 - 0.592i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.805 - 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93813 + 0.636643i\)
\(L(\frac12)\) \(\approx\) \(1.93813 + 0.636643i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 0.866i)T \)
good2 \( 1 + (-1.5 - 0.866i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 3T + 5T^{2} \)
11 \( 1 - 1.73iT - 11T^{2} \)
13 \( 1 + (1.5 + 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.5 - 2.59i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.19iT - 23T^{2} \)
29 \( 1 + (-4.5 + 2.59i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3 - 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.5 - 4.33i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12 - 6.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (4.5 + 2.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.5 - 12.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.5 - 0.866i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.80389300771747457324342231409, −12.38251671581157514297850003327, −10.37798055714551652126354418374, −9.873579597725457193422854875596, −8.715092273493901535443830730038, −6.92615712046326489284773316905, −6.20616212865813750100494825717, −5.42297074508985339728239871501, −4.09932339155468257746074422346, −2.41184879641827223592830948461, 2.21997194182799098016930422096, 3.39720133514273382413083312497, 4.85130786653000608853382043751, 5.89657219822988426669678908810, 6.89213041171976789131685994296, 8.708631301806503453496507094623, 9.703844744779360319034302167829, 10.61745912684713716323891137596, 11.67531767153783021823022308673, 12.76162064228188467154084977381

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