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Properties

Label 2-189-63.11-c2-0-4
Degree $2$
Conductor $189$
Sign $0.876 - 0.481i$
Analytic cond. $5.14987$
Root an. cond. $2.26933$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.0767i·2-s + 3.99·4-s + (3.53 + 2.03i)5-s + (2.97 + 6.33i)7-s − 0.613i·8-s + (0.156 − 0.270i)10-s + (−10.7 + 6.23i)11-s + (−5.97 − 10.3i)13-s + (0.486 − 0.228i)14-s + 15.9·16-s + (14.2 + 8.24i)17-s + (−3.69 − 6.39i)19-s + (14.1 + 8.14i)20-s + (0.478 + 0.828i)22-s + (24.1 + 13.9i)23-s + ⋯
L(s)  = 1  − 0.0383i·2-s + 0.998·4-s + (0.706 + 0.407i)5-s + (0.424 + 0.905i)7-s − 0.0766i·8-s + (0.0156 − 0.0270i)10-s + (−0.980 + 0.566i)11-s + (−0.459 − 0.796i)13-s + (0.0347 − 0.0162i)14-s + 0.995·16-s + (0.840 + 0.485i)17-s + (−0.194 − 0.336i)19-s + (0.705 + 0.407i)20-s + (0.0217 + 0.0376i)22-s + (1.05 + 0.606i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.876 - 0.481i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ 0.876 - 0.481i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.99042 + 0.510604i\)
\(L(\frac12)\) \(\approx\) \(1.99042 + 0.510604i\)
\(L(2)\) 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 \( 1 + (-2.97 - 6.33i)T \)
good2 \( 1 + 0.0767iT - 4T^{2} \)
5 \( 1 + (-3.53 - 2.03i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (10.7 - 6.23i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (5.97 + 10.3i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-14.2 - 8.24i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (3.69 + 6.39i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-24.1 - 13.9i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (32.2 + 18.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 32.5T + 961T^{2} \)
37 \( 1 + (10.3 + 17.9i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-26.1 + 15.0i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (12.7 - 22.1i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + 79.0iT - 2.20e3T^{2} \)
53 \( 1 + (61.5 + 35.5i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + 43.1iT - 3.48e3T^{2} \)
61 \( 1 + 22.4T + 3.72e3T^{2} \)
67 \( 1 + 86.1T + 4.48e3T^{2} \)
71 \( 1 - 102. iT - 5.04e3T^{2} \)
73 \( 1 + (-0.403 + 0.699i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 27.0T + 6.24e3T^{2} \)
83 \( 1 + (36.4 + 21.0i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-36.1 + 20.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-6.66 + 11.5i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.38739783592775098759825905556, −11.39937545752043633430840514987, −10.44247043946582978251502993294, −9.712472687310791306242840035493, −8.171661801038417448953540186162, −7.29768713693517372210440723341, −5.99293908004062917102013408398, −5.22631491253381203083936149083, −2.95441573611057622816638488666, −2.01564039875637858544292093681, 1.43057306921296916799228581853, 2.97460248265508694093327384547, 4.81348934465124256330797429740, 5.93493428317488284130013896854, 7.15861191497770288658136138043, 7.967227812959571593073378348367, 9.393073876630499021954317197128, 10.46021277716331161160441322519, 11.12713200065015806065032305288, 12.23074195979068763934693060506

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