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Properties

Label 2-189-63.23-c2-0-10
Degree $2$
Conductor $189$
Sign $0.614 + 0.788i$
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  + 3.79i·2-s − 10.3·4-s + (−0.282 + 0.163i)5-s + (−2.88 − 6.37i)7-s − 24.1i·8-s + (−0.618 − 1.07i)10-s + (−13.1 − 7.57i)11-s + (−3.96 + 6.86i)13-s + (24.1 − 10.9i)14-s + 49.9·16-s + (−6.90 + 3.98i)17-s + (2.86 − 4.96i)19-s + (2.92 − 1.69i)20-s + (28.6 − 49.7i)22-s + (−1.79 + 1.03i)23-s + ⋯
L(s)  = 1  + 1.89i·2-s − 2.59·4-s + (−0.0564 + 0.0326i)5-s + (−0.411 − 0.911i)7-s − 3.01i·8-s + (−0.0618 − 0.107i)10-s + (−1.19 − 0.688i)11-s + (−0.305 + 0.528i)13-s + (1.72 − 0.780i)14-s + 3.12·16-s + (−0.406 + 0.234i)17-s + (0.150 − 0.261i)19-s + (0.146 − 0.0845i)20-s + (1.30 − 2.25i)22-s + (−0.0781 + 0.0451i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 + 0.788i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.614 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.614 + 0.788i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ 0.614 + 0.788i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0954928 - 0.0466342i\)
\(L(\frac12)\) \(\approx\) \(0.0954928 - 0.0466342i\)
\(L(2)\) 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 \( 1 + (2.88 + 6.37i)T \)
good2 \( 1 - 3.79iT - 4T^{2} \)
5 \( 1 + (0.282 - 0.163i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (13.1 + 7.57i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (3.96 - 6.86i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (6.90 - 3.98i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-2.86 + 4.96i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (1.79 - 1.03i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-5.71 + 3.30i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + 50.5T + 961T^{2} \)
37 \( 1 + (-5.56 + 9.63i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-12.8 - 7.43i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-3.67 - 6.37i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + 66.9iT - 2.20e3T^{2} \)
53 \( 1 + (22.0 - 12.7i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + 39.7iT - 3.48e3T^{2} \)
61 \( 1 + 74.3T + 3.72e3T^{2} \)
67 \( 1 - 42.9T + 4.48e3T^{2} \)
71 \( 1 - 72.5iT - 5.04e3T^{2} \)
73 \( 1 + (45.8 + 79.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 - 23.0T + 6.24e3T^{2} \)
83 \( 1 + (97.9 - 56.5i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-63.4 - 36.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (57.2 + 99.0i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.84166695278409583349687903140, −10.99769625147545357557414766288, −9.836366620791730703838911325937, −8.866926403508191970641016471316, −7.75239985630336122129480425555, −7.12229029512445063120672829160, −6.01374003240029036730539187486, −4.97161351135911537173387171949, −3.71403334135595679198914876399, −0.06008798791478636144535591755, 2.12698383124973857915018224786, 3.06607008148088342425643510900, 4.57617451798766109211903445741, 5.63491757666200308824238943404, 7.79094752964335960283767419200, 8.929627732532966885183983263415, 9.799697487916856359656457055032, 10.54715638592966867630690273385, 11.53574258194178443262300129354, 12.59132885770325588392746191861

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