Properties

Label 2-189-63.58-c1-0-1
Degree $2$
Conductor $189$
Sign $0.888 + 0.458i$
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  − 2-s − 4-s + (−0.5 − 0.866i)5-s + (2 + 1.73i)7-s + 3·8-s + (0.5 + 0.866i)10-s + (2.5 − 4.33i)11-s + (2.5 − 4.33i)13-s + (−2 − 1.73i)14-s − 16-s + (1.5 + 2.59i)17-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)20-s + (−2.5 + 4.33i)22-s + (1.5 + 2.59i)23-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.5·4-s + (−0.223 − 0.387i)5-s + (0.755 + 0.654i)7-s + 1.06·8-s + (0.158 + 0.273i)10-s + (0.753 − 1.30i)11-s + (0.693 − 1.20i)13-s + (−0.534 − 0.462i)14-s − 0.250·16-s + (0.363 + 0.630i)17-s + (−0.114 + 0.198i)19-s + (0.111 + 0.193i)20-s + (−0.533 + 0.923i)22-s + (0.312 + 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.888 + 0.458i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.888 + 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.771514 - 0.187092i\)
\(L(\frac12)\) \(\approx\) \(0.771514 - 0.187092i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 1.73i)T \)
good2 \( 1 + T + 2T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (1.5 + 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.5 - 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.5 - 7.79i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.46212104465039664491971004511, −11.32924784759997815795275612755, −10.51838487147393649811474419264, −9.254913218369034317099183957610, −8.355259586042305130754459436618, −8.041558117334159433696107494868, −6.09809757732958974580992158973, −5.01179670474333687540389094978, −3.55698322445557741853188652806, −1.15954005450070349770990608819, 1.54879141719047672260390690944, 3.98784000574745962223690651716, 4.84405173034031777728197531192, 6.85993090869678789422442766272, 7.51857919503984356996870283016, 8.787510548118890264028300112934, 9.527710891089699547149021574599, 10.61988458834588128445306079576, 11.41339285891468038713768079353, 12.56228453862573756095980609702

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