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Properties

Label 2-189-9.4-c1-0-5
Degree $2$
Conductor $189$
Sign $-0.173 + 0.984i$
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  + (−0.439 − 0.761i)2-s + (0.613 − 1.06i)4-s + (0.673 − 1.16i)5-s + (−0.5 − 0.866i)7-s − 2.83·8-s − 1.18·10-s + (0.826 + 1.43i)11-s + (1.68 − 2.91i)13-s + (−0.439 + 0.761i)14-s + (0.0209 + 0.0362i)16-s − 0.467·17-s − 3.22·19-s + (−0.826 − 1.43i)20-s + (0.726 − 1.25i)22-s + (4.47 − 7.74i)23-s + ⋯
L(s)  = 1  + (−0.310 − 0.538i)2-s + (0.306 − 0.531i)4-s + (0.301 − 0.521i)5-s + (−0.188 − 0.327i)7-s − 1.00·8-s − 0.374·10-s + (0.249 + 0.431i)11-s + (0.467 − 0.809i)13-s + (−0.117 + 0.203i)14-s + (0.00523 + 0.00906i)16-s − 0.113·17-s − 0.740·19-s + (−0.184 − 0.320i)20-s + (0.154 − 0.268i)22-s + (0.932 − 1.61i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.692087 - 0.824798i\)
\(L(\frac12)\) \(\approx\) \(0.692087 - 0.824798i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.439 + 0.761i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.673 + 1.16i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.826 - 1.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.68 + 2.91i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.467T + 17T^{2} \)
19 \( 1 + 3.22T + 19T^{2} \)
23 \( 1 + (-4.47 + 7.74i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.13 - 5.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.61 - 7.99i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.23T + 37T^{2} \)
41 \( 1 + (-1.70 + 2.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.20 - 3.82i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.67 - 8.10i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.573T + 53T^{2} \)
59 \( 1 + (5.19 - 9.00i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.81 + 6.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.298 - 0.516i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.554T + 71T^{2} \)
73 \( 1 - 2.04T + 73T^{2} \)
79 \( 1 + (-1.20 - 2.08i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.52 + 13.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.08T + 89T^{2} \)
97 \( 1 + (-0.949 - 1.64i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.46414059082505661558898573096, −10.92188029914600836593950433077, −10.56958287162613011098782157462, −9.356180252745843499348996705443, −8.621950746548719260976662824586, −7.01614181346704908930936333758, −5.97319602943244698246172172806, −4.70116061828204160052833276461, −2.88912242770314900545470895722, −1.18828745570387212178649372787, 2.50273273181646134525430891596, 3.90727407255354225909979301245, 5.89007315023916187007989011715, 6.61444128359880553100548282014, 7.68767236848735720792094942811, 8.790687227432763431757829816578, 9.591572592453379326014656879601, 11.08059827952534416893018254807, 11.68736117843268124713128679370, 12.89154894888064048106140319647

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