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Properties

Label 2-189-9.7-c1-0-5
Degree $2$
Conductor $189$
Sign $-0.766 + 0.642i$
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  + (1.26 − 2.19i)2-s + (−2.20 − 3.82i)4-s + (−0.439 − 0.761i)5-s + (−0.5 + 0.866i)7-s − 6.10·8-s − 2.22·10-s + (1.93 − 3.35i)11-s + (2.72 + 4.72i)13-s + (1.26 + 2.19i)14-s + (−3.31 + 5.74i)16-s − 1.65·17-s + 2.41·19-s + (−1.93 + 3.35i)20-s + (−4.91 − 8.50i)22-s + (1.58 + 2.73i)23-s + ⋯
L(s)  = 1  + (0.895 − 1.55i)2-s + (−1.10 − 1.91i)4-s + (−0.196 − 0.340i)5-s + (−0.188 + 0.327i)7-s − 2.15·8-s − 0.704·10-s + (0.584 − 1.01i)11-s + (0.756 + 1.30i)13-s + (0.338 + 0.586i)14-s + (−0.829 + 1.43i)16-s − 0.400·17-s + 0.553·19-s + (−0.433 + 0.751i)20-s + (−1.04 − 1.81i)22-s + (0.329 + 0.571i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.567506 - 1.55921i\)
\(L(\frac12)\) \(\approx\) \(0.567506 - 1.55921i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.26 + 2.19i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.439 + 0.761i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.93 + 3.35i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.72 - 4.72i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.65T + 17T^{2} \)
19 \( 1 - 2.41T + 19T^{2} \)
23 \( 1 + (-1.58 - 2.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.02 - 5.23i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.27 - 3.94i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.55T + 37T^{2} \)
41 \( 1 + (0.592 + 1.02i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.0923 - 0.160i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.511 - 0.885i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.29T + 53T^{2} \)
59 \( 1 + (-3.33 - 5.76i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.29 + 2.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.47 - 2.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.68T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 + (-2.97 + 5.15i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.109 - 0.189i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 + (6.25 - 10.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.02349042938062884194491019713, −11.45647123498958188302859570618, −10.64773475219572127724432584781, −9.324835982509263002968368253639, −8.702655859173719319786719824502, −6.58905225828936102131683508168, −5.36421495847388876699018500862, −4.15941807685087078981683397483, −3.15593454568155724252765376838, −1.44847883687483930650634110133, 3.36748418655730523022974973046, 4.48263794856912570888418305475, 5.68671443817530915704681156292, 6.72826094754002162447343403048, 7.49783537951705805238262452709, 8.470469279860520753527497609365, 9.782873302468943231685643102727, 11.16147143179355017271133731892, 12.47998252709940282206743554995, 13.17657245384583613234702466474

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