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Properties

Label 2-189-189.184-c1-0-20
Degree $2$
Conductor $189$
Sign $-0.998 - 0.0547i$
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  + (−0.345 − 1.96i)2-s + (−0.361 − 1.69i)3-s + (−1.84 + 0.672i)4-s + (0.424 − 2.40i)5-s + (−3.19 + 1.29i)6-s + (2.63 + 0.288i)7-s + (−0.0327 − 0.0567i)8-s + (−2.73 + 1.22i)9-s − 4.86·10-s + (0.315 + 1.79i)11-s + (1.80 + 2.88i)12-s + (3.42 + 2.87i)13-s + (−0.344 − 5.25i)14-s + (−4.23 + 0.151i)15-s + (−3.11 + 2.61i)16-s + 1.57·17-s + ⋯
L(s)  = 1  + (−0.244 − 1.38i)2-s + (−0.208 − 0.977i)3-s + (−0.924 + 0.336i)4-s + (0.189 − 1.07i)5-s + (−1.30 + 0.528i)6-s + (0.994 + 0.108i)7-s + (−0.0115 − 0.0200i)8-s + (−0.912 + 0.408i)9-s − 1.53·10-s + (0.0951 + 0.539i)11-s + (0.521 + 0.833i)12-s + (0.949 + 0.796i)13-s + (−0.0920 − 1.40i)14-s + (−1.09 + 0.0391i)15-s + (−0.778 + 0.653i)16-s + 0.381·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.998 - 0.0547i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.998 - 0.0547i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0288138 + 1.05233i\)
\(L(\frac12)\) \(\approx\) \(0.0288138 + 1.05233i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.361 + 1.69i)T \)
7 \( 1 + (-2.63 - 0.288i)T \)
good2 \( 1 + (0.345 + 1.96i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (-0.424 + 2.40i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-0.315 - 1.79i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-3.42 - 2.87i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 - 1.57T + 17T^{2} \)
19 \( 1 + 3.38T + 19T^{2} \)
23 \( 1 + (-0.757 - 0.635i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (0.421 - 0.354i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-4.15 + 1.51i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (5.35 + 9.28i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.76 + 3.99i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (7.65 + 2.78i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-9.10 - 3.31i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (3.29 + 5.71i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.320 - 0.269i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-13.2 - 4.82i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (2.62 - 14.9i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (6.86 - 11.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.764 + 1.32i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.51 + 8.58i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (9.94 - 8.34i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + 8.50T + 89T^{2} \)
97 \( 1 + (-13.9 - 5.08i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.96711430371989192555167134024, −11.39112230969593011404496438572, −10.37045837379305995073568241785, −8.930201121585943125009031873448, −8.477530760951950615276274099552, −6.95554374233563098690964944311, −5.48156758264307210046457114999, −4.14667988681555403964659036476, −2.11336136184321211099472494352, −1.22180208495648886162831007985, 3.17853853267258141968698040479, 4.82028258167199614464312696492, 5.88908238992416803398455377346, 6.70058919467321214409208120057, 8.122418155335825271179353756583, 8.661821608397486055160610136824, 10.16732095048616942078014862778, 10.88696657021618817525962312808, 11.71847362536945493249795356437, 13.69275180915651648203811335369

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