Properties

Label 2-189-189.47-c1-0-17
Degree $2$
Conductor $189$
Sign $0.949 + 0.314i$
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  + (1.66 + 0.294i)2-s + (0.685 − 1.59i)3-s + (0.822 + 0.299i)4-s + (1.39 + 0.507i)5-s + (1.61 − 2.45i)6-s + (−0.356 + 2.62i)7-s + (−1.65 − 0.953i)8-s + (−2.05 − 2.18i)9-s + (2.18 + 1.25i)10-s + (−0.0445 − 0.122i)11-s + (1.03 − 1.10i)12-s + (0.311 − 0.855i)13-s + (−1.36 + 4.27i)14-s + (1.76 − 1.87i)15-s + (−3.81 − 3.20i)16-s + (−2.81 + 4.87i)17-s + ⋯
L(s)  = 1  + (1.18 + 0.208i)2-s + (0.395 − 0.918i)3-s + (0.411 + 0.149i)4-s + (0.624 + 0.227i)5-s + (0.658 − 1.00i)6-s + (−0.134 + 0.990i)7-s + (−0.584 − 0.337i)8-s + (−0.686 − 0.727i)9-s + (0.689 + 0.398i)10-s + (−0.0134 − 0.0368i)11-s + (0.300 − 0.318i)12-s + (0.0863 − 0.237i)13-s + (−0.365 + 1.14i)14-s + (0.455 − 0.483i)15-s + (−0.954 − 0.800i)16-s + (−0.683 + 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.949 + 0.314i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.949 + 0.314i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15835 - 0.347668i\)
\(L(\frac12)\) \(\approx\) \(2.15835 - 0.347668i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.685 + 1.59i)T \)
7 \( 1 + (0.356 - 2.62i)T \)
good2 \( 1 + (-1.66 - 0.294i)T + (1.87 + 0.684i)T^{2} \)
5 \( 1 + (-1.39 - 0.507i)T + (3.83 + 3.21i)T^{2} \)
11 \( 1 + (0.0445 + 0.122i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (-0.311 + 0.855i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (2.81 - 4.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0831 - 0.0479i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.39 + 1.12i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-0.238 - 0.656i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (3.11 - 8.55i)T + (-23.7 - 19.9i)T^{2} \)
37 \( 1 + 3.45T + 37T^{2} \)
41 \( 1 + (-2.04 - 0.744i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.37 + 7.82i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-11.8 + 4.29i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (1.26 - 0.731i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.86 + 4.08i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (2.67 + 7.33i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (2.49 + 14.1i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-2.38 + 1.37i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.02iT - 73T^{2} \)
79 \( 1 + (-0.0703 + 0.398i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-4.54 + 1.65i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-7.98 - 13.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.2 + 1.81i)T + (91.1 + 33.1i)T^{2} \)
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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

−12.64194878508611683117275336413, −12.19609086285689813837508908594, −10.79326913782609165799398343908, −9.236397747323214892389302751477, −8.538102158766919914976468611407, −6.90155751840925895591174002864, −6.12921362370372912784908888067, −5.24927674869534808491434352444, −3.46981158566430900904098559479, −2.24098573096724202369894471723, 2.67391336171548797610767905285, 3.95610857435143473148960150751, 4.79014985789803026115938682419, 5.83429096923652580375943591724, 7.35195546969150615690656424863, 8.964220546203061188326999184489, 9.590104564378459663777664935035, 10.84800448350677808838657185488, 11.59075544516636846199894317895, 13.13955935005332677857672801750

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