Properties

Label 2-189-189.5-c1-0-3
Degree $2$
Conductor $189$
Sign $-0.0261 - 0.999i$
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  + (0.357 − 0.0629i)2-s + (−1.26 − 1.18i)3-s + (−1.75 + 0.639i)4-s + (−0.363 + 2.06i)5-s + (−0.525 − 0.343i)6-s + (1.11 + 2.39i)7-s + (−1.21 + 0.701i)8-s + (0.189 + 2.99i)9-s + 0.758i·10-s + (1.41 − 0.249i)11-s + (2.97 + 1.27i)12-s + (−1.90 + 2.27i)13-s + (0.550 + 0.785i)14-s + (2.90 − 2.17i)15-s + (2.47 − 2.07i)16-s − 4.77·17-s + ⋯
L(s)  = 1  + (0.252 − 0.0445i)2-s + (−0.729 − 0.684i)3-s + (−0.877 + 0.319i)4-s + (−0.162 + 0.921i)5-s + (−0.214 − 0.140i)6-s + (0.422 + 0.906i)7-s + (−0.429 + 0.247i)8-s + (0.0630 + 0.998i)9-s + 0.239i·10-s + (0.427 − 0.0753i)11-s + (0.858 + 0.367i)12-s + (−0.528 + 0.630i)13-s + (0.147 + 0.210i)14-s + (0.749 − 0.560i)15-s + (0.618 − 0.518i)16-s − 1.15·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0261 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0261 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.0261 - 0.999i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.0261 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.479476 + 0.492206i\)
\(L(\frac12)\) \(\approx\) \(0.479476 + 0.492206i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.26 + 1.18i)T \)
7 \( 1 + (-1.11 - 2.39i)T \)
good2 \( 1 + (-0.357 + 0.0629i)T + (1.87 - 0.684i)T^{2} \)
5 \( 1 + (0.363 - 2.06i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-1.41 + 0.249i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (1.90 - 2.27i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + 4.77T + 17T^{2} \)
19 \( 1 - 3.09iT - 19T^{2} \)
23 \( 1 + (1.57 - 1.87i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (4.37 + 5.21i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.879 + 2.41i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (-5.23 - 9.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.50 + 5.45i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-10.0 - 3.66i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.563 - 0.204i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-8.71 + 5.03i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-11.0 - 9.23i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-2.19 + 6.03i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.588 + 3.33i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-6.21 - 3.59i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.53 - 0.886i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.456 + 2.59i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (5.95 - 4.99i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 + (2.10 - 5.78i)T + (-74.3 - 62.3i)T^{2} \)
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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

−12.74656565908670068970659735552, −11.75939132901926226747758134882, −11.32954364290727183337042196219, −9.879572443154274741247505460165, −8.699542816460676902044973557902, −7.64001028052451661950416792737, −6.48940863480927734889537135153, −5.43637291564391209576364322536, −4.16985783315600692443993370911, −2.35245012353865536219479025001, 0.64506725755978690395495132092, 3.98675048050073325064595420222, 4.66860531401099034265004489800, 5.51897686667070038294163371668, 6.99594465741927586479811035813, 8.595385154391858510333038585411, 9.335644012989428935546308779075, 10.38593231328345933803796670297, 11.23574045158297199918514042275, 12.50035681965413273265804355732

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