# Properties

 Label 2-882-3.2-c2-0-24 Degree $2$ Conductor $882$ Sign $-0.816 + 0.577i$ Analytic cond. $24.0327$ Root an. cond. $4.90232$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 1.41i·2-s − 2.00·4-s − 8.60i·5-s − 2.82i·8-s + 12.1·10-s − 2.82i·11-s − 12.1·13-s + 4.00·16-s − 25.8i·17-s + 24.3·19-s + 17.2i·20-s + 4.00·22-s + 42.4i·23-s − 49·25-s − 17.2i·26-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 0.500·4-s − 1.72i·5-s − 0.353i·8-s + 1.21·10-s − 0.257i·11-s − 0.935·13-s + 0.250·16-s − 1.51i·17-s + 1.28·19-s + 0.860i·20-s + 0.181·22-s + 1.84i·23-s − 1.95·25-s − 0.661i·26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$882$$    =    $$2 \cdot 3^{2} \cdot 7^{2}$$ Sign: $-0.816 + 0.577i$ Analytic conductor: $$24.0327$$ Root analytic conductor: $$4.90232$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{882} (197, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 882,\ (\ :1),\ -0.816 + 0.577i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.6476952475$$ $$L(\frac12)$$ $$\approx$$ $$0.6476952475$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 1.41iT$$
3 $$1$$
7 $$1$$
good5 $$1 + 8.60iT - 25T^{2}$$
11 $$1 + 2.82iT - 121T^{2}$$
13 $$1 + 12.1T + 169T^{2}$$
17 $$1 + 25.8iT - 289T^{2}$$
19 $$1 - 24.3T + 361T^{2}$$
23 $$1 - 42.4iT - 529T^{2}$$
29 $$1 + 15.5iT - 841T^{2}$$
31 $$1 + 24.3T + 961T^{2}$$
37 $$1 + 6T + 1.36e3T^{2}$$
41 $$1 + 25.8iT - 1.68e3T^{2}$$
43 $$1 + 68T + 1.84e3T^{2}$$
47 $$1 - 68.8iT - 2.20e3T^{2}$$
53 $$1 + 41.0iT - 2.80e3T^{2}$$
59 $$1 - 68.8iT - 3.48e3T^{2}$$
61 $$1 + 97.3T + 3.72e3T^{2}$$
67 $$1 + 104T + 4.48e3T^{2}$$
71 $$1 + 70.7iT - 5.04e3T^{2}$$
73 $$1 + 60.8T + 5.32e3T^{2}$$
79 $$1 + 20T + 6.24e3T^{2}$$
83 $$1 - 6.88e3T^{2}$$
89 $$1 + 94.6iT - 7.92e3T^{2}$$
97 $$1 - 158.T + 9.40e3T^{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

−9.305147565829497401559365771471, −8.939651097246258510497194214267, −7.64171679816264929049934802563, −7.43589090533826086707910553609, −5.85164225374559524640314830051, −5.15951976411244748106373266657, −4.62880457375277762205533627978, −3.27215240928678424396207865802, −1.44949666130378794019637614118, −0.21251314587217680263919261291, 1.85510960251154594786740767522, 2.85537934973616689009891257080, 3.61632056576438744305955260752, 4.80901252622570268956694178078, 6.04057963136690073635759412224, 6.89810729838789633614437076043, 7.65586115506888548046826406500, 8.688499754477104325322466232888, 9.864841267880452423455848858897, 10.33725631629655156777986851961