# Properties

 Label 2-1050-3.2-c2-0-5 Degree $2$ Conductor $1050$ Sign $-0.569 - 0.821i$ Analytic cond. $28.6104$ Root an. cond. $5.34887$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.41i·2-s + (−2.46 + 1.70i)3-s − 2.00·4-s + (2.41 + 3.48i)6-s − 2.64·7-s + 2.82i·8-s + (3.15 − 8.42i)9-s − 2.66i·11-s + (4.93 − 3.41i)12-s + 5.56·13-s + 3.74i·14-s + 4.00·16-s + 18.5i·17-s + (−11.9 − 4.46i)18-s − 0.634·19-s + ⋯
 L(s)  = 1 − 0.707i·2-s + (−0.821 + 0.569i)3-s − 0.500·4-s + (0.402 + 0.581i)6-s − 0.377·7-s + 0.353i·8-s + (0.350 − 0.936i)9-s − 0.242i·11-s + (0.410 − 0.284i)12-s + 0.427·13-s + 0.267i·14-s + 0.250·16-s + 1.09i·17-s + (−0.662 − 0.248i)18-s − 0.0334·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 - 0.821i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1050$$    =    $$2 \cdot 3 \cdot 5^{2} \cdot 7$$ Sign: $-0.569 - 0.821i$ Analytic conductor: $$28.6104$$ Root analytic conductor: $$5.34887$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{1050} (701, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1050,\ (\ :1),\ -0.569 - 0.821i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.2567289965$$ $$L(\frac12)$$ $$\approx$$ $$0.2567289965$$ $$L(2)$$ 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 + (2.46 - 1.70i)T$$
5 $$1$$
7 $$1 + 2.64T$$
good11 $$1 + 2.66iT - 121T^{2}$$
13 $$1 - 5.56T + 169T^{2}$$
17 $$1 - 18.5iT - 289T^{2}$$
19 $$1 + 0.634T + 361T^{2}$$
23 $$1 + 15.6iT - 529T^{2}$$
29 $$1 + 43.3iT - 841T^{2}$$
31 $$1 - 13.5T + 961T^{2}$$
37 $$1 + 35.0T + 1.36e3T^{2}$$
41 $$1 + 15.6iT - 1.68e3T^{2}$$
43 $$1 - 64.4T + 1.84e3T^{2}$$
47 $$1 - 52.5iT - 2.20e3T^{2}$$
53 $$1 - 51.6iT - 2.80e3T^{2}$$
59 $$1 - 32.9iT - 3.48e3T^{2}$$
61 $$1 + 104.T + 3.72e3T^{2}$$
67 $$1 + 113.T + 4.48e3T^{2}$$
71 $$1 - 36.8iT - 5.04e3T^{2}$$
73 $$1 + 144.T + 5.32e3T^{2}$$
79 $$1 + 57.9T + 6.24e3T^{2}$$
83 $$1 - 45.0iT - 6.88e3T^{2}$$
89 $$1 + 80.9iT - 7.92e3T^{2}$$
97 $$1 + 96.2T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$