# Properties

 Degree $2$ Conductor $1050$ Sign $0.943 + 0.330i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s − 0.999·6-s + (−1.73 − 2i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−1.5 + 2.59i)11-s + (0.866 − 0.499i)12-s − 5i·13-s + (2.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.499i)18-s + (2.5 + 4.33i)19-s + (−0.499 − 2.59i)21-s − 3i·22-s + ⋯
 L(s)  = 1 + (−0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s − 0.408·6-s + (−0.654 − 0.755i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.452 + 0.783i)11-s + (0.249 − 0.144i)12-s − 1.38i·13-s + (0.668 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.204 − 0.117i)18-s + (0.573 + 0.993i)19-s + (−0.109 − 0.566i)21-s − 0.639i·22-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1050$$    =    $$2 \cdot 3 \cdot 5^{2} \cdot 7$$ Sign: $0.943 + 0.330i$ Motivic weight: $$1$$ Character: $\chi_{1050} (949, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1050,\ (\ :1/2),\ 0.943 + 0.330i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.238619274$$ $$L(\frac12)$$ $$\approx$$ $$1.238619274$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.866 - 0.5i)T$$
3 $$1 + (-0.866 - 0.5i)T$$
5 $$1$$
7 $$1 + (1.73 + 2i)T$$
good11 $$1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + 5iT - 13T^{2}$$
17 $$1 + (8.5 + 14.7i)T^{2}$$
19 $$1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (-7.79 + 4.5i)T + (11.5 - 19.9i)T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + (-5 + 8.66i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (0.866 - 0.5i)T + (18.5 - 32.0i)T^{2}$$
41 $$1 - 9T + 41T^{2}$$
43 $$1 + 8iT - 43T^{2}$$
47 $$1 + (-2.59 + 1.5i)T + (23.5 - 40.7i)T^{2}$$
53 $$1 + (2.59 + 1.5i)T + (26.5 + 45.8i)T^{2}$$
59 $$1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (6.92 + 4i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 + 6T + 71T^{2}$$
73 $$1 + (-1.73 - i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 83T^{2}$$
89 $$1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 - 8iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$