# Properties

 Degree $2$ Conductor $1050$ Sign $-0.0460 - 0.998i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s + (0.618 + 1.61i)3-s + 4-s + (0.618 + 1.61i)6-s + (0.381 + 2.61i)7-s + 8-s + (−2.23 + 2.00i)9-s − 4.47i·11-s + (0.618 + 1.61i)12-s + 1.23·13-s + (0.381 + 2.61i)14-s + 16-s + 5.23i·17-s + (−2.23 + 2.00i)18-s + 8.47i·19-s + ⋯
 L(s)  = 1 + 0.707·2-s + (0.356 + 0.934i)3-s + 0.5·4-s + (0.252 + 0.660i)6-s + (0.144 + 0.989i)7-s + 0.353·8-s + (−0.745 + 0.666i)9-s − 1.34i·11-s + (0.178 + 0.467i)12-s + 0.342·13-s + (0.102 + 0.699i)14-s + 0.250·16-s + 1.26i·17-s + (−0.527 + 0.471i)18-s + 1.94i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0460 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0460 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1050$$    =    $$2 \cdot 3 \cdot 5^{2} \cdot 7$$ Sign: $-0.0460 - 0.998i$ Motivic weight: $$1$$ Character: $\chi_{1050} (1049, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1050,\ (\ :1/2),\ -0.0460 - 0.998i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.738780289$$ $$L(\frac12)$$ $$\approx$$ $$2.738780289$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - T$$
3 $$1 + (-0.618 - 1.61i)T$$
5 $$1$$
7 $$1 + (-0.381 - 2.61i)T$$
good11 $$1 + 4.47iT - 11T^{2}$$
13 $$1 - 1.23T + 13T^{2}$$
17 $$1 - 5.23iT - 17T^{2}$$
19 $$1 - 8.47iT - 19T^{2}$$
23 $$1 - 4T + 23T^{2}$$
29 $$1 + 7.70iT - 29T^{2}$$
31 $$1 - 2.76iT - 31T^{2}$$
37 $$1 - 0.763iT - 37T^{2}$$
41 $$1 + 2.47T + 41T^{2}$$
43 $$1 - 4.94iT - 43T^{2}$$
47 $$1 + 6.47iT - 47T^{2}$$
53 $$1 + 0.472T + 53T^{2}$$
59 $$1 + 4.47T + 59T^{2}$$
61 $$1 + 7.23iT - 61T^{2}$$
67 $$1 + 12iT - 67T^{2}$$
71 $$1 - 7.23iT - 71T^{2}$$
73 $$1 - 11.2T + 73T^{2}$$
79 $$1 - 8.94T + 79T^{2}$$
83 $$1 + 14.6iT - 83T^{2}$$
89 $$1 - 5.52T + 89T^{2}$$
97 $$1 - 0.763T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$