# Properties

 Label 2-1050-7.6-c2-0-25 Degree $2$ Conductor $1050$ Sign $0.974 - 0.225i$ 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 − 1.41·2-s + 1.73i·3-s + 2.00·4-s − 2.44i·6-s + (1.57 + 6.81i)7-s − 2.82·8-s − 2.99·9-s − 8.15·11-s + 3.46i·12-s − 14.6i·13-s + (−2.23 − 9.64i)14-s + 4.00·16-s + 5.81i·17-s + 4.24·18-s − 33.7i·19-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.577i·3-s + 0.500·4-s − 0.408i·6-s + (0.225 + 0.974i)7-s − 0.353·8-s − 0.333·9-s − 0.741·11-s + 0.288i·12-s − 1.12i·13-s + (−0.159 − 0.688i)14-s + 0.250·16-s + 0.342i·17-s + 0.235·18-s − 1.77i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1050$$    =    $$2 \cdot 3 \cdot 5^{2} \cdot 7$$ Sign: $0.974 - 0.225i$ Analytic conductor: $$28.6104$$ Root analytic conductor: $$5.34887$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{1050} (601, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1050,\ (\ :1),\ 0.974 - 0.225i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.248738644$$ $$L(\frac12)$$ $$\approx$$ $$1.248738644$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 1.41T$$
3 $$1 - 1.73iT$$
5 $$1$$
7 $$1 + (-1.57 - 6.81i)T$$
good11 $$1 + 8.15T + 121T^{2}$$
13 $$1 + 14.6iT - 169T^{2}$$
17 $$1 - 5.81iT - 289T^{2}$$
19 $$1 + 33.7iT - 361T^{2}$$
23 $$1 - 37.2T + 529T^{2}$$
29 $$1 - 9.25T + 841T^{2}$$
31 $$1 + 19.2iT - 961T^{2}$$
37 $$1 - 63.4T + 1.36e3T^{2}$$
41 $$1 - 8.25iT - 1.68e3T^{2}$$
43 $$1 + 42.0T + 1.84e3T^{2}$$
47 $$1 - 23.3iT - 2.20e3T^{2}$$
53 $$1 - 71.3T + 2.80e3T^{2}$$
59 $$1 - 42.9iT - 3.48e3T^{2}$$
61 $$1 + 34.2iT - 3.72e3T^{2}$$
67 $$1 - 4.99T + 4.48e3T^{2}$$
71 $$1 + 38.8T + 5.04e3T^{2}$$
73 $$1 + 124. iT - 5.32e3T^{2}$$
79 $$1 - 56.1T + 6.24e3T^{2}$$
83 $$1 - 90.3iT - 6.88e3T^{2}$$
89 $$1 + 16.2iT - 7.92e3T^{2}$$
97 $$1 - 82.7iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$