# Properties

 Label 2-1050-35.34-c2-0-41 Degree $2$ Conductor $1050$ Sign $-0.00582 + 0.999i$ 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 − 1.73·3-s − 2.00·4-s − 2.44i·6-s + (3.16 − 6.24i)7-s − 2.82i·8-s + 2.99·9-s − 1.75·11-s + 3.46·12-s + 18.7·13-s + (8.82 + 4.47i)14-s + 4.00·16-s − 23.4·17-s + 4.24i·18-s − 23.0i·19-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 0.577·3-s − 0.500·4-s − 0.408i·6-s + (0.452 − 0.891i)7-s − 0.353i·8-s + 0.333·9-s − 0.159·11-s + 0.288·12-s + 1.44·13-s + (0.630 + 0.319i)14-s + 0.250·16-s − 1.38·17-s + 0.235i·18-s − 1.21i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00582 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.00582 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1050$$    =    $$2 \cdot 3 \cdot 5^{2} \cdot 7$$ Sign: $-0.00582 + 0.999i$ Analytic conductor: $$28.6104$$ Root analytic conductor: $$5.34887$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{1050} (349, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1050,\ (\ :1),\ -0.00582 + 0.999i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.7578519508$$ $$L(\frac12)$$ $$\approx$$ $$0.7578519508$$ $$L(2)$$ 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 + 1.73T$$
5 $$1$$
7 $$1 + (-3.16 + 6.24i)T$$
good11 $$1 + 1.75T + 121T^{2}$$
13 $$1 - 18.7T + 169T^{2}$$
17 $$1 + 23.4T + 289T^{2}$$
19 $$1 + 23.0iT - 361T^{2}$$
23 $$1 - 18.7iT - 529T^{2}$$
29 $$1 + 30T + 841T^{2}$$
31 $$1 + 8.60iT - 961T^{2}$$
37 $$1 - 70.9iT - 1.36e3T^{2}$$
41 $$1 + 41.3iT - 1.68e3T^{2}$$
43 $$1 - 10.4iT - 1.84e3T^{2}$$
47 $$1 + 38.6T + 2.20e3T^{2}$$
53 $$1 + 37.0iT - 2.80e3T^{2}$$
59 $$1 + 97.4iT - 3.48e3T^{2}$$
61 $$1 + 16.7iT - 3.72e3T^{2}$$
67 $$1 + 60.9iT - 4.48e3T^{2}$$
71 $$1 + 110.T + 5.04e3T^{2}$$
73 $$1 + 56.7T + 5.32e3T^{2}$$
79 $$1 - 69.8T + 6.24e3T^{2}$$
83 $$1 - 6.43T + 6.88e3T^{2}$$
89 $$1 + 42.0iT - 7.92e3T^{2}$$
97 $$1 + 51.7T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$