# Properties

 Label 2-1050-35.12-c1-0-22 Degree $2$ Conductor $1050$ Sign $-0.810 + 0.585i$ Analytic cond. $8.38429$ Root an. cond. $2.89556$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.965 − 0.258i)2-s + (−0.258 + 0.965i)3-s + (0.866 − 0.499i)4-s + i·6-s + (−2.63 + 0.258i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−2.54 − 4.40i)11-s + (0.258 + 0.965i)12-s + (−2.02 − 2.02i)13-s + (−2.47 + 0.931i)14-s + (0.500 − 0.866i)16-s + (−6.71 − 1.79i)17-s + (−0.965 − 0.258i)18-s + (−1.79 + 3.11i)19-s + ⋯
 L(s)  = 1 + (0.683 − 0.183i)2-s + (−0.149 + 0.557i)3-s + (0.433 − 0.249i)4-s + 0.408i·6-s + (−0.995 + 0.0978i)7-s + (0.249 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.767 − 1.32i)11-s + (0.0747 + 0.278i)12-s + (−0.561 − 0.561i)13-s + (−0.661 + 0.248i)14-s + (0.125 − 0.216i)16-s + (−1.62 − 0.436i)17-s + (−0.227 − 0.0610i)18-s + (−0.412 + 0.714i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1050$$    =    $$2 \cdot 3 \cdot 5^{2} \cdot 7$$ Sign: $-0.810 + 0.585i$ Analytic conductor: $$8.38429$$ Root analytic conductor: $$2.89556$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1050} (607, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1050,\ (\ :1/2),\ -0.810 + 0.585i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.6109033030$$ $$L(\frac12)$$ $$\approx$$ $$0.6109033030$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.965 + 0.258i)T$$
3 $$1 + (0.258 - 0.965i)T$$
5 $$1$$
7 $$1 + (2.63 - 0.258i)T$$
good11 $$1 + (2.54 + 4.40i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + (2.02 + 2.02i)T + 13iT^{2}$$
17 $$1 + (6.71 + 1.79i)T + (14.7 + 8.5i)T^{2}$$
19 $$1 + (1.79 - 3.11i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (-1.79 - 6.71i)T + (-19.9 + 11.5i)T^{2}$$
29 $$1 + 8.81iT - 29T^{2}$$
31 $$1 + (-1.61 + 0.931i)T + (15.5 - 26.8i)T^{2}$$
37 $$1 + (7.50 - 2.01i)T + (32.0 - 18.5i)T^{2}$$
41 $$1 - 4.13iT - 41T^{2}$$
43 $$1 + (-7.84 + 7.84i)T - 43iT^{2}$$
47 $$1 + (0.482 + 1.79i)T + (-40.7 + 23.5i)T^{2}$$
53 $$1 + (0.879 + 0.235i)T + (45.8 + 26.5i)T^{2}$$
59 $$1 + (2.08 + 3.61i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (5.08 + 2.93i)T + (30.5 + 52.8i)T^{2}$$
67 $$1 + (0.762 - 2.84i)T + (-58.0 - 33.5i)T^{2}$$
71 $$1 - 7.86T + 71T^{2}$$
73 $$1 + (3.50 - 13.0i)T + (-63.2 - 36.5i)T^{2}$$
79 $$1 + (10.4 + 6.04i)T + (39.5 + 68.4i)T^{2}$$
83 $$1 + (-1.99 - 1.99i)T + 83iT^{2}$$
89 $$1 + (-3.82 + 6.62i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (13.4 - 13.4i)T - 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$