# Properties

 Degree $2$ Conductor $210$ Sign $-0.441 - 0.897i$ 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 + (−1.86 + 1.23i)5-s − 0.999·6-s + (−1.73 + 2i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−2.23 + 0.133i)10-s + (2.5 + 4.33i)11-s + (−0.866 − 0.499i)12-s − i·13-s + (−2.5 + 0.866i)14-s + (1 − 2i)15-s + (−0.5 + 0.866i)16-s + (−1.73 + i)17-s + ⋯
 L(s)  = 1 + (0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.834 + 0.550i)5-s − 0.408·6-s + (−0.654 + 0.755i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.705 + 0.0423i)10-s + (0.753 + 1.30i)11-s + (−0.249 − 0.144i)12-s − 0.277i·13-s + (−0.668 + 0.231i)14-s + (0.258 − 0.516i)15-s + (−0.125 + 0.216i)16-s + (−0.420 + 0.242i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$210$$    =    $$2 \cdot 3 \cdot 5 \cdot 7$$ Sign: $-0.441 - 0.897i$ Motivic weight: $$1$$ Character: $\chi_{210} (79, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 210,\ (\ :1/2),\ -0.441 - 0.897i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.605913 + 0.972923i$$ $$L(\frac12)$$ $$\approx$$ $$0.605913 + 0.972923i$$ $$L(\frac{3}{2})$$ 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 + (1.86 - 1.23i)T$$
7 $$1 + (1.73 - 2i)T$$
good11 $$1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + iT - 13T^{2}$$
17 $$1 + (1.73 - i)T + (8.5 - 14.7i)T^{2}$$
19 $$1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (-2.59 - 1.5i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (-4.33 - 2.5i)T + (18.5 + 32.0i)T^{2}$$
41 $$1 + 9T + 41T^{2}$$
43 $$1 + 10iT - 43T^{2}$$
47 $$1 + (-11.2 - 6.5i)T + (23.5 + 40.7i)T^{2}$$
53 $$1 + (-0.866 + 0.5i)T + (26.5 - 45.8i)T^{2}$$
59 $$1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-5.19 + 3i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 + 2T + 71T^{2}$$
73 $$1 + (3.46 - 2i)T + (36.5 - 63.2i)T^{2}$$
79 $$1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 - 10iT - 83T^{2}$$
89 $$1 + (-5 + 8.66i)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}$$