# Properties

 Degree $2$ Conductor $5070$ Sign $-0.832 - 0.554i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·2-s + 3-s − 4-s − i·5-s − i·6-s + 4i·7-s + i·8-s + 9-s − 10-s − 12-s + 4·14-s − i·15-s + 16-s − 6·17-s − i·18-s − 4i·19-s + ⋯
 L(s)  = 1 − 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.447i·5-s − 0.408i·6-s + 1.51i·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s − 0.288·12-s + 1.06·14-s − 0.258i·15-s + 0.250·16-s − 1.45·17-s − 0.235i·18-s − 0.917i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$5070$$    =    $$2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Sign: $-0.832 - 0.554i$ Motivic weight: $$1$$ Character: $\chi_{5070} (1351, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$1$$ Selberg data: $$(2,\ 5070,\ (\ :1/2),\ -0.832 - 0.554i)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + iT$$
3 $$1 - T$$
5 $$1 + iT$$
13 $$1$$
good7 $$1 - 4iT - 7T^{2}$$
11 $$1 - 11T^{2}$$
17 $$1 + 6T + 17T^{2}$$
19 $$1 + 4iT - 19T^{2}$$
23 $$1 + 23T^{2}$$
29 $$1 + 6T + 29T^{2}$$
31 $$1 - 8iT - 31T^{2}$$
37 $$1 + 2iT - 37T^{2}$$
41 $$1 + 6iT - 41T^{2}$$
43 $$1 - 4T + 43T^{2}$$
47 $$1 - 47T^{2}$$
53 $$1 + 6T + 53T^{2}$$
59 $$1 - 59T^{2}$$
61 $$1 + 10T + 61T^{2}$$
67 $$1 + 4iT - 67T^{2}$$
71 $$1 - 71T^{2}$$
73 $$1 + 2iT - 73T^{2}$$
79 $$1 - 8T + 79T^{2}$$
83 $$1 - 12iT - 83T^{2}$$
89 $$1 + 18iT - 89T^{2}$$
97 $$1 - 2iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$