# Properties

 Label 2-5070-13.12-c1-0-21 Degree $2$ Conductor $5070$ Sign $-0.277 - 0.960i$ Analytic cond. $40.4841$ Root an. cond. $6.36271$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·2-s − 3-s − 4-s + i·5-s − i·6-s − 2i·7-s − i·8-s + 9-s − 10-s + 0.464i·11-s + 12-s + 2·14-s − i·15-s + 16-s − 4·17-s + i·18-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s − 0.755i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s + 0.139i·11-s + 0.288·12-s + 0.534·14-s − 0.258i·15-s + 0.250·16-s − 0.970·17-s + 0.235i·18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$5070$$    =    $$2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Sign: $-0.277 - 0.960i$ Analytic conductor: $$40.4841$$ Root analytic conductor: $$6.36271$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{5070} (1351, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 5070,\ (\ :1/2),\ -0.277 - 0.960i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.054056787$$ $$L(\frac12)$$ $$\approx$$ $$1.054056787$$ $$L(\frac{3}{2})$$ 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 + 2iT - 7T^{2}$$
11 $$1 - 0.464iT - 11T^{2}$$
17 $$1 + 4T + 17T^{2}$$
19 $$1 - 0.535iT - 19T^{2}$$
23 $$1 - 0.267T + 23T^{2}$$
29 $$1 - 3.73T + 29T^{2}$$
31 $$1 - 1.73iT - 31T^{2}$$
37 $$1 - 1.19iT - 37T^{2}$$
41 $$1 + 2iT - 41T^{2}$$
43 $$1 + 1.92T + 43T^{2}$$
47 $$1 + 10.4iT - 47T^{2}$$
53 $$1 + 12.9T + 53T^{2}$$
59 $$1 - 1.53iT - 59T^{2}$$
61 $$1 - 10.3T + 61T^{2}$$
67 $$1 - 4.53iT - 67T^{2}$$
71 $$1 - 8.39iT - 71T^{2}$$
73 $$1 - 2iT - 73T^{2}$$
79 $$1 + 0.0717T + 79T^{2}$$
83 $$1 + 4.92iT - 83T^{2}$$
89 $$1 - 7.46iT - 89T^{2}$$
97 $$1 - 7.46iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$