# Properties

 Degree $2$ Conductor $6012$ Sign $0.0713 - 0.997i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3.47·5-s + 3.93·7-s − 4.34i·11-s + 0.788i·13-s − 5.32·17-s + 4.26·19-s − 6.69·23-s + 7.06·25-s + 5.99i·29-s − 6.79·31-s − 13.6·35-s + 4.66i·37-s + 4.62·41-s − 11.6i·43-s + 12.6i·47-s + ⋯
 L(s)  = 1 − 1.55·5-s + 1.48·7-s − 1.30i·11-s + 0.218i·13-s − 1.29·17-s + 0.977·19-s − 1.39·23-s + 1.41·25-s + 1.11i·29-s − 1.22·31-s − 2.30·35-s + 0.766i·37-s + 0.721·41-s − 1.76i·43-s + 1.84i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0713 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0713 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$6012$$    =    $$2^{2} \cdot 3^{2} \cdot 167$$ Sign: $0.0713 - 0.997i$ Motivic weight: $$1$$ Character: $\chi_{6012} (3005, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 6012,\ (\ :1/2),\ 0.0713 - 0.997i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.8776382258$$ $$L(\frac12)$$ $$\approx$$ $$0.8776382258$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
167 $$1 + (-9.99 - 8.19i)T$$
good5 $$1 + 3.47T + 5T^{2}$$
7 $$1 - 3.93T + 7T^{2}$$
11 $$1 + 4.34iT - 11T^{2}$$
13 $$1 - 0.788iT - 13T^{2}$$
17 $$1 + 5.32T + 17T^{2}$$
19 $$1 - 4.26T + 19T^{2}$$
23 $$1 + 6.69T + 23T^{2}$$
29 $$1 - 5.99iT - 29T^{2}$$
31 $$1 + 6.79T + 31T^{2}$$
37 $$1 - 4.66iT - 37T^{2}$$
41 $$1 - 4.62T + 41T^{2}$$
43 $$1 + 11.6iT - 43T^{2}$$
47 $$1 - 12.6iT - 47T^{2}$$
53 $$1 - 0.406T + 53T^{2}$$
59 $$1 - 6.56T + 59T^{2}$$
61 $$1 + 4.94T + 61T^{2}$$
67 $$1 - 3.40iT - 67T^{2}$$
71 $$1 + 0.507T + 71T^{2}$$
73 $$1 + 2.74iT - 73T^{2}$$
79 $$1 - 7.44iT - 79T^{2}$$
83 $$1 - 6.51T + 83T^{2}$$
89 $$1 + 15.1iT - 89T^{2}$$
97 $$1 + 3.32T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$