# Properties

 Degree $2$ Conductor $1224$ Sign $-0.816 - 0.577i$ Motivic weight $0$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 5-s + 1.41i·7-s − 11-s − 13-s − 17-s − 19-s + 23-s + 1.41i·31-s − 1.41i·35-s − 1.41i·37-s − 41-s + 43-s + 1.41i·47-s − 1.00·49-s + 1.41i·53-s + ⋯
 L(s)  = 1 − 5-s + 1.41i·7-s − 11-s − 13-s − 17-s − 19-s + 23-s + 1.41i·31-s − 1.41i·35-s − 1.41i·37-s − 41-s + 43-s + 1.41i·47-s − 1.00·49-s + 1.41i·53-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1224$$    =    $$2^{3} \cdot 3^{2} \cdot 17$$ Sign: $-0.816 - 0.577i$ Motivic weight: $$0$$ Character: $\chi_{1224} (305, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1224,\ (\ :0),\ -0.816 - 0.577i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.3823805483$$ $$L(\frac12)$$ $$\approx$$ $$0.3823805483$$ $$L(1)$$ 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$$
17 $$1 + T$$
good5 $$1 + T + T^{2}$$
7 $$1 - 1.41iT - T^{2}$$
11 $$1 + T + T^{2}$$
13 $$1 + T + T^{2}$$
19 $$1 + T + T^{2}$$
23 $$1 - T + T^{2}$$
29 $$1 + T^{2}$$
31 $$1 - 1.41iT - T^{2}$$
37 $$1 + 1.41iT - T^{2}$$
41 $$1 + T + T^{2}$$
43 $$1 - T + T^{2}$$
47 $$1 - 1.41iT - T^{2}$$
53 $$1 - 1.41iT - T^{2}$$
59 $$1 - T^{2}$$
61 $$1 - 1.41iT - T^{2}$$
67 $$1 + T^{2}$$
71 $$1 + T^{2}$$
73 $$1 + 1.41iT - T^{2}$$
79 $$1 - T^{2}$$
83 $$1 - 1.41iT - T^{2}$$
89 $$1 + 1.41iT - T^{2}$$
97 $$1 - T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$