# Properties

 Degree $2$ Conductor $3024$ Sign $-0.188 + 0.981i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.5 − 2.59i)7-s − 5.19i·13-s + 7·19-s + 5·25-s − 4·31-s − 11·37-s + 10.3i·43-s + (−6.5 − 2.59i)49-s − 15.5i·61-s − 15.5i·67-s − 15.5i·73-s + 5.19i·79-s + (−13.5 − 2.59i)91-s − 5.19i·97-s − 13·103-s + ⋯
 L(s)  = 1 + (0.188 − 0.981i)7-s − 1.44i·13-s + 1.60·19-s + 25-s − 0.718·31-s − 1.80·37-s + 1.58i·43-s + (−0.928 − 0.371i)49-s − 1.99i·61-s − 1.90i·67-s − 1.82i·73-s + 0.584i·79-s + (−1.41 − 0.272i)91-s − 0.527i·97-s − 1.28·103-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.188 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3024$$    =    $$2^{4} \cdot 3^{3} \cdot 7$$ Sign: $-0.188 + 0.981i$ Motivic weight: $$1$$ Character: $\chi_{3024} (1567, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3024,\ (\ :1/2),\ -0.188 + 0.981i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.629162861$$ $$L(\frac12)$$ $$\approx$$ $$1.629162861$$ $$L(\frac{3}{2})$$ 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$$
7 $$1 + (-0.5 + 2.59i)T$$
good5 $$1 - 5T^{2}$$
11 $$1 - 11T^{2}$$
13 $$1 + 5.19iT - 13T^{2}$$
17 $$1 - 17T^{2}$$
19 $$1 - 7T + 19T^{2}$$
23 $$1 - 23T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + 4T + 31T^{2}$$
37 $$1 + 11T + 37T^{2}$$
41 $$1 - 41T^{2}$$
43 $$1 - 10.3iT - 43T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 + 53T^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 + 15.5iT - 61T^{2}$$
67 $$1 + 15.5iT - 67T^{2}$$
71 $$1 - 71T^{2}$$
73 $$1 + 15.5iT - 73T^{2}$$
79 $$1 - 5.19iT - 79T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 - 89T^{2}$$
97 $$1 + 5.19iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$