# Properties

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

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.01i·5-s − 2.64·7-s − 2.01i·11-s − 7.34i·17-s − 3.35·19-s + 9.36i·23-s + 0.937·25-s + 2.29·31-s − 5.33i·35-s + 11.9·37-s − 9.36i·41-s + 7.00·49-s + 4.06·55-s − 16.7i·71-s + 5.33i·77-s + ⋯
 L(s)  = 1 + 0.901i·5-s − 0.999·7-s − 0.607i·11-s − 1.78i·17-s − 0.769·19-s + 1.95i·23-s + 0.187·25-s + 0.411·31-s − 0.901i·35-s + 1.96·37-s − 1.46i·41-s + 49-s + 0.547·55-s − 1.98i·71-s + 0.607i·77-s + ⋯

## Functional equation

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

## Invariants

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

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.423413890$$ $$L(\frac12)$$ $$\approx$$ $$1.423413890$$ $$L(\frac{3}{2})$$ 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 + 2.64T$$
good5 $$1 - 2.01iT - 5T^{2}$$
11 $$1 + 2.01iT - 11T^{2}$$
13 $$1 - 13T^{2}$$
17 $$1 + 7.34iT - 17T^{2}$$
19 $$1 + 3.35T + 19T^{2}$$
23 $$1 - 9.36iT - 23T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 - 2.29T + 31T^{2}$$
37 $$1 - 11.9T + 37T^{2}$$
41 $$1 + 9.36iT - 41T^{2}$$
43 $$1 - 43T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 + 53T^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 - 61T^{2}$$
67 $$1 - 67T^{2}$$
71 $$1 + 16.7iT - 71T^{2}$$
73 $$1 - 73T^{2}$$
79 $$1 - 79T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 - 17.4iT - 89T^{2}$$
97 $$1 - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$