# Properties

 Label 2-4608-8.5-c1-0-5 Degree $2$ Conductor $4608$ Sign $-1$ Analytic cond. $36.7950$ Root an. cond. $6.06589$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.41i·5-s − 1.41·7-s + 2i·11-s − 2·17-s + 4i·19-s + 2.82·23-s + 2.99·25-s + 9.89i·29-s − 7.07·31-s − 2.00i·35-s − 8.48i·37-s + 6·41-s − 8i·43-s − 2.82·47-s − 5·49-s + ⋯
 L(s)  = 1 + 0.632i·5-s − 0.534·7-s + 0.603i·11-s − 0.485·17-s + 0.917i·19-s + 0.589·23-s + 0.599·25-s + 1.83i·29-s − 1.27·31-s − 0.338i·35-s − 1.39i·37-s + 0.937·41-s − 1.21i·43-s − 0.412·47-s − 0.714·49-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$4608$$    =    $$2^{9} \cdot 3^{2}$$ Sign: $-1$ Analytic conductor: $$36.7950$$ Root analytic conductor: $$6.06589$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{4608} (2305, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 4608,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.5811132559$$ $$L(\frac12)$$ $$\approx$$ $$0.5811132559$$ $$L(\frac{3}{2})$$ 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$$
good5 $$1 - 1.41iT - 5T^{2}$$
7 $$1 + 1.41T + 7T^{2}$$
11 $$1 - 2iT - 11T^{2}$$
13 $$1 - 13T^{2}$$
17 $$1 + 2T + 17T^{2}$$
19 $$1 - 4iT - 19T^{2}$$
23 $$1 - 2.82T + 23T^{2}$$
29 $$1 - 9.89iT - 29T^{2}$$
31 $$1 + 7.07T + 31T^{2}$$
37 $$1 + 8.48iT - 37T^{2}$$
41 $$1 - 6T + 41T^{2}$$
43 $$1 + 8iT - 43T^{2}$$
47 $$1 + 2.82T + 47T^{2}$$
53 $$1 - 1.41iT - 53T^{2}$$
59 $$1 - 12iT - 59T^{2}$$
61 $$1 + 14.1iT - 61T^{2}$$
67 $$1 - 8iT - 67T^{2}$$
71 $$1 + 14.1T + 71T^{2}$$
73 $$1 - 8T + 73T^{2}$$
79 $$1 + 4.24T + 79T^{2}$$
83 $$1 + 6iT - 83T^{2}$$
89 $$1 - 2T + 89T^{2}$$
97 $$1 + 14T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$