# Properties

 Label 2-4608-16.13-c1-0-47 Degree $2$ Conductor $4608$ Sign $0.382 + 0.923i$ 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 + i)5-s + 4i·7-s + (−4 + 4i)11-s + (−3 − 3i)13-s + 6·17-s + (−4 − 4i)19-s − 8i·23-s + 3i·25-s + (−3 − 3i)29-s − 4·31-s + (−4 − 4i)35-s + (−1 + i)37-s − 2i·41-s + (−4 + 4i)43-s − 8·47-s + ⋯
 L(s)  = 1 + (−0.447 + 0.447i)5-s + 1.51i·7-s + (−1.20 + 1.20i)11-s + (−0.832 − 0.832i)13-s + 1.45·17-s + (−0.917 − 0.917i)19-s − 1.66i·23-s + 0.600i·25-s + (−0.557 − 0.557i)29-s − 0.718·31-s + (−0.676 − 0.676i)35-s + (−0.164 + 0.164i)37-s − 0.312i·41-s + (−0.609 + 0.609i)43-s − 1.16·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$4608$$    =    $$2^{9} \cdot 3^{2}$$ Sign: $0.382 + 0.923i$ Analytic conductor: $$36.7950$$ Root analytic conductor: $$6.06589$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{4608} (1153, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 4608,\ (\ :1/2),\ 0.382 + 0.923i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.5391581270$$ $$L(\frac12)$$ $$\approx$$ $$0.5391581270$$ $$L(\frac{3}{2})$$ 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 - i)T - 5iT^{2}$$
7 $$1 - 4iT - 7T^{2}$$
11 $$1 + (4 - 4i)T - 11iT^{2}$$
13 $$1 + (3 + 3i)T + 13iT^{2}$$
17 $$1 - 6T + 17T^{2}$$
19 $$1 + (4 + 4i)T + 19iT^{2}$$
23 $$1 + 8iT - 23T^{2}$$
29 $$1 + (3 + 3i)T + 29iT^{2}$$
31 $$1 + 4T + 31T^{2}$$
37 $$1 + (1 - i)T - 37iT^{2}$$
41 $$1 + 2iT - 41T^{2}$$
43 $$1 + (4 - 4i)T - 43iT^{2}$$
47 $$1 + 8T + 47T^{2}$$
53 $$1 + (-7 + 7i)T - 53iT^{2}$$
59 $$1 - 59iT^{2}$$
61 $$1 + (-3 - 3i)T + 61iT^{2}$$
67 $$1 + (-8 - 8i)T + 67iT^{2}$$
71 $$1 - 71T^{2}$$
73 $$1 - 10iT - 73T^{2}$$
79 $$1 - 12T + 79T^{2}$$
83 $$1 + (-4 - 4i)T + 83iT^{2}$$
89 $$1 + 16iT - 89T^{2}$$
97 $$1 - 8T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$