# Properties

 Label 2-72e2-24.11-c1-0-77 Degree $2$ Conductor $5184$ Sign $0.258 + 0.965i$ Analytic cond. $41.3944$ Root an. cond. $6.43385$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.24·5-s − 3.77i·7-s + 2.57i·11-s + 2.77i·13-s − 4.81i·17-s + 5.77·19-s + 1.91·23-s + 0.0289·25-s − 10.2·29-s − 6.22i·31-s − 8.45i·35-s − 1.96i·37-s − 10.6i·41-s + 5.01·43-s + 6.36·47-s + ⋯
 L(s)  = 1 + 1.00·5-s − 1.42i·7-s + 0.776i·11-s + 0.768i·13-s − 1.16i·17-s + 1.32·19-s + 0.398·23-s + 0.00579·25-s − 1.90·29-s − 1.11i·31-s − 1.42i·35-s − 0.322i·37-s − 1.65i·41-s + 0.764·43-s + 0.929·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$5184$$    =    $$2^{6} \cdot 3^{4}$$ Sign: $0.258 + 0.965i$ Analytic conductor: $$41.3944$$ Root analytic conductor: $$6.43385$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{5184} (2591, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 5184,\ (\ :1/2),\ 0.258 + 0.965i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.265908484$$ $$L(\frac12)$$ $$\approx$$ $$2.265908484$$ $$L(\frac{3}{2})$$ 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 - 2.24T + 5T^{2}$$
7 $$1 + 3.77iT - 7T^{2}$$
11 $$1 - 2.57iT - 11T^{2}$$
13 $$1 - 2.77iT - 13T^{2}$$
17 $$1 + 4.81iT - 17T^{2}$$
19 $$1 - 5.77T + 19T^{2}$$
23 $$1 - 1.91T + 23T^{2}$$
29 $$1 + 10.2T + 29T^{2}$$
31 $$1 + 6.22iT - 31T^{2}$$
37 $$1 + 1.96iT - 37T^{2}$$
41 $$1 + 10.6iT - 41T^{2}$$
43 $$1 - 5.01T + 43T^{2}$$
47 $$1 - 6.36T + 47T^{2}$$
53 $$1 + 2.35T + 53T^{2}$$
59 $$1 - 5.22iT - 59T^{2}$$
61 $$1 - 2.26iT - 61T^{2}$$
67 $$1 - 12.5T + 67T^{2}$$
71 $$1 + 1.19T + 71T^{2}$$
73 $$1 - 8.57T + 73T^{2}$$
79 $$1 + 2.39iT - 79T^{2}$$
83 $$1 + 15.3iT - 83T^{2}$$
89 $$1 - 14.7iT - 89T^{2}$$
97 $$1 + 13.7T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$