# Properties

 Degree $2$ Conductor $576$ Sign $1$ Motivic weight $7$ Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 82·5-s + 456·7-s + 2.52e3·11-s + 1.07e4·13-s + 1.11e4·17-s + 4.12e3·19-s + 8.17e4·23-s − 7.14e4·25-s + 9.97e4·29-s + 4.04e4·31-s − 3.73e4·35-s + 4.19e5·37-s − 1.41e5·41-s − 6.90e5·43-s − 6.82e5·47-s − 6.15e5·49-s + 1.81e6·53-s − 2.06e5·55-s + 9.66e5·59-s − 1.88e6·61-s − 8.83e5·65-s + 2.96e6·67-s − 2.54e6·71-s − 1.68e6·73-s + 1.15e6·77-s − 4.03e6·79-s + 5.38e6·83-s + ⋯
 L(s)  = 1 − 0.293·5-s + 0.502·7-s + 0.571·11-s + 1.36·13-s + 0.550·17-s + 0.137·19-s + 1.40·23-s − 0.913·25-s + 0.759·29-s + 0.244·31-s − 0.147·35-s + 1.36·37-s − 0.320·41-s − 1.32·43-s − 0.958·47-s − 0.747·49-s + 1.67·53-s − 0.167·55-s + 0.612·59-s − 1.06·61-s − 0.399·65-s + 1.20·67-s − 0.844·71-s − 0.505·73-s + 0.287·77-s − 0.921·79-s + 1.03·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$576$$    =    $$2^{6} \cdot 3^{2}$$ Sign: $1$ Motivic weight: $$7$$ Character: $\chi_{576} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 576,\ (\ :7/2),\ 1)$$

## Particular Values

 $$L(4)$$ $$\approx$$ $$2.917943943$$ $$L(\frac12)$$ $$\approx$$ $$2.917943943$$ $$L(\frac{9}{2})$$ 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 + 82 T + p^{7} T^{2}$$
7 $$1 - 456 T + p^{7} T^{2}$$
11 $$1 - 2524 T + p^{7} T^{2}$$
13 $$1 - 10778 T + p^{7} T^{2}$$
17 $$1 - 11150 T + p^{7} T^{2}$$
19 $$1 - 4124 T + p^{7} T^{2}$$
23 $$1 - 81704 T + p^{7} T^{2}$$
29 $$1 - 99798 T + p^{7} T^{2}$$
31 $$1 - 40480 T + p^{7} T^{2}$$
37 $$1 - 419442 T + p^{7} T^{2}$$
41 $$1 + 141402 T + p^{7} T^{2}$$
43 $$1 + 690428 T + p^{7} T^{2}$$
47 $$1 + 682032 T + p^{7} T^{2}$$
53 $$1 - 1813118 T + p^{7} T^{2}$$
59 $$1 - 966028 T + p^{7} T^{2}$$
61 $$1 + 1887670 T + p^{7} T^{2}$$
67 $$1 - 2965868 T + p^{7} T^{2}$$
71 $$1 + 2548232 T + p^{7} T^{2}$$
73 $$1 + 1680326 T + p^{7} T^{2}$$
79 $$1 + 4038064 T + p^{7} T^{2}$$
83 $$1 - 5385764 T + p^{7} T^{2}$$
89 $$1 - 6473046 T + p^{7} T^{2}$$
97 $$1 + 6065758 T + p^{7} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$