# Properties

 Label 2-48e2-4.3-c2-0-16 Degree $2$ Conductor $2304$ Sign $1$ Analytic cond. $62.7794$ Root an. cond. $7.92334$ Motivic weight $2$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 8·5-s − 24·13-s − 30·17-s + 39·25-s − 40·29-s − 24·37-s + 18·41-s + 49·49-s + 56·53-s − 120·61-s + 192·65-s − 110·73-s + 240·85-s + 78·89-s − 130·97-s − 40·101-s − 120·109-s + 30·113-s + ⋯
 L(s)  = 1 − 8/5·5-s − 1.84·13-s − 1.76·17-s + 1.55·25-s − 1.37·29-s − 0.648·37-s + 0.439·41-s + 49-s + 1.05·53-s − 1.96·61-s + 2.95·65-s − 1.50·73-s + 2.82·85-s + 0.876·89-s − 1.34·97-s − 0.396·101-s − 1.10·109-s + 0.265·113-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2304$$    =    $$2^{8} \cdot 3^{2}$$ Sign: $1$ Analytic conductor: $$62.7794$$ Root analytic conductor: $$7.92334$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: $\chi_{2304} (1279, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 2304,\ (\ :1),\ 1)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.3926445416$$ $$L(\frac12)$$ $$\approx$$ $$0.3926445416$$ $$L(2)$$ 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 + 8 T + p^{2} T^{2}$$
7 $$( 1 - p T )( 1 + p T )$$
11 $$( 1 - p T )( 1 + p T )$$
13 $$1 + 24 T + p^{2} T^{2}$$
17 $$1 + 30 T + p^{2} T^{2}$$
19 $$( 1 - p T )( 1 + p T )$$
23 $$( 1 - p T )( 1 + p T )$$
29 $$1 + 40 T + p^{2} T^{2}$$
31 $$( 1 - p T )( 1 + p T )$$
37 $$1 + 24 T + p^{2} T^{2}$$
41 $$1 - 18 T + p^{2} T^{2}$$
43 $$( 1 - p T )( 1 + p T )$$
47 $$( 1 - p T )( 1 + p T )$$
53 $$1 - 56 T + p^{2} T^{2}$$
59 $$( 1 - p T )( 1 + p T )$$
61 $$1 + 120 T + p^{2} T^{2}$$
67 $$( 1 - p T )( 1 + p T )$$
71 $$( 1 - p T )( 1 + p T )$$
73 $$1 + 110 T + p^{2} T^{2}$$
79 $$( 1 - p T )( 1 + p T )$$
83 $$( 1 - p T )( 1 + p T )$$
89 $$1 - 78 T + p^{2} T^{2}$$
97 $$1 + 130 T + p^{2} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$