# Properties

 Label 2-48e2-1.1-c3-0-92 Degree $2$ Conductor $2304$ Sign $-1$ Analytic cond. $135.940$ Root an. cond. $11.6593$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.82·5-s + 14.1·7-s − 20·11-s + 39.5·13-s + 34·17-s + 52·19-s − 62.2·23-s − 117·25-s − 200.·29-s − 110.·31-s − 40.0·35-s − 271.·37-s + 26·41-s + 252·43-s − 345.·47-s − 142.·49-s + 681.·53-s + 56.5·55-s − 364·59-s + 735.·61-s − 112.·65-s + 628·67-s + 333.·71-s + 338·73-s − 282.·77-s − 789.·79-s − 1.03e3·83-s + ⋯
 L(s)  = 1 − 0.252·5-s + 0.763·7-s − 0.548·11-s + 0.844·13-s + 0.485·17-s + 0.627·19-s − 0.564·23-s − 0.936·25-s − 1.28·29-s − 0.639·31-s − 0.193·35-s − 1.20·37-s + 0.0990·41-s + 0.893·43-s − 1.07·47-s − 0.416·49-s + 1.76·53-s + 0.138·55-s − 0.803·59-s + 1.54·61-s − 0.213·65-s + 1.14·67-s + 0.557·71-s + 0.541·73-s − 0.418·77-s − 1.12·79-s − 1.37·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2304$$    =    $$2^{8} \cdot 3^{2}$$ Sign: $-1$ Analytic conductor: $$135.940$$ Root analytic conductor: $$11.6593$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 2304,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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 + 2.82T + 125T^{2}$$
7 $$1 - 14.1T + 343T^{2}$$
11 $$1 + 20T + 1.33e3T^{2}$$
13 $$1 - 39.5T + 2.19e3T^{2}$$
17 $$1 - 34T + 4.91e3T^{2}$$
19 $$1 - 52T + 6.85e3T^{2}$$
23 $$1 + 62.2T + 1.21e4T^{2}$$
29 $$1 + 200.T + 2.43e4T^{2}$$
31 $$1 + 110.T + 2.97e4T^{2}$$
37 $$1 + 271.T + 5.06e4T^{2}$$
41 $$1 - 26T + 6.89e4T^{2}$$
43 $$1 - 252T + 7.95e4T^{2}$$
47 $$1 + 345.T + 1.03e5T^{2}$$
53 $$1 - 681.T + 1.48e5T^{2}$$
59 $$1 + 364T + 2.05e5T^{2}$$
61 $$1 - 735.T + 2.26e5T^{2}$$
67 $$1 - 628T + 3.00e5T^{2}$$
71 $$1 - 333.T + 3.57e5T^{2}$$
73 $$1 - 338T + 3.89e5T^{2}$$
79 $$1 + 789.T + 4.93e5T^{2}$$
83 $$1 + 1.03e3T + 5.71e5T^{2}$$
89 $$1 + 234T + 7.04e5T^{2}$$
97 $$1 + 178T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$