# Properties

 Label 2-96e2-1.1-c1-0-90 Degree $2$ Conductor $9216$ Sign $1$ Analytic cond. $73.5901$ Root an. cond. $8.57846$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.79·5-s + 2.15·7-s + 2.54·11-s + 1.95·13-s − 0.224·17-s + 0.224·19-s + 2.82·23-s + 9.42·25-s − 2.62·29-s + 1.84·31-s + 8.19·35-s − 5.18·37-s − 5.88·41-s + 10.9·43-s − 2.82·47-s − 2.33·49-s + 10.6·53-s + 9.65·55-s − 5.65·59-s − 8.46·61-s + 7.43·65-s + 14.7·67-s + 4.31·71-s + 5.97·73-s + 5.48·77-s − 15.0·79-s + 14.3·83-s + ⋯
 L(s)  = 1 + 1.69·5-s + 0.816·7-s + 0.766·11-s + 0.542·13-s − 0.0545·17-s + 0.0515·19-s + 0.589·23-s + 1.88·25-s − 0.487·29-s + 0.330·31-s + 1.38·35-s − 0.853·37-s − 0.918·41-s + 1.67·43-s − 0.412·47-s − 0.334·49-s + 1.45·53-s + 1.30·55-s − 0.736·59-s − 1.08·61-s + 0.921·65-s + 1.80·67-s + 0.512·71-s + 0.699·73-s + 0.625·77-s − 1.68·79-s + 1.57·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$9216$$    =    $$2^{10} \cdot 3^{2}$$ Sign: $1$ Analytic conductor: $$73.5901$$ Root analytic conductor: $$8.57846$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{9216} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 9216,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$3.948744236$$ $$L(\frac12)$$ $$\approx$$ $$3.948744236$$ $$L(\frac{3}{2})$$ 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 - 3.79T + 5T^{2}$$
7 $$1 - 2.15T + 7T^{2}$$
11 $$1 - 2.54T + 11T^{2}$$
13 $$1 - 1.95T + 13T^{2}$$
17 $$1 + 0.224T + 17T^{2}$$
19 $$1 - 0.224T + 19T^{2}$$
23 $$1 - 2.82T + 23T^{2}$$
29 $$1 + 2.62T + 29T^{2}$$
31 $$1 - 1.84T + 31T^{2}$$
37 $$1 + 5.18T + 37T^{2}$$
41 $$1 + 5.88T + 41T^{2}$$
43 $$1 - 10.9T + 43T^{2}$$
47 $$1 + 2.82T + 47T^{2}$$
53 $$1 - 10.6T + 53T^{2}$$
59 $$1 + 5.65T + 59T^{2}$$
61 $$1 + 8.46T + 61T^{2}$$
67 $$1 - 14.7T + 67T^{2}$$
71 $$1 - 4.31T + 71T^{2}$$
73 $$1 - 5.97T + 73T^{2}$$
79 $$1 + 15.0T + 79T^{2}$$
83 $$1 - 14.3T + 83T^{2}$$
89 $$1 - 1.42T + 89T^{2}$$
97 $$1 + 16.3T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$