# Properties

 Label 2-2736-19.18-c2-0-16 Degree $2$ Conductor $2736$ Sign $1$ Analytic cond. $74.5506$ Root an. cond. $8.63426$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.725·5-s − 13.8·7-s − 20.3·11-s − 18.9·17-s + 19·19-s − 30·23-s − 24.4·25-s + 10.0·35-s − 53.8·43-s − 86.5·47-s + 142.·49-s + 14.7·55-s + 5.12·61-s − 112.·73-s + 281.·77-s + 90·83-s + 13.7·85-s − 13.7·95-s + 102·101-s + 21.7·115-s + 261.·119-s + ⋯
 L(s)  = 1 − 0.145·5-s − 1.97·7-s − 1.85·11-s − 1.11·17-s + 19-s − 1.30·23-s − 0.978·25-s + 0.286·35-s − 1.25·43-s − 1.84·47-s + 2.90·49-s + 0.268·55-s + 0.0839·61-s − 1.53·73-s + 3.65·77-s + 1.08·83-s + 0.161·85-s − 0.145·95-s + 1.00·101-s + 0.189·115-s + 2.19·119-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2736$$    =    $$2^{4} \cdot 3^{2} \cdot 19$$ Sign: $1$ Analytic conductor: $$74.5506$$ Root analytic conductor: $$8.63426$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{2736} (721, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 2736,\ (\ :1),\ 1)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.2240552783$$ $$L(\frac12)$$ $$\approx$$ $$0.2240552783$$ $$L(2)$$ 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$$
19 $$1 - 19T$$
good5 $$1 + 0.725T + 25T^{2}$$
7 $$1 + 13.8T + 49T^{2}$$
11 $$1 + 20.3T + 121T^{2}$$
13 $$1 - 169T^{2}$$
17 $$1 + 18.9T + 289T^{2}$$
23 $$1 + 30T + 529T^{2}$$
29 $$1 - 841T^{2}$$
31 $$1 - 961T^{2}$$
37 $$1 - 1.36e3T^{2}$$
41 $$1 - 1.68e3T^{2}$$
43 $$1 + 53.8T + 1.84e3T^{2}$$
47 $$1 + 86.5T + 2.20e3T^{2}$$
53 $$1 - 2.80e3T^{2}$$
59 $$1 - 3.48e3T^{2}$$
61 $$1 - 5.12T + 3.72e3T^{2}$$
67 $$1 - 4.48e3T^{2}$$
71 $$1 - 5.04e3T^{2}$$
73 $$1 + 112.T + 5.32e3T^{2}$$
79 $$1 - 6.24e3T^{2}$$
83 $$1 - 90T + 6.88e3T^{2}$$
89 $$1 - 7.92e3T^{2}$$
97 $$1 - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$