# Properties

 Label 2-2736-19.18-c2-0-98 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 no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 4·5-s + 10·7-s + 10·11-s − 24.2i·13-s − 10·17-s − 19·19-s − 20·23-s − 9·25-s + 34.6i·29-s − 17.3i·31-s − 40·35-s − 10.3i·37-s − 34.6i·41-s + 10·43-s − 80·47-s + ⋯
 L(s)  = 1 − 0.800·5-s + 1.42·7-s + 0.909·11-s − 1.86i·13-s − 0.588·17-s − 19-s − 0.869·23-s − 0.359·25-s + 1.19i·29-s − 0.558i·31-s − 1.14·35-s − 0.280i·37-s − 0.844i·41-s + 0.232·43-s − 1.70·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\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: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2736,\ (\ :1),\ -1)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.3731935293$$ $$L(\frac12)$$ $$\approx$$ $$0.3731935293$$ $$L(2)$$ 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 + 4T + 25T^{2}$$
7 $$1 - 10T + 49T^{2}$$
11 $$1 - 10T + 121T^{2}$$
13 $$1 + 24.2iT - 169T^{2}$$
17 $$1 + 10T + 289T^{2}$$
23 $$1 + 20T + 529T^{2}$$
29 $$1 - 34.6iT - 841T^{2}$$
31 $$1 + 17.3iT - 961T^{2}$$
37 $$1 + 10.3iT - 1.36e3T^{2}$$
41 $$1 + 34.6iT - 1.68e3T^{2}$$
43 $$1 - 10T + 1.84e3T^{2}$$
47 $$1 + 80T + 2.20e3T^{2}$$
53 $$1 + 41.5iT - 2.80e3T^{2}$$
59 $$1 - 34.6iT - 3.48e3T^{2}$$
61 $$1 + 10T + 3.72e3T^{2}$$
67 $$1 - 76.2iT - 4.48e3T^{2}$$
71 $$1 - 103. iT - 5.04e3T^{2}$$
73 $$1 + 10T + 5.32e3T^{2}$$
79 $$1 - 17.3iT - 6.24e3T^{2}$$
83 $$1 - 70T + 6.88e3T^{2}$$
89 $$1 - 103. iT - 7.92e3T^{2}$$
97 $$1 - 76.2iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$