# Properties

 Label 2-2736-19.18-c2-0-93 Degree $2$ Conductor $2736$ Sign $-0.526 + 0.850i$ 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 + 7-s + 14·11-s − 16.1i·13-s − 23·17-s + (−10 + 16.1i)19-s − 23-s − 9·25-s − 48.4i·29-s − 32.3i·31-s + 4·35-s − 32.3i·37-s + 32.3i·41-s − 68·43-s + 26·47-s + ⋯
 L(s)  = 1 + 0.800·5-s + 0.142·7-s + 1.27·11-s − 1.24i·13-s − 1.35·17-s + (−0.526 + 0.850i)19-s − 0.0434·23-s − 0.359·25-s − 1.67i·29-s − 1.04i·31-s + 0.114·35-s − 0.873i·37-s + 0.788i·41-s − 1.58·43-s + 0.553·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 + 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2736$$    =    $$2^{4} \cdot 3^{2} \cdot 19$$ Sign: $-0.526 + 0.850i$ 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: $$(2,\ 2736,\ (\ :1),\ -0.526 + 0.850i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.510767588$$ $$L(\frac12)$$ $$\approx$$ $$1.510767588$$ $$L(2)$$ 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 + (10 - 16.1i)T$$
good5 $$1 - 4T + 25T^{2}$$
7 $$1 - T + 49T^{2}$$
11 $$1 - 14T + 121T^{2}$$
13 $$1 + 16.1iT - 169T^{2}$$
17 $$1 + 23T + 289T^{2}$$
23 $$1 + T + 529T^{2}$$
29 $$1 + 48.4iT - 841T^{2}$$
31 $$1 + 32.3iT - 961T^{2}$$
37 $$1 + 32.3iT - 1.36e3T^{2}$$
41 $$1 - 32.3iT - 1.68e3T^{2}$$
43 $$1 + 68T + 1.84e3T^{2}$$
47 $$1 - 26T + 2.20e3T^{2}$$
53 $$1 + 80.7iT - 2.80e3T^{2}$$
59 $$1 + 16.1iT - 3.48e3T^{2}$$
61 $$1 + 40T + 3.72e3T^{2}$$
67 $$1 + 16.1iT - 4.48e3T^{2}$$
71 $$1 - 32.3iT - 5.04e3T^{2}$$
73 $$1 + 7T + 5.32e3T^{2}$$
79 $$1 - 96.9iT - 6.24e3T^{2}$$
83 $$1 - 32T + 6.88e3T^{2}$$
89 $$1 + 129. iT - 7.92e3T^{2}$$
97 $$1 - 96.9iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$