# Properties

 Label 2-1216-76.11-c0-0-0 Degree $2$ Conductor $1216$ Sign $0.305 - 0.952i$ Analytic cond. $0.606863$ Root an. cond. $0.779014$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.866 − 0.5i)3-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)13-s + (0.866 − 0.499i)15-s + (−0.5 + 0.866i)17-s − i·19-s + (−0.866 + 0.5i)23-s + i·27-s + (0.5 + 0.866i)29-s + 2i·31-s − 0.999i·39-s + (−0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (0.866 − 0.5i)47-s + 49-s + ⋯
 L(s)  = 1 + (−0.866 − 0.5i)3-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)13-s + (0.866 − 0.499i)15-s + (−0.5 + 0.866i)17-s − i·19-s + (−0.866 + 0.5i)23-s + i·27-s + (0.5 + 0.866i)29-s + 2i·31-s − 0.999i·39-s + (−0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (0.866 − 0.5i)47-s + 49-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1216$$    =    $$2^{6} \cdot 19$$ Sign: $0.305 - 0.952i$ Analytic conductor: $$0.606863$$ Root analytic conductor: $$0.779014$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1216} (1151, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1216,\ (\ :0),\ 0.305 - 0.952i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.5788813692$$ $$L(\frac12)$$ $$\approx$$ $$0.5788813692$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
19 $$1 + iT$$
good3 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
5 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
7 $$1 - T^{2}$$
11 $$1 - T^{2}$$
13 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
17 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
23 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}$$
29 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
31 $$1 - 2iT - T^{2}$$
37 $$1 + T^{2}$$
41 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
43 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
47 $$1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}$$
53 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
59 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
61 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
67 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}$$
71 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
73 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
79 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
83 $$1 - 2iT - T^{2}$$
89 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
97 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$