# Properties

 Label 2-1368-456.125-c0-0-3 Degree $2$ Conductor $1368$ Sign $0.787 + 0.616i$ Analytic cond. $0.682720$ Root an. cond. $0.826269$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s − 7-s + (0.707 − 0.707i)8-s + 1.41·11-s + (0.866 − 0.5i)13-s + (−0.965 + 0.258i)14-s + (0.500 − 0.866i)16-s + (−1.22 − 0.707i)17-s + i·19-s + (1.36 − 0.366i)22-s + (−1.22 + 0.707i)23-s + (0.5 + 0.866i)25-s + (0.707 − 0.707i)26-s + (−0.866 + 0.499i)28-s + (−0.707 − 1.22i)29-s + ⋯
 L(s)  = 1 + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s − 7-s + (0.707 − 0.707i)8-s + 1.41·11-s + (0.866 − 0.5i)13-s + (−0.965 + 0.258i)14-s + (0.500 − 0.866i)16-s + (−1.22 − 0.707i)17-s + i·19-s + (1.36 − 0.366i)22-s + (−1.22 + 0.707i)23-s + (0.5 + 0.866i)25-s + (0.707 − 0.707i)26-s + (−0.866 + 0.499i)28-s + (−0.707 − 1.22i)29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1368$$    =    $$2^{3} \cdot 3^{2} \cdot 19$$ Sign: $0.787 + 0.616i$ Analytic conductor: $$0.682720$$ Root analytic conductor: $$0.826269$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1368} (125, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1368,\ (\ :0),\ 0.787 + 0.616i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.897892527$$ $$L(\frac12)$$ $$\approx$$ $$1.897892527$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.965 + 0.258i)T$$
3 $$1$$
19 $$1 - iT$$
good5 $$1 + (-0.5 - 0.866i)T^{2}$$
7 $$1 + T + T^{2}$$
11 $$1 - 1.41T + T^{2}$$
13 $$1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}$$
17 $$1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2}$$
23 $$1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}$$
29 $$1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2}$$
31 $$1 - T + T^{2}$$
37 $$1 + iT - T^{2}$$
41 $$1 + (0.5 + 0.866i)T^{2}$$
43 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
47 $$1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}$$
53 $$1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2}$$
59 $$1 + (-0.5 - 0.866i)T^{2}$$
61 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}$$
67 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}$$
71 $$1 + (0.5 + 0.866i)T^{2}$$
73 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
79 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
83 $$1 + 1.41T + T^{2}$$
89 $$1 + (0.5 - 0.866i)T^{2}$$
97 $$1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$