# Properties

 Label 2-124-124.67-c0-0-0 Degree $2$ Conductor $124$ Sign $0.390 - 0.920i$ Analytic cond. $0.0618840$ Root an. cond. $0.248765$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·2-s + (0.866 + 0.5i)3-s − 4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.866 − 0.5i)7-s − i·8-s + (0.866 − 0.5i)10-s + (−0.866 + 0.5i)11-s + (−0.866 − 0.5i)12-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s − 0.999i·15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯
 L(s)  = 1 + i·2-s + (0.866 + 0.5i)3-s − 4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.866 − 0.5i)7-s − i·8-s + (0.866 − 0.5i)10-s + (−0.866 + 0.5i)11-s + (−0.866 − 0.5i)12-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s − 0.999i·15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$124$$    =    $$2^{2} \cdot 31$$ Sign: $0.390 - 0.920i$ Analytic conductor: $$0.0618840$$ Root analytic conductor: $$0.248765$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{124} (67, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 124,\ (\ :0),\ 0.390 - 0.920i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.6137468847$$ $$L(\frac12)$$ $$\approx$$ $$0.6137468847$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - iT$$
31 $$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 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
11 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)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}$$
19 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
23 $$1 - T^{2}$$
29 $$1 + T^{2}$$
37 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)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 - 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 + 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 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}$$
89 $$1 + T^{2}$$
97 $$1 + T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$