# Properties

 Label 2-43-43.42-c8-0-8 Degree $2$ Conductor $43$ Sign $0.381 + 0.924i$ Analytic cond. $17.5172$ Root an. cond. $4.18536$ Motivic weight $8$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 31.0i·2-s + 76.9i·3-s − 710.·4-s − 309. i·5-s + 2.39e3·6-s + 1.60e3i·7-s + 1.41e4i·8-s + 638.·9-s − 9.61e3·10-s − 7.35e3·11-s − 5.47e4i·12-s + 4.21e4·13-s + 4.98e4·14-s + 2.37e4·15-s + 2.57e5·16-s − 5.70e4·17-s + ⋯
 L(s)  = 1 − 1.94i·2-s + 0.950i·3-s − 2.77·4-s − 0.494i·5-s + 1.84·6-s + 0.668i·7-s + 3.45i·8-s + 0.0972·9-s − 0.961·10-s − 0.502·11-s − 2.63i·12-s + 1.47·13-s + 1.29·14-s + 0.469·15-s + 3.93·16-s − 0.683·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$43$$ Sign: $0.381 + 0.924i$ Analytic conductor: $$17.5172$$ Root analytic conductor: $$4.18536$$ Motivic weight: $$8$$ Rational: no Arithmetic: yes Character: $\chi_{43} (42, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 43,\ (\ :4),\ 0.381 + 0.924i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$1.26735 - 0.848317i$$ $$L(\frac12)$$ $$\approx$$ $$1.26735 - 0.848317i$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad43 $$1 + (-1.30e6 - 3.16e6i)T$$
good2 $$1 + 31.0iT - 256T^{2}$$
3 $$1 - 76.9iT - 6.56e3T^{2}$$
5 $$1 + 309. iT - 3.90e5T^{2}$$
7 $$1 - 1.60e3iT - 5.76e6T^{2}$$
11 $$1 + 7.35e3T + 2.14e8T^{2}$$
13 $$1 - 4.21e4T + 8.15e8T^{2}$$
17 $$1 + 5.70e4T + 6.97e9T^{2}$$
19 $$1 + 1.27e5iT - 1.69e10T^{2}$$
23 $$1 - 2.87e5T + 7.83e10T^{2}$$
29 $$1 - 2.92e5iT - 5.00e11T^{2}$$
31 $$1 + 4.78e5T + 8.52e11T^{2}$$
37 $$1 - 1.22e6iT - 3.51e12T^{2}$$
41 $$1 - 5.52e6T + 7.98e12T^{2}$$
47 $$1 + 1.51e6T + 2.38e13T^{2}$$
53 $$1 - 7.83e6T + 6.22e13T^{2}$$
59 $$1 - 1.53e7T + 1.46e14T^{2}$$
61 $$1 - 3.41e6iT - 1.91e14T^{2}$$
67 $$1 + 3.26e7T + 4.06e14T^{2}$$
71 $$1 + 4.27e7iT - 6.45e14T^{2}$$
73 $$1 - 5.32e7iT - 8.06e14T^{2}$$
79 $$1 - 4.22e7T + 1.51e15T^{2}$$
83 $$1 + 4.72e7T + 2.25e15T^{2}$$
89 $$1 - 7.79e7iT - 3.93e15T^{2}$$
97 $$1 - 5.35e7T + 7.83e15T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$