# Properties

 Label 2-43-43.42-c8-0-21 Degree $2$ Conductor $43$ Sign $-0.171 + 0.985i$ 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 + 11.2i·2-s − 77.9i·3-s + 129.·4-s − 278. i·5-s + 875.·6-s − 3.16e3i·7-s + 4.33e3i·8-s + 479.·9-s + 3.12e3·10-s − 1.63e4·11-s − 1.01e4i·12-s − 529.·13-s + 3.54e4·14-s − 2.16e4·15-s − 1.53e4·16-s − 1.22e5·17-s + ⋯
 L(s)  = 1 + 0.701i·2-s − 0.962i·3-s + 0.507·4-s − 0.444i·5-s + 0.675·6-s − 1.31i·7-s + 1.05i·8-s + 0.0730·9-s + 0.312·10-s − 1.11·11-s − 0.488i·12-s − 0.0185·13-s + 0.923·14-s − 0.428·15-s − 0.234·16-s − 1.46·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$43$$ Sign: $-0.171 + 0.985i$ 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: $$(2,\ 43,\ (\ :4),\ -0.171 + 0.985i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$1.08199 - 1.28688i$$ $$L(\frac12)$$ $$\approx$$ $$1.08199 - 1.28688i$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad43 $$1 + (5.86e5 - 3.36e6i)T$$
good2 $$1 - 11.2iT - 256T^{2}$$
3 $$1 + 77.9iT - 6.56e3T^{2}$$
5 $$1 + 278. iT - 3.90e5T^{2}$$
7 $$1 + 3.16e3iT - 5.76e6T^{2}$$
11 $$1 + 1.63e4T + 2.14e8T^{2}$$
13 $$1 + 529.T + 8.15e8T^{2}$$
17 $$1 + 1.22e5T + 6.97e9T^{2}$$
19 $$1 + 1.26e5iT - 1.69e10T^{2}$$
23 $$1 + 3.32e4T + 7.83e10T^{2}$$
29 $$1 + 1.14e6iT - 5.00e11T^{2}$$
31 $$1 - 7.37e5T + 8.52e11T^{2}$$
37 $$1 + 1.37e6iT - 3.51e12T^{2}$$
41 $$1 + 5.06e6T + 7.98e12T^{2}$$
47 $$1 + 3.40e6T + 2.38e13T^{2}$$
53 $$1 - 8.04e6T + 6.22e13T^{2}$$
59 $$1 - 1.77e7T + 1.46e14T^{2}$$
61 $$1 - 8.79e6iT - 1.91e14T^{2}$$
67 $$1 - 2.55e6T + 4.06e14T^{2}$$
71 $$1 + 1.00e7iT - 6.45e14T^{2}$$
73 $$1 - 7.54e6iT - 8.06e14T^{2}$$
79 $$1 - 1.30e7T + 1.51e15T^{2}$$
83 $$1 - 7.73e7T + 2.25e15T^{2}$$
89 $$1 + 3.53e7iT - 3.93e15T^{2}$$
97 $$1 - 1.12e8T + 7.83e15T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$