# Properties

 Label 2-43-43.42-c6-0-0 Degree $2$ Conductor $43$ Sign $-0.931 + 0.364i$ Analytic cond. $9.89232$ Root an. cond. $3.14520$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 11.9i·2-s − 41.5i·3-s − 77.9·4-s + 111. i·5-s + 495.·6-s + 49.3i·7-s − 166. i·8-s − 998.·9-s − 1.32e3·10-s − 2.54e3·11-s + 3.24e3i·12-s − 277.·13-s − 587.·14-s + 4.63e3·15-s − 3.00e3·16-s − 532.·17-s + ⋯
 L(s)  = 1 + 1.48i·2-s − 1.53i·3-s − 1.21·4-s + 0.892i·5-s + 2.29·6-s + 0.143i·7-s − 0.325i·8-s − 1.36·9-s − 1.32·10-s − 1.91·11-s + 1.87i·12-s − 0.126·13-s − 0.214·14-s + 1.37·15-s − 0.733·16-s − 0.108·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$43$$ Sign: $-0.931 + 0.364i$ Analytic conductor: $$9.89232$$ Root analytic conductor: $$3.14520$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{43} (42, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 43,\ (\ :3),\ -0.931 + 0.364i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.0956478 - 0.506582i$$ $$L(\frac12)$$ $$\approx$$ $$0.0956478 - 0.506582i$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad43 $$1 + (-7.40e4 + 2.89e4i)T$$
good2 $$1 - 11.9iT - 64T^{2}$$
3 $$1 + 41.5iT - 729T^{2}$$
5 $$1 - 111. iT - 1.56e4T^{2}$$
7 $$1 - 49.3iT - 1.17e5T^{2}$$
11 $$1 + 2.54e3T + 1.77e6T^{2}$$
13 $$1 + 277.T + 4.82e6T^{2}$$
17 $$1 + 532.T + 2.41e7T^{2}$$
19 $$1 - 6.00e3iT - 4.70e7T^{2}$$
23 $$1 + 1.85e4T + 1.48e8T^{2}$$
29 $$1 - 2.70e4iT - 5.94e8T^{2}$$
31 $$1 - 2.61e4T + 8.87e8T^{2}$$
37 $$1 + 2.17e3iT - 2.56e9T^{2}$$
41 $$1 + 8.26e4T + 4.75e9T^{2}$$
47 $$1 + 1.68e4T + 1.07e10T^{2}$$
53 $$1 + 8.04e4T + 2.21e10T^{2}$$
59 $$1 - 1.51e5T + 4.21e10T^{2}$$
61 $$1 - 3.48e5iT - 5.15e10T^{2}$$
67 $$1 - 2.61e5T + 9.04e10T^{2}$$
71 $$1 + 4.95e5iT - 1.28e11T^{2}$$
73 $$1 + 3.04e5iT - 1.51e11T^{2}$$
79 $$1 + 2.89e5T + 2.43e11T^{2}$$
83 $$1 - 4.35e5T + 3.26e11T^{2}$$
89 $$1 - 1.28e6iT - 4.96e11T^{2}$$
97 $$1 + 2.91e5T + 8.32e11T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$