# Properties

 Degree 2 Conductor 43 Sign $-0.530 - 0.847i$ Motivic weight 8 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 10.2i·2-s − 141. i·3-s + 151.·4-s + 508. i·5-s − 1.44e3·6-s + 610. i·7-s − 4.16e3i·8-s − 1.33e4·9-s + 5.19e3·10-s − 2.33e4·11-s − 2.13e4i·12-s − 3.82e4·13-s + 6.24e3·14-s + 7.17e4·15-s − 3.80e3·16-s + 8.27e3·17-s + ⋯
 L(s)  = 1 − 0.638i·2-s − 1.74i·3-s + 0.591·4-s + 0.813i·5-s − 1.11·6-s + 0.254i·7-s − 1.01i·8-s − 2.03·9-s + 0.519·10-s − 1.59·11-s − 1.03i·12-s − 1.33·13-s + 0.162·14-s + 1.41·15-s − 0.0580·16-s + 0.0990·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $-0.530 - 0.847i$ motivic weight = $$8$$ character : $\chi_{43} (42, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :4),\ -0.530 - 0.847i)$$ $$L(\frac{9}{2})$$ $$\approx$$ $$0.423560 + 0.764295i$$ $$L(\frac12)$$ $$\approx$$ $$0.423560 + 0.764295i$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 43$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad43 $$1 + (1.81e6 + 2.89e6i)T$$
good2 $$1 + 10.2iT - 256T^{2}$$
3 $$1 + 141. iT - 6.56e3T^{2}$$
5 $$1 - 508. iT - 3.90e5T^{2}$$
7 $$1 - 610. iT - 5.76e6T^{2}$$
11 $$1 + 2.33e4T + 2.14e8T^{2}$$
13 $$1 + 3.82e4T + 8.15e8T^{2}$$
17 $$1 - 8.27e3T + 6.97e9T^{2}$$
19 $$1 - 4.39e4iT - 1.69e10T^{2}$$
23 $$1 + 1.09e5T + 7.83e10T^{2}$$
29 $$1 + 1.33e6iT - 5.00e11T^{2}$$
31 $$1 + 6.24e5T + 8.52e11T^{2}$$
37 $$1 + 1.28e4iT - 3.51e12T^{2}$$
41 $$1 - 4.22e6T + 7.98e12T^{2}$$
47 $$1 - 1.43e6T + 2.38e13T^{2}$$
53 $$1 + 1.03e7T + 6.22e13T^{2}$$
59 $$1 - 8.60e6T + 1.46e14T^{2}$$
61 $$1 + 1.46e7iT - 1.91e14T^{2}$$
67 $$1 + 9.05e6T + 4.06e14T^{2}$$
71 $$1 + 1.39e7iT - 6.45e14T^{2}$$
73 $$1 - 4.77e7iT - 8.06e14T^{2}$$
79 $$1 + 2.98e7T + 1.51e15T^{2}$$
83 $$1 + 3.68e7T + 2.25e15T^{2}$$
89 $$1 + 9.08e7iT - 3.93e15T^{2}$$
97 $$1 + 4.00e7T + 7.83e15T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}