# Properties

 Degree 2 Conductor 43 Sign $-0.994 - 0.104i$ Motivic weight 6 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 7.14i·2-s + 19.7i·3-s + 13.0·4-s + 151. i·5-s − 141.·6-s − 62.8i·7-s + 549. i·8-s + 338.·9-s − 1.08e3·10-s − 497.·11-s + 256. i·12-s + 850.·13-s + 449.·14-s − 2.99e3·15-s − 3.09e3·16-s − 8.04e3·17-s + ⋯
 L(s)  = 1 + 0.892i·2-s + 0.731i·3-s + 0.203·4-s + 1.21i·5-s − 0.653·6-s − 0.183i·7-s + 1.07i·8-s + 0.464·9-s − 1.08·10-s − 0.374·11-s + 0.148i·12-s + 0.387·13-s + 0.163·14-s − 0.888·15-s − 0.755·16-s − 1.63·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.104i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.994 - 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $-0.994 - 0.104i$ motivic weight = $$6$$ character : $\chi_{43} (42, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :3),\ -0.994 - 0.104i)$$ $$L(\frac{7}{2})$$ $$\approx$$ $$0.0962520 + 1.83380i$$ $$L(\frac12)$$ $$\approx$$ $$0.0962520 + 1.83380i$$ $$L(4)$$ 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 + (-7.90e4 - 8.32e3i)T$$
good2 $$1 - 7.14iT - 64T^{2}$$
3 $$1 - 19.7iT - 729T^{2}$$
5 $$1 - 151. iT - 1.56e4T^{2}$$
7 $$1 + 62.8iT - 1.17e5T^{2}$$
11 $$1 + 497.T + 1.77e6T^{2}$$
13 $$1 - 850.T + 4.82e6T^{2}$$
17 $$1 + 8.04e3T + 2.41e7T^{2}$$
19 $$1 + 1.14e4iT - 4.70e7T^{2}$$
23 $$1 - 6.20e3T + 1.48e8T^{2}$$
29 $$1 - 2.38e4iT - 5.94e8T^{2}$$
31 $$1 - 2.69e4T + 8.87e8T^{2}$$
37 $$1 - 2.35e4iT - 2.56e9T^{2}$$
41 $$1 - 8.73e4T + 4.75e9T^{2}$$
47 $$1 + 3.90e4T + 1.07e10T^{2}$$
53 $$1 + 1.78e5T + 2.21e10T^{2}$$
59 $$1 - 1.99e4T + 4.21e10T^{2}$$
61 $$1 + 3.49e4iT - 5.15e10T^{2}$$
67 $$1 + 2.99e5T + 9.04e10T^{2}$$
71 $$1 - 3.62e5iT - 1.28e11T^{2}$$
73 $$1 - 8.99e4iT - 1.51e11T^{2}$$
79 $$1 - 4.34e5T + 2.43e11T^{2}$$
83 $$1 - 6.62e5T + 3.26e11T^{2}$$
89 $$1 + 1.26e6iT - 4.96e11T^{2}$$
97 $$1 - 8.37e4T + 8.32e11T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}