# Properties

 Degree 2 Conductor 43 Sign $0.731 - 0.681i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 7.49i·2-s + 14.5i·3-s − 40.1·4-s + 9.40i·5-s + 108.·6-s + 53.5i·7-s + 180. i·8-s − 129.·9-s + 70.4·10-s + 151.·11-s − 582. i·12-s − 319.·13-s + 401.·14-s − 136.·15-s + 713.·16-s + 32.2·17-s + ⋯
 L(s)  = 1 − 1.87i·2-s + 1.61i·3-s − 2.50·4-s + 0.376i·5-s + 3.02·6-s + 1.09i·7-s + 2.82i·8-s − 1.60·9-s + 0.704·10-s + 1.24·11-s − 4.04i·12-s − 1.89·13-s + 2.04·14-s − 0.606·15-s + 2.78·16-s + 0.111·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 - 0.681i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $0.731 - 0.681i$ motivic weight = $$4$$ character : $\chi_{43} (42, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :2),\ 0.731 - 0.681i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$0.910207 + 0.358450i$$ $$L(\frac12)$$ $$\approx$$ $$0.910207 + 0.358450i$$ $$L(3)$$ 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 + (-1.35e3 + 1.26e3i)T$$
good2 $$1 + 7.49iT - 16T^{2}$$
3 $$1 - 14.5iT - 81T^{2}$$
5 $$1 - 9.40iT - 625T^{2}$$
7 $$1 - 53.5iT - 2.40e3T^{2}$$
11 $$1 - 151.T + 1.46e4T^{2}$$
13 $$1 + 319.T + 2.85e4T^{2}$$
17 $$1 - 32.2T + 8.35e4T^{2}$$
19 $$1 - 304. iT - 1.30e5T^{2}$$
23 $$1 + 199.T + 2.79e5T^{2}$$
29 $$1 - 268. iT - 7.07e5T^{2}$$
31 $$1 - 665.T + 9.23e5T^{2}$$
37 $$1 + 1.17e3iT - 1.87e6T^{2}$$
41 $$1 - 212.T + 2.82e6T^{2}$$
47 $$1 + 3.10e3T + 4.87e6T^{2}$$
53 $$1 - 2.66e3T + 7.89e6T^{2}$$
59 $$1 - 2.74e3T + 1.21e7T^{2}$$
61 $$1 - 5.88e3iT - 1.38e7T^{2}$$
67 $$1 + 1.09e3T + 2.01e7T^{2}$$
71 $$1 - 5.84e3iT - 2.54e7T^{2}$$
73 $$1 + 663. iT - 2.83e7T^{2}$$
79 $$1 - 6.32e3T + 3.89e7T^{2}$$
83 $$1 - 8.35e3T + 4.74e7T^{2}$$
89 $$1 - 4.25e3iT - 6.27e7T^{2}$$
97 $$1 + 3.58e3T + 8.85e7T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}