# Properties

 Degree 2 Conductor 43 Sign $0.942 + 0.335i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.65i·2-s − 4.18i·3-s + 2.64·4-s − 45.6i·5-s + 15.3·6-s − 34.3i·7-s + 68.1i·8-s + 63.4·9-s + 166.·10-s − 103.·11-s − 11.0i·12-s + 134.·13-s + 125.·14-s − 191.·15-s − 206.·16-s + 240.·17-s + ⋯
 L(s)  = 1 + 0.913i·2-s − 0.465i·3-s + 0.165·4-s − 1.82i·5-s + 0.425·6-s − 0.700i·7-s + 1.06i·8-s + 0.783·9-s + 1.66·10-s − 0.852·11-s − 0.0768i·12-s + 0.798·13-s + 0.640·14-s − 0.850·15-s − 0.807·16-s + 0.831·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $0.942 + 0.335i$ motivic weight = $$4$$ character : $\chi_{43} (42, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :2),\ 0.942 + 0.335i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$1.64771 - 0.284405i$$ $$L(\frac12)$$ $$\approx$$ $$1.64771 - 0.284405i$$ $$L(3)$$ 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 + (-1.74e3 - 619. i)T$$
good2 $$1 - 3.65iT - 16T^{2}$$
3 $$1 + 4.18iT - 81T^{2}$$
5 $$1 + 45.6iT - 625T^{2}$$
7 $$1 + 34.3iT - 2.40e3T^{2}$$
11 $$1 + 103.T + 1.46e4T^{2}$$
13 $$1 - 134.T + 2.85e4T^{2}$$
17 $$1 - 240.T + 8.35e4T^{2}$$
19 $$1 - 100. iT - 1.30e5T^{2}$$
23 $$1 + 475.T + 2.79e5T^{2}$$
29 $$1 - 159. iT - 7.07e5T^{2}$$
31 $$1 - 1.11e3T + 9.23e5T^{2}$$
37 $$1 - 2.45e3iT - 1.87e6T^{2}$$
41 $$1 - 1.06e3T + 2.82e6T^{2}$$
47 $$1 + 2.93e3T + 4.87e6T^{2}$$
53 $$1 + 825.T + 7.89e6T^{2}$$
59 $$1 + 1.24e3T + 1.21e7T^{2}$$
61 $$1 - 6.20e3iT - 1.38e7T^{2}$$
67 $$1 - 265.T + 2.01e7T^{2}$$
71 $$1 - 3.86e3iT - 2.54e7T^{2}$$
73 $$1 + 8.27e3iT - 2.83e7T^{2}$$
79 $$1 - 4.95e3T + 3.89e7T^{2}$$
83 $$1 + 1.35e4T + 4.74e7T^{2}$$
89 $$1 + 5.35e3iT - 6.27e7T^{2}$$
97 $$1 - 3.19e3T + 8.85e7T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}