# Properties

 Degree 2 Conductor 43 Sign $-0.518 - 0.855i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.58i·2-s + 3.59i·3-s − 2.67·4-s − 7.02i·5-s − 9.28·6-s + 0.845i·7-s + 3.42i·8-s − 3.91·9-s + 18.1·10-s + 14.5·11-s − 9.60i·12-s − 15.5·13-s − 2.18·14-s + 25.2·15-s − 19.5·16-s + 6.95·17-s + ⋯
 L(s)  = 1 + 1.29i·2-s + 1.19i·3-s − 0.668·4-s − 1.40i·5-s − 1.54·6-s + 0.120i·7-s + 0.428i·8-s − 0.435·9-s + 1.81·10-s + 1.32·11-s − 0.800i·12-s − 1.19·13-s − 0.156·14-s + 1.68·15-s − 1.22·16-s + 0.409·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $-0.518 - 0.855i$ motivic weight = $$2$$ character : $\chi_{43} (42, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :1),\ -0.518 - 0.855i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.553630 + 0.982598i$$ $$L(\frac12)$$ $$\approx$$ $$0.553630 + 0.982598i$$ $$L(2)$$ 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 + (-22.2 - 36.7i)T$$
good2 $$1 - 2.58iT - 4T^{2}$$
3 $$1 - 3.59iT - 9T^{2}$$
5 $$1 + 7.02iT - 25T^{2}$$
7 $$1 - 0.845iT - 49T^{2}$$
11 $$1 - 14.5T + 121T^{2}$$
13 $$1 + 15.5T + 169T^{2}$$
17 $$1 - 6.95T + 289T^{2}$$
19 $$1 + 30.8iT - 361T^{2}$$
23 $$1 - 17.5T + 529T^{2}$$
29 $$1 + 7.86iT - 841T^{2}$$
31 $$1 + 57.7T + 961T^{2}$$
37 $$1 + 32.5iT - 1.36e3T^{2}$$
41 $$1 + 18.4T + 1.68e3T^{2}$$
47 $$1 + 25.7T + 2.20e3T^{2}$$
53 $$1 + 79.9T + 2.80e3T^{2}$$
59 $$1 - 18.4T + 3.48e3T^{2}$$
61 $$1 - 76.1iT - 3.72e3T^{2}$$
67 $$1 - 9.00T + 4.48e3T^{2}$$
71 $$1 - 51.6iT - 5.04e3T^{2}$$
73 $$1 - 77.4iT - 5.32e3T^{2}$$
79 $$1 - 7.04T + 6.24e3T^{2}$$
83 $$1 - 83.8T + 6.88e3T^{2}$$
89 $$1 + 91.8iT - 7.92e3T^{2}$$
97 $$1 + 155.T + 9.40e3T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}