# Properties

 Degree 2 Conductor 43 Sign $0.985 + 0.171i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s + (−0.5 − 0.866i)3-s − 4-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−1.5 + 2.59i)7-s − 3·8-s + (1 − 1.73i)9-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)12-s + (2.5 − 4.33i)13-s + (−1.5 + 2.59i)14-s + (0.499 − 0.866i)15-s − 16-s + (−1.5 + 2.59i)17-s + (1 − 1.73i)18-s + ⋯
 L(s)  = 1 + 0.707·2-s + (−0.288 − 0.499i)3-s − 0.5·4-s + (0.223 + 0.387i)5-s + (−0.204 − 0.353i)6-s + (−0.566 + 0.981i)7-s − 1.06·8-s + (0.333 − 0.577i)9-s + (0.158 + 0.273i)10-s + (0.144 + 0.249i)12-s + (0.693 − 1.20i)13-s + (−0.400 + 0.694i)14-s + (0.129 − 0.223i)15-s − 0.250·16-s + (−0.363 + 0.630i)17-s + (0.235 − 0.408i)18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $0.985 + 0.171i$ motivic weight = $$1$$ character : $\chi_{43} (36, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :1/2),\ 0.985 + 0.171i)$$ $$L(1)$$ $$\approx$$ $$0.880980 - 0.0761557i$$ $$L(\frac12)$$ $$\approx$$ $$0.880980 - 0.0761557i$$ $$L(\frac{3}{2})$$ 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 + (4 + 5.19i)T$$
good2 $$1 - T + 2T^{2}$$
3 $$1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2}$$
5 $$1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2}$$
7 $$1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2}$$
11 $$1 + 11T^{2}$$
13 $$1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (-3.5 - 6.06i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (-4.5 - 7.79i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + 10T + 41T^{2}$$
47 $$1 + 8T + 47T^{2}$$
53 $$1 + (-2.5 - 4.33i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 - 12T + 59T^{2}$$
61 $$1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + (-0.5 + 0.866i)T + (-35.5 - 61.4i)T^{2}$$
73 $$1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (-2.5 + 4.33i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 + (-0.5 - 0.866i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + 2T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}