# Properties

 Degree 2 Conductor 43 Sign $0.819 - 0.572i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.61·2-s + (0.190 + 0.330i)3-s + 4.85·4-s + (1.61 + 2.80i)5-s + (−0.5 − 0.866i)6-s + (−0.118 + 0.204i)7-s − 7.47·8-s + (1.42 − 2.47i)9-s + (−4.23 − 7.33i)10-s − 1.38·11-s + (0.927 + 1.60i)12-s + (−1.80 + 3.13i)13-s + (0.309 − 0.535i)14-s + (−0.618 + 1.07i)15-s + 9.85·16-s + (2.54 − 4.40i)17-s + ⋯
 L(s)  = 1 − 1.85·2-s + (0.110 + 0.190i)3-s + 2.42·4-s + (0.723 + 1.25i)5-s + (−0.204 − 0.353i)6-s + (−0.0446 + 0.0772i)7-s − 2.64·8-s + (0.475 − 0.823i)9-s + (−1.33 − 2.32i)10-s − 0.416·11-s + (0.267 + 0.463i)12-s + (−0.501 + 0.869i)13-s + (0.0825 − 0.143i)14-s + (−0.159 + 0.276i)15-s + 2.46·16-s + (0.617 − 1.06i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $0.819 - 0.572i$ motivic weight = $$1$$ character : $\chi_{43} (36, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :1/2),\ 0.819 - 0.572i)$$ $$L(1)$$ $$\approx$$ $$0.411383 + 0.129474i$$ $$L(\frac12)$$ $$\approx$$ $$0.411383 + 0.129474i$$ $$L(\frac{3}{2})$$ 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 + (6.5 + 0.866i)T$$
good2 $$1 + 2.61T + 2T^{2}$$
3 $$1 + (-0.190 - 0.330i)T + (-1.5 + 2.59i)T^{2}$$
5 $$1 + (-1.61 - 2.80i)T + (-2.5 + 4.33i)T^{2}$$
7 $$1 + (0.118 - 0.204i)T + (-3.5 - 6.06i)T^{2}$$
11 $$1 + 1.38T + 11T^{2}$$
13 $$1 + (1.80 - 3.13i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 + (-2.54 + 4.40i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (1.61 + 2.80i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (3.30 + 5.73i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (0.927 + 1.60i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 - 0.527T + 41T^{2}$$
47 $$1 - 7.85T + 47T^{2}$$
53 $$1 + (-1.80 - 3.13i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + 6.09T + 59T^{2}$$
61 $$1 + (-1.92 + 3.33i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (1.42 + 2.47i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + (6.89 - 11.9i)T + (-35.5 - 61.4i)T^{2}$$
73 $$1 + (-2.42 + 4.20i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (-1.80 + 3.13i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (-6.51 - 11.2i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 + (2.42 + 4.20i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + 4.76T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}