# Properties

 Degree 2 Conductor 43 Sign $-0.377 + 0.925i$ Motivic weight 6 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 11.0i·2-s + 50.0i·3-s − 57.2·4-s − 30.3i·5-s − 551.·6-s + 75.1i·7-s + 74.1i·8-s − 1.77e3·9-s + 333.·10-s + 604.·11-s − 2.86e3i·12-s + 2.63e3·13-s − 827.·14-s + 1.51e3·15-s − 4.48e3·16-s + 4.98e3·17-s + ⋯
 L(s)  = 1 + 1.37i·2-s + 1.85i·3-s − 0.894·4-s − 0.242i·5-s − 2.55·6-s + 0.219i·7-s + 0.144i·8-s − 2.43·9-s + 0.333·10-s + 0.454·11-s − 1.65i·12-s + 1.19·13-s − 0.301·14-s + 0.449·15-s − 1.09·16-s + 1.01·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $-0.377 + 0.925i$ motivic weight = $$6$$ character : $\chi_{43} (42, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :3),\ -0.377 + 0.925i)$$ $$L(\frac{7}{2})$$ $$\approx$$ $$0.852844 - 1.26927i$$ $$L(\frac12)$$ $$\approx$$ $$0.852844 - 1.26927i$$ $$L(4)$$ 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 + (-3.00e4 + 7.36e4i)T$$
good2 $$1 - 11.0iT - 64T^{2}$$
3 $$1 - 50.0iT - 729T^{2}$$
5 $$1 + 30.3iT - 1.56e4T^{2}$$
7 $$1 - 75.1iT - 1.17e5T^{2}$$
11 $$1 - 604.T + 1.77e6T^{2}$$
13 $$1 - 2.63e3T + 4.82e6T^{2}$$
17 $$1 - 4.98e3T + 2.41e7T^{2}$$
19 $$1 - 3.03e3iT - 4.70e7T^{2}$$
23 $$1 + 1.24e4T + 1.48e8T^{2}$$
29 $$1 + 4.44e3iT - 5.94e8T^{2}$$
31 $$1 + 2.93e4T + 8.87e8T^{2}$$
37 $$1 - 8.30e4iT - 2.56e9T^{2}$$
41 $$1 + 3.34e4T + 4.75e9T^{2}$$
47 $$1 + 1.57e5T + 1.07e10T^{2}$$
53 $$1 - 2.12e5T + 2.21e10T^{2}$$
59 $$1 + 2.63e5T + 4.21e10T^{2}$$
61 $$1 + 1.33e5iT - 5.15e10T^{2}$$
67 $$1 + 1.57e5T + 9.04e10T^{2}$$
71 $$1 - 4.85e5iT - 1.28e11T^{2}$$
73 $$1 + 4.85e4iT - 1.51e11T^{2}$$
79 $$1 - 7.62e5T + 2.43e11T^{2}$$
83 $$1 - 2.68e5T + 3.26e11T^{2}$$
89 $$1 - 1.32e6iT - 4.96e11T^{2}$$
97 $$1 - 1.43e6T + 8.32e11T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}