# Properties

 Degree 2 Conductor 43 Sign $-0.441 - 0.897i$ Motivic weight 6 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.98i·2-s + 41.6i·3-s + 55.0·4-s + 126. i·5-s + 124.·6-s + 69.5i·7-s − 355. i·8-s − 1.00e3·9-s + 377.·10-s − 100.·11-s + 2.29e3i·12-s − 1.72e3·13-s + 207.·14-s − 5.26e3·15-s + 2.46e3·16-s − 115.·17-s + ⋯
 L(s)  = 1 − 0.373i·2-s + 1.54i·3-s + 0.860·4-s + 1.01i·5-s + 0.576·6-s + 0.202i·7-s − 0.694i·8-s − 1.38·9-s + 0.377·10-s − 0.0755·11-s + 1.32i·12-s − 0.784·13-s + 0.0757·14-s − 1.55·15-s + 0.600·16-s − 0.0235·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $-0.441 - 0.897i$ motivic weight = $$6$$ character : $\chi_{43} (42, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :3),\ -0.441 - 0.897i)$$ $$L(\frac{7}{2})$$ $$\approx$$ $$0.987282 + 1.58710i$$ $$L(\frac12)$$ $$\approx$$ $$0.987282 + 1.58710i$$ $$L(4)$$ 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 + (-3.51e4 - 7.13e4i)T$$
good2 $$1 + 2.98iT - 64T^{2}$$
3 $$1 - 41.6iT - 729T^{2}$$
5 $$1 - 126. iT - 1.56e4T^{2}$$
7 $$1 - 69.5iT - 1.17e5T^{2}$$
11 $$1 + 100.T + 1.77e6T^{2}$$
13 $$1 + 1.72e3T + 4.82e6T^{2}$$
17 $$1 + 115.T + 2.41e7T^{2}$$
19 $$1 - 9.21e3iT - 4.70e7T^{2}$$
23 $$1 - 6.39e3T + 1.48e8T^{2}$$
29 $$1 + 1.54e4iT - 5.94e8T^{2}$$
31 $$1 + 4.20e4T + 8.87e8T^{2}$$
37 $$1 + 3.82e4iT - 2.56e9T^{2}$$
41 $$1 - 7.86e4T + 4.75e9T^{2}$$
47 $$1 - 1.38e5T + 1.07e10T^{2}$$
53 $$1 - 5.48e4T + 2.21e10T^{2}$$
59 $$1 + 1.77e5T + 4.21e10T^{2}$$
61 $$1 - 3.72e5iT - 5.15e10T^{2}$$
67 $$1 - 5.75e5T + 9.04e10T^{2}$$
71 $$1 + 3.67e5iT - 1.28e11T^{2}$$
73 $$1 + 4.75e4iT - 1.51e11T^{2}$$
79 $$1 - 2.34e5T + 2.43e11T^{2}$$
83 $$1 + 6.87e4T + 3.26e11T^{2}$$
89 $$1 + 3.35e4iT - 4.96e11T^{2}$$
97 $$1 - 3.41e5T + 8.32e11T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}