# Properties

 Degree 2 Conductor 43 Sign $-0.776 + 0.630i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.75i·2-s − 12.3i·3-s + 8.40·4-s − 21.9i·5-s − 33.9·6-s + 63.3i·7-s − 67.2i·8-s − 71.0·9-s − 60.5·10-s − 1.17·11-s − 103. i·12-s − 173.·13-s + 174.·14-s − 271.·15-s − 51.0·16-s + 469.·17-s + ⋯
 L(s)  = 1 − 0.689i·2-s − 1.37i·3-s + 0.525·4-s − 0.879i·5-s − 0.944·6-s + 1.29i·7-s − 1.05i·8-s − 0.877·9-s − 0.605·10-s − 0.00967·11-s − 0.719i·12-s − 1.02·13-s + 0.890·14-s − 1.20·15-s − 0.199·16-s + 1.62·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $-0.776 + 0.630i$ motivic weight = $$4$$ character : $\chi_{43} (42, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :2),\ -0.776 + 0.630i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$0.557954 - 1.57290i$$ $$L(\frac12)$$ $$\approx$$ $$0.557954 - 1.57290i$$ $$L(3)$$ 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 + (1.43e3 - 1.16e3i)T$$
good2 $$1 + 2.75iT - 16T^{2}$$
3 $$1 + 12.3iT - 81T^{2}$$
5 $$1 + 21.9iT - 625T^{2}$$
7 $$1 - 63.3iT - 2.40e3T^{2}$$
11 $$1 + 1.17T + 1.46e4T^{2}$$
13 $$1 + 173.T + 2.85e4T^{2}$$
17 $$1 - 469.T + 8.35e4T^{2}$$
19 $$1 - 27.8iT - 1.30e5T^{2}$$
23 $$1 - 79.3T + 2.79e5T^{2}$$
29 $$1 - 696. iT - 7.07e5T^{2}$$
31 $$1 - 1.19e3T + 9.23e5T^{2}$$
37 $$1 + 1.53e3iT - 1.87e6T^{2}$$
41 $$1 + 2.45e3T + 2.82e6T^{2}$$
47 $$1 - 1.69e3T + 4.87e6T^{2}$$
53 $$1 + 1.99e3T + 7.89e6T^{2}$$
59 $$1 - 3.56e3T + 1.21e7T^{2}$$
61 $$1 - 5.44e3iT - 1.38e7T^{2}$$
67 $$1 - 4.93e3T + 2.01e7T^{2}$$
71 $$1 - 9.66e3iT - 2.54e7T^{2}$$
73 $$1 - 5.60e3iT - 2.83e7T^{2}$$
79 $$1 + 1.14e4T + 3.89e7T^{2}$$
83 $$1 - 5.52e3T + 4.74e7T^{2}$$
89 $$1 - 2.55e3iT - 6.27e7T^{2}$$
97 $$1 - 3.65e3T + 8.85e7T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}