# Properties

 Degree 2 Conductor 43 Sign $-0.836 - 0.548i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 6.72i·2-s − 8.31i·3-s − 29.1·4-s + 1.48i·5-s − 55.9·6-s − 13.7i·7-s + 88.6i·8-s + 11.7·9-s + 9.96·10-s − 10.2·11-s + 242. i·12-s + 98.4·13-s − 92.7·14-s + 12.3·15-s + 128.·16-s − 286.·17-s + ⋯
 L(s)  = 1 − 1.68i·2-s − 0.924i·3-s − 1.82·4-s + 0.0592i·5-s − 1.55·6-s − 0.281i·7-s + 1.38i·8-s + 0.145·9-s + 0.0996·10-s − 0.0843·11-s + 1.68i·12-s + 0.582·13-s − 0.473·14-s + 0.0548·15-s + 0.503·16-s − 0.991·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $-0.836 - 0.548i$ motivic weight = $$4$$ character : $\chi_{43} (42, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :2),\ -0.836 - 0.548i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$0.357835 + 1.19851i$$ $$L(\frac12)$$ $$\approx$$ $$0.357835 + 1.19851i$$ $$L(3)$$ 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 + (1.54e3 + 1.01e3i)T$$
good2 $$1 + 6.72iT - 16T^{2}$$
3 $$1 + 8.31iT - 81T^{2}$$
5 $$1 - 1.48iT - 625T^{2}$$
7 $$1 + 13.7iT - 2.40e3T^{2}$$
11 $$1 + 10.2T + 1.46e4T^{2}$$
13 $$1 - 98.4T + 2.85e4T^{2}$$
17 $$1 + 286.T + 8.35e4T^{2}$$
19 $$1 + 367. iT - 1.30e5T^{2}$$
23 $$1 + 242.T + 2.79e5T^{2}$$
29 $$1 + 1.14e3iT - 7.07e5T^{2}$$
31 $$1 - 895.T + 9.23e5T^{2}$$
37 $$1 - 2.29e3iT - 1.87e6T^{2}$$
41 $$1 - 1.69e3T + 2.82e6T^{2}$$
47 $$1 - 743.T + 4.87e6T^{2}$$
53 $$1 - 99.3T + 7.89e6T^{2}$$
59 $$1 - 3.28e3T + 1.21e7T^{2}$$
61 $$1 - 3.22e3iT - 1.38e7T^{2}$$
67 $$1 - 5.55e3T + 2.01e7T^{2}$$
71 $$1 + 2.95e3iT - 2.54e7T^{2}$$
73 $$1 - 3.61e3iT - 2.83e7T^{2}$$
79 $$1 - 2.38e3T + 3.89e7T^{2}$$
83 $$1 + 6.42e3T + 4.74e7T^{2}$$
89 $$1 + 3.29e3iT - 6.27e7T^{2}$$
97 $$1 + 9.32e3T + 8.85e7T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}