# Properties

 Degree 2 Conductor 43 Sign $1$ Motivic weight 5 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.50·2-s + 16.8·3-s − 25.7·4-s + 47.4·5-s − 42.2·6-s + 67.4·7-s + 144.·8-s + 42.2·9-s − 118.·10-s + 81.3·11-s − 434.·12-s + 1.05e3·13-s − 168.·14-s + 801.·15-s + 462.·16-s + 251.·17-s − 105.·18-s + 1.61e3·19-s − 1.22e3·20-s + 1.13e3·21-s − 203.·22-s − 32.7·23-s + 2.43e3·24-s − 872.·25-s − 2.64e3·26-s − 3.39e3·27-s − 1.73e3·28-s + ⋯
 L(s)  = 1 − 0.441·2-s + 1.08·3-s − 0.804·4-s + 0.849·5-s − 0.478·6-s + 0.520·7-s + 0.797·8-s + 0.173·9-s − 0.375·10-s + 0.202·11-s − 0.871·12-s + 1.73·13-s − 0.229·14-s + 0.919·15-s + 0.452·16-s + 0.210·17-s − 0.0768·18-s + 1.02·19-s − 0.683·20-s + 0.563·21-s − 0.0896·22-s − 0.0129·23-s + 0.864·24-s − 0.279·25-s − 0.767·26-s − 0.895·27-s − 0.418·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $1$ motivic weight = $$5$$ character : $\chi_{43} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :5/2),\ 1)$$ $$L(3)$$ $$\approx$$ $$1.87525$$ $$L(\frac12)$$ $$\approx$$ $$1.87525$$ $$L(\frac{7}{2})$$ 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 - 1.84e3T$$
good2 $$1 + 2.50T + 32T^{2}$$
3 $$1 - 16.8T + 243T^{2}$$
5 $$1 - 47.4T + 3.12e3T^{2}$$
7 $$1 - 67.4T + 1.68e4T^{2}$$
11 $$1 - 81.3T + 1.61e5T^{2}$$
13 $$1 - 1.05e3T + 3.71e5T^{2}$$
17 $$1 - 251.T + 1.41e6T^{2}$$
19 $$1 - 1.61e3T + 2.47e6T^{2}$$
23 $$1 + 32.7T + 6.43e6T^{2}$$
29 $$1 + 2.58e3T + 2.05e7T^{2}$$
31 $$1 + 7.20e3T + 2.86e7T^{2}$$
37 $$1 + 6.17e3T + 6.93e7T^{2}$$
41 $$1 - 1.55e4T + 1.15e8T^{2}$$
47 $$1 + 1.69e3T + 2.29e8T^{2}$$
53 $$1 + 2.56e4T + 4.18e8T^{2}$$
59 $$1 - 2.45e4T + 7.14e8T^{2}$$
61 $$1 - 8.20e3T + 8.44e8T^{2}$$
67 $$1 + 1.23e4T + 1.35e9T^{2}$$
71 $$1 - 1.87e4T + 1.80e9T^{2}$$
73 $$1 + 1.21e4T + 2.07e9T^{2}$$
79 $$1 + 5.23e4T + 3.07e9T^{2}$$
83 $$1 - 2.89e4T + 3.93e9T^{2}$$
89 $$1 - 1.17e5T + 5.58e9T^{2}$$
97 $$1 + 1.47e5T + 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}