# 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 − 8.86·2-s + 1.50·3-s + 46.5·4-s − 37.8·5-s − 13.3·6-s − 124.·7-s − 129.·8-s − 240.·9-s + 335.·10-s + 590.·11-s + 69.9·12-s + 434.·13-s + 1.10e3·14-s − 56.8·15-s − 343.·16-s + 1.92e3·17-s + 2.13e3·18-s + 654.·19-s − 1.76e3·20-s − 187.·21-s − 5.23e3·22-s + 2.80e3·23-s − 194.·24-s − 1.69e3·25-s − 3.85e3·26-s − 726.·27-s − 5.81e3·28-s + ⋯
 L(s)  = 1 − 1.56·2-s + 0.0963·3-s + 1.45·4-s − 0.676·5-s − 0.150·6-s − 0.962·7-s − 0.714·8-s − 0.990·9-s + 1.06·10-s + 1.47·11-s + 0.140·12-s + 0.713·13-s + 1.50·14-s − 0.0651·15-s − 0.335·16-s + 1.61·17-s + 1.55·18-s + 0.415·19-s − 0.985·20-s − 0.0926·21-s − 2.30·22-s + 1.10·23-s − 0.0688·24-s − 0.542·25-s − 1.11·26-s − 0.191·27-s − 1.40·28-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 = $$(2,\ 43,\ (\ :5/2),\ 1)$$ $$L(3)$$ $$\approx$$ $$0.605902$$ $$L(\frac12)$$ $$\approx$$ $$0.605902$$ $$L(\frac{7}{2})$$ 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 + 8.86T + 32T^{2}$$
3 $$1 - 1.50T + 243T^{2}$$
5 $$1 + 37.8T + 3.12e3T^{2}$$
7 $$1 + 124.T + 1.68e4T^{2}$$
11 $$1 - 590.T + 1.61e5T^{2}$$
13 $$1 - 434.T + 3.71e5T^{2}$$
17 $$1 - 1.92e3T + 1.41e6T^{2}$$
19 $$1 - 654.T + 2.47e6T^{2}$$
23 $$1 - 2.80e3T + 6.43e6T^{2}$$
29 $$1 + 1.45e3T + 2.05e7T^{2}$$
31 $$1 - 4.41e3T + 2.86e7T^{2}$$
37 $$1 - 3.75e3T + 6.93e7T^{2}$$
41 $$1 - 1.97e3T + 1.15e8T^{2}$$
47 $$1 - 2.20e3T + 2.29e8T^{2}$$
53 $$1 - 2.49e4T + 4.18e8T^{2}$$
59 $$1 + 4.27e4T + 7.14e8T^{2}$$
61 $$1 + 2.10e4T + 8.44e8T^{2}$$
67 $$1 + 2.52e4T + 1.35e9T^{2}$$
71 $$1 - 4.80e4T + 1.80e9T^{2}$$
73 $$1 - 5.88e4T + 2.07e9T^{2}$$
79 $$1 - 9.27e4T + 3.07e9T^{2}$$
83 $$1 + 1.84e3T + 3.93e9T^{2}$$
89 $$1 + 7.03e4T + 5.58e9T^{2}$$
97 $$1 + 8.88e4T + 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}