# 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 + 5.38·2-s + 25.0·3-s − 2.96·4-s − 0.456·5-s + 134.·6-s + 166.·7-s − 188.·8-s + 384.·9-s − 2.46·10-s − 65.6·11-s − 74.3·12-s − 689.·13-s + 897.·14-s − 11.4·15-s − 920.·16-s + 737.·17-s + 2.07e3·18-s − 609.·19-s + 1.35·20-s + 4.17e3·21-s − 353.·22-s + 1.31e3·23-s − 4.71e3·24-s − 3.12e3·25-s − 3.71e3·26-s + 3.53e3·27-s − 494.·28-s + ⋯
 L(s)  = 1 + 0.952·2-s + 1.60·3-s − 0.0927·4-s − 0.00816·5-s + 1.53·6-s + 1.28·7-s − 1.04·8-s + 1.58·9-s − 0.00778·10-s − 0.163·11-s − 0.148·12-s − 1.13·13-s + 1.22·14-s − 0.0131·15-s − 0.898·16-s + 0.618·17-s + 1.50·18-s − 0.387·19-s + 0.000757·20-s + 2.06·21-s − 0.155·22-s + 0.517·23-s − 1.67·24-s − 0.999·25-s − 1.07·26-s + 0.934·27-s − 0.119·28-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 = $$(2,\ 43,\ (\ :5/2),\ 1)$$ $$L(3)$$ $$\approx$$ $$3.73317$$ $$L(\frac12)$$ $$\approx$$ $$3.73317$$ $$L(\frac{7}{2})$$ 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 - 5.38T + 32T^{2}$$
3 $$1 - 25.0T + 243T^{2}$$
5 $$1 + 0.456T + 3.12e3T^{2}$$
7 $$1 - 166.T + 1.68e4T^{2}$$
11 $$1 + 65.6T + 1.61e5T^{2}$$
13 $$1 + 689.T + 3.71e5T^{2}$$
17 $$1 - 737.T + 1.41e6T^{2}$$
19 $$1 + 609.T + 2.47e6T^{2}$$
23 $$1 - 1.31e3T + 6.43e6T^{2}$$
29 $$1 + 8.96e3T + 2.05e7T^{2}$$
31 $$1 - 5.85e3T + 2.86e7T^{2}$$
37 $$1 + 55.9T + 6.93e7T^{2}$$
41 $$1 + 1.04e4T + 1.15e8T^{2}$$
47 $$1 - 1.94e3T + 2.29e8T^{2}$$
53 $$1 - 3.01e4T + 4.18e8T^{2}$$
59 $$1 - 5.21e4T + 7.14e8T^{2}$$
61 $$1 + 1.40e4T + 8.44e8T^{2}$$
67 $$1 - 5.14e4T + 1.35e9T^{2}$$
71 $$1 - 1.12e4T + 1.80e9T^{2}$$
73 $$1 + 6.11e4T + 2.07e9T^{2}$$
79 $$1 - 9.60e4T + 3.07e9T^{2}$$
83 $$1 - 3.03e4T + 3.93e9T^{2}$$
89 $$1 + 6.50e4T + 5.58e9T^{2}$$
97 $$1 - 1.20e4T + 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}