# Properties

 Degree 2 Conductor 43 Sign $-1$ Motivic weight 9 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − 9.62·2-s + 189.·3-s − 419.·4-s + 636.·5-s − 1.82e3·6-s − 3.28e3·7-s + 8.96e3·8-s + 1.63e4·9-s − 6.12e3·10-s − 7.18e4·11-s − 7.96e4·12-s − 5.64e4·13-s + 3.16e4·14-s + 1.20e5·15-s + 1.28e5·16-s + 2.27e5·17-s − 1.57e5·18-s − 2.58e5·19-s − 2.66e5·20-s − 6.24e5·21-s + 6.92e5·22-s − 2.05e5·23-s + 1.70e6·24-s − 1.54e6·25-s + 5.44e5·26-s − 6.26e5·27-s + 1.37e6·28-s + ⋯
 L(s)  = 1 − 0.425·2-s + 1.35·3-s − 0.818·4-s + 0.455·5-s − 0.576·6-s − 0.517·7-s + 0.774·8-s + 0.832·9-s − 0.193·10-s − 1.48·11-s − 1.10·12-s − 0.548·13-s + 0.220·14-s + 0.616·15-s + 0.489·16-s + 0.660·17-s − 0.354·18-s − 0.454·19-s − 0.372·20-s − 0.700·21-s + 0.630·22-s − 0.153·23-s + 1.04·24-s − 0.792·25-s + 0.233·26-s − 0.226·27-s + 0.423·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $-1$ motivic weight = $$9$$ character : $\chi_{43} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$1$$ Selberg data = $$(2,\ 43,\ (\ :9/2),\ -1)$$ $$L(5)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{11}{2})$$ 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 + 3.41e6T$$
good2 $$1 + 9.62T + 512T^{2}$$
3 $$1 - 189.T + 1.96e4T^{2}$$
5 $$1 - 636.T + 1.95e6T^{2}$$
7 $$1 + 3.28e3T + 4.03e7T^{2}$$
11 $$1 + 7.18e4T + 2.35e9T^{2}$$
13 $$1 + 5.64e4T + 1.06e10T^{2}$$
17 $$1 - 2.27e5T + 1.18e11T^{2}$$
19 $$1 + 2.58e5T + 3.22e11T^{2}$$
23 $$1 + 2.05e5T + 1.80e12T^{2}$$
29 $$1 - 5.68e5T + 1.45e13T^{2}$$
31 $$1 + 6.46e6T + 2.64e13T^{2}$$
37 $$1 + 1.41e7T + 1.29e14T^{2}$$
41 $$1 + 1.35e7T + 3.27e14T^{2}$$
47 $$1 - 2.14e7T + 1.11e15T^{2}$$
53 $$1 - 3.02e7T + 3.29e15T^{2}$$
59 $$1 - 1.16e8T + 8.66e15T^{2}$$
61 $$1 + 1.09e8T + 1.16e16T^{2}$$
67 $$1 - 1.24e8T + 2.72e16T^{2}$$
71 $$1 - 2.04e8T + 4.58e16T^{2}$$
73 $$1 - 1.16e8T + 5.88e16T^{2}$$
79 $$1 - 4.47e8T + 1.19e17T^{2}$$
83 $$1 + 5.31e8T + 1.86e17T^{2}$$
89 $$1 - 2.15e7T + 3.50e17T^{2}$$
97 $$1 - 9.39e8T + 7.60e17T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}