# 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 − 4.38·2-s + 117.·3-s − 492.·4-s − 1.23e3·5-s − 517.·6-s + 1.25e4·7-s + 4.40e3·8-s − 5.77e3·9-s + 5.42e3·10-s + 2.79e4·11-s − 5.81e4·12-s − 1.39e5·13-s − 5.50e4·14-s − 1.45e5·15-s + 2.32e5·16-s − 4.08e5·17-s + 2.53e4·18-s − 1.57e5·19-s + 6.09e5·20-s + 1.47e6·21-s − 1.22e5·22-s − 7.30e5·23-s + 5.19e5·24-s − 4.22e5·25-s + 6.12e5·26-s − 3.00e6·27-s − 6.18e6·28-s + ⋯
 L(s)  = 1 − 0.193·2-s + 0.840·3-s − 0.962·4-s − 0.885·5-s − 0.162·6-s + 1.97·7-s + 0.380·8-s − 0.293·9-s + 0.171·10-s + 0.574·11-s − 0.808·12-s − 1.35·13-s − 0.382·14-s − 0.744·15-s + 0.888·16-s − 1.18·17-s + 0.0568·18-s − 0.277·19-s + 0.851·20-s + 1.66·21-s − 0.111·22-s − 0.544·23-s + 0.319·24-s − 0.216·25-s + 0.262·26-s − 1.08·27-s − 1.90·28-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 + 4.38T + 512T^{2}$$
3 $$1 - 117.T + 1.96e4T^{2}$$
5 $$1 + 1.23e3T + 1.95e6T^{2}$$
7 $$1 - 1.25e4T + 4.03e7T^{2}$$
11 $$1 - 2.79e4T + 2.35e9T^{2}$$
13 $$1 + 1.39e5T + 1.06e10T^{2}$$
17 $$1 + 4.08e5T + 1.18e11T^{2}$$
19 $$1 + 1.57e5T + 3.22e11T^{2}$$
23 $$1 + 7.30e5T + 1.80e12T^{2}$$
29 $$1 + 6.14e6T + 1.45e13T^{2}$$
31 $$1 - 6.18e6T + 2.64e13T^{2}$$
37 $$1 + 2.16e7T + 1.29e14T^{2}$$
41 $$1 - 2.42e7T + 3.27e14T^{2}$$
47 $$1 + 3.21e7T + 1.11e15T^{2}$$
53 $$1 + 6.96e7T + 3.29e15T^{2}$$
59 $$1 + 1.12e8T + 8.66e15T^{2}$$
61 $$1 - 8.03e7T + 1.16e16T^{2}$$
67 $$1 + 5.78e7T + 2.72e16T^{2}$$
71 $$1 - 3.11e8T + 4.58e16T^{2}$$
73 $$1 + 3.23e7T + 5.88e16T^{2}$$
79 $$1 - 2.36e8T + 1.19e17T^{2}$$
83 $$1 - 4.61e8T + 1.86e17T^{2}$$
89 $$1 - 6.16e8T + 3.50e17T^{2}$$
97 $$1 + 1.00e9T + 7.60e17T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}