# Properties

 Degree 2 Conductor 43 Sign $1$ Motivic weight 10 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 + 1.02e3·4-s + 5.90e4·9-s − 1.85e4·11-s + 3.03e5·13-s + 1.04e6·16-s − 2.76e6·17-s + 4.12e6·23-s + 9.76e6·25-s + 5.72e7·31-s + 6.04e7·36-s + 1.42e8·41-s − 1.47e8·43-s − 1.89e7·44-s + 4.51e8·47-s + 2.82e8·49-s + 3.11e8·52-s − 3.39e7·53-s − 9.90e8·59-s + 1.07e9·64-s − 1.50e9·67-s − 2.83e9·68-s − 2.64e9·79-s + 3.48e9·81-s − 6.75e9·83-s + 4.22e9·92-s − 1.50e10·97-s − 1.09e9·99-s + ⋯
 L(s)  = 1 + 4-s + 9-s − 0.114·11-s + 0.818·13-s + 16-s − 1.94·17-s + 0.641·23-s + 25-s + 1.99·31-s + 36-s + 1.23·41-s − 43-s − 0.114·44-s + 1.96·47-s + 49-s + 0.818·52-s − 0.0812·53-s − 1.38·59-s + 64-s − 1.11·67-s − 1.94·68-s − 0.858·79-s + 81-s − 1.71·83-s + 0.641·92-s − 1.74·97-s − 0.114·99-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $1$ motivic weight = $$10$$ character : $\chi_{43} (42, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :5),\ 1)$$ $$L(\frac{11}{2})$$ $$\approx$$ $$2.80969$$ $$L(\frac12)$$ $$\approx$$ $$2.80969$$ $$L(6)$$ 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 + p^{5} T$$
good2 $$( 1 - p^{5} T )( 1 + p^{5} T )$$
3 $$( 1 - p^{5} T )( 1 + p^{5} T )$$
5 $$( 1 - p^{5} T )( 1 + p^{5} T )$$
7 $$( 1 - p^{5} T )( 1 + p^{5} T )$$
11 $$1 + 18501 T + p^{10} T^{2}$$
13 $$1 - 303943 T + p^{10} T^{2}$$
17 $$1 + 2764089 T + p^{10} T^{2}$$
19 $$( 1 - p^{5} T )( 1 + p^{5} T )$$
23 $$1 - 4126443 T + p^{10} T^{2}$$
29 $$( 1 - p^{5} T )( 1 + p^{5} T )$$
31 $$1 - 57253099 T + p^{10} T^{2}$$
37 $$( 1 - p^{5} T )( 1 + p^{5} T )$$
41 $$1 - 142671399 T + p^{10} T^{2}$$
47 $$1 - 451176882 T + p^{10} T^{2}$$
53 $$1 + 33972057 T + p^{10} T^{2}$$
59 $$1 + 990191574 T + p^{10} T^{2}$$
61 $$( 1 - p^{5} T )( 1 + p^{5} T )$$
67 $$1 + 1504819589 T + p^{10} T^{2}$$
71 $$( 1 - p^{5} T )( 1 + p^{5} T )$$
73 $$( 1 - p^{5} T )( 1 + p^{5} T )$$
79 $$1 + 2641416974 T + p^{10} T^{2}$$
83 $$1 + 6757639557 T + p^{10} T^{2}$$
89 $$( 1 - p^{5} T )( 1 + p^{5} T )$$
97 $$1 + 15006753793 T + p^{10} T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}