# Properties

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

# Origins

## Dirichlet series

 L(s)  = 1 + 4·4-s + 9·9-s − 21·11-s − 17·13-s + 16·16-s − 9·17-s + 3·23-s + 25·25-s + 19·31-s + 36·36-s + 39·41-s − 43·43-s − 84·44-s − 78·47-s + 49·49-s − 68·52-s + 63·53-s − 54·59-s + 64·64-s + 91·67-s − 36·68-s − 14·79-s + 81·81-s + 123·83-s + 12·92-s − 193·97-s − 189·99-s + ⋯
 L(s)  = 1 + 4-s + 9-s − 1.90·11-s − 1.30·13-s + 16-s − 0.529·17-s + 3/23·23-s + 25-s + 0.612·31-s + 36-s + 0.951·41-s − 43-s − 1.90·44-s − 1.65·47-s + 49-s − 1.30·52-s + 1.18·53-s − 0.915·59-s + 64-s + 1.35·67-s − 0.529·68-s − 0.177·79-s + 81-s + 1.48·83-s + 3/23·92-s − 1.98·97-s − 1.90·99-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : $\chi_{43} (42, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :1),\ 1)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$1.23364$$ $$L(\frac12)$$ $$\approx$$ $$1.23364$$ $$L(2)$$ 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 + p T$$
good2 $$( 1 - p T )( 1 + p T )$$
3 $$( 1 - p T )( 1 + p T )$$
5 $$( 1 - p T )( 1 + p T )$$
7 $$( 1 - p T )( 1 + p T )$$
11 $$1 + 21 T + p^{2} T^{2}$$
13 $$1 + 17 T + p^{2} T^{2}$$
17 $$1 + 9 T + p^{2} T^{2}$$
19 $$( 1 - p T )( 1 + p T )$$
23 $$1 - 3 T + p^{2} T^{2}$$
29 $$( 1 - p T )( 1 + p T )$$
31 $$1 - 19 T + p^{2} T^{2}$$
37 $$( 1 - p T )( 1 + p T )$$
41 $$1 - 39 T + p^{2} T^{2}$$
47 $$1 + 78 T + p^{2} T^{2}$$
53 $$1 - 63 T + p^{2} T^{2}$$
59 $$1 + 54 T + p^{2} T^{2}$$
61 $$( 1 - p T )( 1 + p T )$$
67 $$1 - 91 T + p^{2} T^{2}$$
71 $$( 1 - p T )( 1 + p T )$$
73 $$( 1 - p T )( 1 + p T )$$
79 $$1 + 14 T + p^{2} T^{2}$$
83 $$1 - 123 T + p^{2} T^{2}$$
89 $$( 1 - p T )( 1 + p T )$$
97 $$1 + 193 T + p^{2} T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}