# Properties

 Degree 2 Conductor $3^{2} \cdot 43$ Sign $1$ Motivic weight 2 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 + 4·4-s + 21·11-s − 17·13-s + 16·16-s + 9·17-s − 3·23-s + 25·25-s + 19·31-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 − 123·83-s − 12·92-s − 193·97-s + 100·100-s − 159·101-s − 181·103-s − 42·107-s + ⋯
 L(s)  = 1 + 4-s + 1.90·11-s − 1.30·13-s + 16-s + 9/17·17-s − 0.130·23-s + 25-s + 0.612·31-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 + 9/17·68-s − 0.177·79-s − 1.48·83-s − 0.130·92-s − 1.98·97-s + 100-s − 1.57·101-s − 1.75·103-s − 0.392·107-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$387$$    =    $$3^{2} \cdot 43$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : $\chi_{387} (343, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 387,\ (\ :1),\ 1)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$2.30508$$ $$L(\frac12)$$ $$\approx$$ $$2.30508$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;43\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{3,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1$$
43 $$1 + p T$$
good2 $$( 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}