# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{2} \cdot 7^{2}$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 + 4·11-s + 8·23-s − 5·25-s − 2·29-s − 6·37-s + 12·43-s + 10·53-s − 4·67-s + 16·71-s − 8·79-s − 20·107-s + 18·109-s − 2·113-s + ⋯
 L(s)  = 1 + 1.20·11-s + 1.66·23-s − 25-s − 0.371·29-s − 0.986·37-s + 1.82·43-s + 1.37·53-s − 0.488·67-s + 1.89·71-s − 0.900·79-s − 1.93·107-s + 1.72·109-s − 0.188·113-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$7056$$    =    $$2^{4} \cdot 3^{2} \cdot 7^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{7056} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 7056,\ (\ :1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$2.232396067$$ $$L(\frac12)$$ $$\approx$$ $$2.232396067$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
7 $$1$$
good5 $$1 + p T^{2}$$
11 $$1 - 4 T + p T^{2}$$
13 $$1 + p T^{2}$$
17 $$1 + p T^{2}$$
19 $$1 + p T^{2}$$
23 $$1 - 8 T + p T^{2}$$
29 $$1 + 2 T + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 + 6 T + p T^{2}$$
41 $$1 + p T^{2}$$
43 $$1 - 12 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 - 10 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 + p T^{2}$$
67 $$1 + 4 T + p T^{2}$$
71 $$1 - 16 T + p T^{2}$$
73 $$1 + p T^{2}$$
79 $$1 + 8 T + p T^{2}$$
83 $$1 + p T^{2}$$
89 $$1 + p T^{2}$$
97 $$1 + p T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}