# Properties

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

# Origins

## Dirichlet series

 L(s)  = 1 + 5·13-s − 19-s − 5·25-s + 11·31-s + 11·37-s − 13·43-s + 14·61-s + 5·67-s + 17·73-s + 17·79-s + 14·97-s − 13·103-s − 19·109-s + ⋯
 L(s)  = 1 + 1.38·13-s − 0.229·19-s − 25-s + 1.97·31-s + 1.80·37-s − 1.98·43-s + 1.79·61-s + 0.610·67-s + 1.98·73-s + 1.91·79-s + 1.42·97-s − 1.28·103-s − 1.81·109-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1764$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{1764} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 1764,\ (\ :1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$1.824793164$$ $$L(\frac12)$$ $$\approx$$ $$1.824793164$$ $$L(\frac{3}{2})$$ 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 + p T^{2}$$
13 $$1 - 5 T + p T^{2}$$
17 $$1 + p T^{2}$$
19 $$1 + T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 + p T^{2}$$
31 $$1 - 11 T + p T^{2}$$
37 $$1 - 11 T + p T^{2}$$
41 $$1 + p T^{2}$$
43 $$1 + 13 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 - 14 T + p T^{2}$$
67 $$1 - 5 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 - 17 T + p T^{2}$$
79 $$1 - 17 T + p T^{2}$$
83 $$1 + p T^{2}$$
89 $$1 + p T^{2}$$
97 $$1 - 14 T + p T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}