# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{2} \cdot 7^{2}$ Sign $0.605 + 0.795i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + 0.999i·8-s + (−1.73 − i)11-s + (−0.5 − 0.866i)16-s + 1.99·22-s + (1.73 − i)23-s + (0.5 − 0.866i)25-s + (0.866 + 0.499i)32-s + (−1 − 1.73i)37-s + (−1.73 + 0.999i)44-s + (−0.999 + 1.73i)46-s + 0.999i·50-s − 0.999·64-s − 2i·71-s + ⋯
 L(s)  = 1 + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + 0.999i·8-s + (−1.73 − i)11-s + (−0.5 − 0.866i)16-s + 1.99·22-s + (1.73 − i)23-s + (0.5 − 0.866i)25-s + (0.866 + 0.499i)32-s + (−1 − 1.73i)37-s + (−1.73 + 0.999i)44-s + (−0.999 + 1.73i)46-s + 0.999i·50-s − 0.999·64-s − 2i·71-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1764$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{2}$$ $$\varepsilon$$ = $0.605 + 0.795i$ motivic weight = $$0$$ character : $\chi_{1764} (1243, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 1764,\ (\ :0),\ 0.605 + 0.795i)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.5869705535$$ $$L(\frac12)$$ $$\approx$$ $$0.5869705535$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$$F_p(T)$$ is a polynomial of degree 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 + (0.866 - 0.5i)T$$
3 $$1$$
7 $$1$$
good5 $$1 + (-0.5 + 0.866i)T^{2}$$
11 $$1 + (1.73 + i)T + (0.5 + 0.866i)T^{2}$$
13 $$1 + T^{2}$$
17 $$1 + (-0.5 - 0.866i)T^{2}$$
19 $$1 + (0.5 - 0.866i)T^{2}$$
23 $$1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2}$$
29 $$1 + T^{2}$$
31 $$1 + (0.5 + 0.866i)T^{2}$$
37 $$1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}$$
41 $$1 + T^{2}$$
43 $$1 - T^{2}$$
47 $$1 + (0.5 - 0.866i)T^{2}$$
53 $$1 + (-0.5 - 0.866i)T^{2}$$
59 $$1 + (0.5 + 0.866i)T^{2}$$
61 $$1 + (-0.5 + 0.866i)T^{2}$$
67 $$1 + (0.5 + 0.866i)T^{2}$$
71 $$1 + 2iT - T^{2}$$
73 $$1 + (-0.5 - 0.866i)T^{2}$$
79 $$1 + (0.5 - 0.866i)T^{2}$$
83 $$1 - T^{2}$$
89 $$1 + (-0.5 + 0.866i)T^{2}$$
97 $$1 + T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}