# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{2} \cdot 5^{2}$ Sign $0.584 - 0.811i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + i·2-s − 4-s − i·8-s + 16-s + (1 − i)17-s + (−1 + i)19-s + (1 − i)23-s + i·32-s + (1 + i)34-s + (−1 − i)38-s + (1 + i)46-s + (1 − i)47-s − i·49-s + 2·53-s + (1 + i)61-s + ⋯
 L(s)  = 1 + i·2-s − 4-s − i·8-s + 16-s + (1 − i)17-s + (−1 + i)19-s + (1 − i)23-s + i·32-s + (1 + i)34-s + (−1 − i)38-s + (1 + i)46-s + (1 − i)47-s − i·49-s + 2·53-s + (1 + i)61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3600$$    =    $$2^{4} \cdot 3^{2} \cdot 5^{2}$$ $$\varepsilon$$ = $0.584 - 0.811i$ motivic weight = $$0$$ character : $\chi_{3600} (1693, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 3600,\ (\ :0),\ 0.584 - 0.811i)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$1.144775467$$ $$L(\frac12)$$ $$\approx$$ $$1.144775467$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 - iT$$
3 $$1$$
5 $$1$$
good7 $$1 + iT^{2}$$
11 $$1 - iT^{2}$$
13 $$1 - T^{2}$$
17 $$1 + (-1 + i)T - iT^{2}$$
19 $$1 + (1 - i)T - iT^{2}$$
23 $$1 + (-1 + i)T - iT^{2}$$
29 $$1 - iT^{2}$$
31 $$1 + T^{2}$$
37 $$1 - T^{2}$$
41 $$1 - T^{2}$$
43 $$1 + T^{2}$$
47 $$1 + (-1 + i)T - iT^{2}$$
53 $$1 - 2T + T^{2}$$
59 $$1 + iT^{2}$$
61 $$1 + (-1 - i)T + iT^{2}$$
67 $$1 + T^{2}$$
71 $$1 - T^{2}$$
73 $$1 - iT^{2}$$
79 $$1 - 2iT - T^{2}$$
83 $$1 - 2iT - T^{2}$$
89 $$1 + T^{2}$$
97 $$1 - iT^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}