# Properties

 Degree 2 Conductor $2^{3} \cdot 3^{2} \cdot 5^{2}$ Sign $0.816 - 0.577i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 7-s + 1.41i·11-s − 13-s + 1.41i·17-s + 19-s − 1.41i·23-s + 1.41i·29-s + 31-s − 43-s − 1.41i·47-s − 1.41i·59-s + 61-s + 67-s + 1.41i·77-s + 1.41i·83-s + ⋯
 L(s)  = 1 + 7-s + 1.41i·11-s − 13-s + 1.41i·17-s + 19-s − 1.41i·23-s + 1.41i·29-s + 31-s − 43-s − 1.41i·47-s − 1.41i·59-s + 61-s + 67-s + 1.41i·77-s + 1.41i·83-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1800$$    =    $$2^{3} \cdot 3^{2} \cdot 5^{2}$$ $$\varepsilon$$ = $0.816 - 0.577i$ motivic weight = $$0$$ character : $\chi_{1800} (1601, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 1800,\ (\ :0),\ 0.816 - 0.577i)$ $L(\frac{1}{2})$ $\approx$ $1.230051638$ $L(\frac12)$ $\approx$ $1.230051638$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;5\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
3 $$1$$
5 $$1$$
good7 $$1 - T + T^{2}$$
11 $$1 - 1.41iT - T^{2}$$
13 $$1 + T + T^{2}$$
17 $$1 - 1.41iT - T^{2}$$
19 $$1 - T + T^{2}$$
23 $$1 + 1.41iT - T^{2}$$
29 $$1 - 1.41iT - T^{2}$$
31 $$1 - T + T^{2}$$
37 $$1 + T^{2}$$
41 $$1 - T^{2}$$
43 $$1 + T + T^{2}$$
47 $$1 + 1.41iT - T^{2}$$
53 $$1 - T^{2}$$
59 $$1 + 1.41iT - T^{2}$$
61 $$1 - T + T^{2}$$
67 $$1 - T + T^{2}$$
71 $$1 - T^{2}$$
73 $$1 + T^{2}$$
79 $$1 + T^{2}$$
83 $$1 - 1.41iT - T^{2}$$
89 $$1 - T^{2}$$
97 $$1 + T + T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}