# Properties

 Degree 2 Conductor $2^{5} \cdot 5^{2}$ Sign $-0.229 - 0.973i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1 + i)3-s + (1 + i)7-s − i·9-s − 2·21-s + (−1 + i)23-s + 2i·29-s + (1 − i)43-s + (−1 − i)47-s + i·49-s + (1 − i)63-s + (1 + i)67-s − 2i·69-s + 81-s + (1 − i)83-s + (−2 − 2i)87-s + ⋯
 L(s)  = 1 + (−1 + i)3-s + (1 + i)7-s − i·9-s − 2·21-s + (−1 + i)23-s + 2i·29-s + (1 − i)43-s + (−1 − i)47-s + i·49-s + (1 − i)63-s + (1 + i)67-s − 2i·69-s + 81-s + (1 − i)83-s + (−2 − 2i)87-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$800$$    =    $$2^{5} \cdot 5^{2}$$ $$\varepsilon$$ = $-0.229 - 0.973i$ motivic weight = $$0$$ character : $\chi_{800} (257, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 800,\ (\ :0),\ -0.229 - 0.973i)$ $L(\frac{1}{2})$ $\approx$ $0.7316422166$ $L(\frac12)$ $\approx$ $0.7316422166$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
5 $$1$$
good3 $$1 + (1 - i)T - iT^{2}$$
7 $$1 + (-1 - i)T + iT^{2}$$
11 $$1 + T^{2}$$
13 $$1 - iT^{2}$$
17 $$1 + iT^{2}$$
19 $$1 - T^{2}$$
23 $$1 + (1 - i)T - iT^{2}$$
29 $$1 - 2iT - T^{2}$$
31 $$1 + T^{2}$$
37 $$1 + iT^{2}$$
41 $$1 + T^{2}$$
43 $$1 + (-1 + i)T - iT^{2}$$
47 $$1 + (1 + i)T + iT^{2}$$
53 $$1 - iT^{2}$$
59 $$1 - T^{2}$$
61 $$1 + T^{2}$$
67 $$1 + (-1 - i)T + iT^{2}$$
71 $$1 + T^{2}$$
73 $$1 - iT^{2}$$
79 $$1 - T^{2}$$
83 $$1 + (-1 + i)T - iT^{2}$$
89 $$1 + 2iT - 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}