# Properties

 Degree 2 Conductor $2^{2} \cdot 5$ Sign $0.995 - 0.0898i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1 + i)3-s + (−3 − 4i)5-s + (−7 + 7i)7-s − 7i·9-s + 10·11-s + (9 + 9i)13-s + (1 − 7i)15-s + (1 − i)17-s + 8i·19-s − 14·21-s + (−23 − 23i)23-s + (−7 + 24i)25-s + (16 − 16i)27-s + 8i·29-s − 14·31-s + ⋯
 L(s)  = 1 + (0.333 + 0.333i)3-s + (−0.600 − 0.800i)5-s + (−1 + i)7-s − 0.777i·9-s + 0.909·11-s + (0.692 + 0.692i)13-s + (0.0666 − 0.466i)15-s + (0.0588 − 0.0588i)17-s + 0.421i·19-s − 0.666·21-s + (−1 − i)23-s + (−0.280 + 0.959i)25-s + (0.592 − 0.592i)27-s + 0.275i·29-s − 0.451·31-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0898i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0898i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$20$$    =    $$2^{2} \cdot 5$$ $$\varepsilon$$ = $0.995 - 0.0898i$ motivic weight = $$2$$ character : $\chi_{20} (13, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 20,\ (\ :1),\ 0.995 - 0.0898i)$ $L(\frac{3}{2})$ $\approx$ $0.855497 + 0.0384920i$ $L(\frac12)$ $\approx$ $0.855497 + 0.0384920i$ $L(2)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5\}$, $$F_p$$ 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 + (3 + 4i)T$$
good3 $$1 + (-1 - i)T + 9iT^{2}$$
7 $$1 + (7 - 7i)T - 49iT^{2}$$
11 $$1 - 10T + 121T^{2}$$
13 $$1 + (-9 - 9i)T + 169iT^{2}$$
17 $$1 + (-1 + i)T - 289iT^{2}$$
19 $$1 - 8iT - 361T^{2}$$
23 $$1 + (23 + 23i)T + 529iT^{2}$$
29 $$1 - 8iT - 841T^{2}$$
31 $$1 + 14T + 961T^{2}$$
37 $$1 + (-33 + 33i)T - 1.36e3iT^{2}$$
41 $$1 + 14T + 1.68e3T^{2}$$
43 $$1 + (15 + 15i)T + 1.84e3iT^{2}$$
47 $$1 + (39 - 39i)T - 2.20e3iT^{2}$$
53 $$1 + (7 + 7i)T + 2.80e3iT^{2}$$
59 $$1 - 56iT - 3.48e3T^{2}$$
61 $$1 - 42T + 3.72e3T^{2}$$
67 $$1 + (7 - 7i)T - 4.48e3iT^{2}$$
71 $$1 - 98T + 5.04e3T^{2}$$
73 $$1 + (-49 - 49i)T + 5.32e3iT^{2}$$
79 $$1 + 96iT - 6.24e3T^{2}$$
83 $$1 + (63 + 63i)T + 6.88e3iT^{2}$$
89 $$1 - 112iT - 7.92e3T^{2}$$
97 $$1 + (-33 + 33i)T - 9.40e3iT^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}