# Properties

 Degree 2 Conductor $2^{2} \cdot 5$ Sign $0.742 + 0.669i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (2 − 2i)2-s − 8i·4-s + (−2 + 11i)5-s + (−16 − 16i)8-s + 27i·9-s + (18 + 26i)10-s + (−37 − 37i)13-s − 64·16-s + (99 − 99i)17-s + (54 + 54i)18-s + (88 + 16i)20-s + (−117 − 44i)25-s − 148·26-s + 284i·29-s + (−128 + 128i)32-s + ⋯
 L(s)  = 1 + (0.707 − 0.707i)2-s − i·4-s + (−0.178 + 0.983i)5-s + (−0.707 − 0.707i)8-s + i·9-s + (0.569 + 0.822i)10-s + (−0.789 − 0.789i)13-s − 16-s + (1.41 − 1.41i)17-s + (0.707 + 0.707i)18-s + (0.983 + 0.178i)20-s + (−0.936 − 0.351i)25-s − 1.11·26-s + 1.81i·29-s + (−0.707 + 0.707i)32-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$20$$    =    $$2^{2} \cdot 5$$ $$\varepsilon$$ = $0.742 + 0.669i$ motivic weight = $$3$$ character : $\chi_{20} (3, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 20,\ (\ :3/2),\ 0.742 + 0.669i)$ $L(2)$ $\approx$ $1.31007 - 0.503184i$ $L(\frac12)$ $\approx$ $1.31007 - 0.503184i$ $L(\frac{5}{2})$ 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 + (-2 + 2i)T$$
5 $$1 + (2 - 11i)T$$
good3 $$1 - 27iT^{2}$$
7 $$1 + 343iT^{2}$$
11 $$1 - 1.33e3T^{2}$$
13 $$1 + (37 + 37i)T + 2.19e3iT^{2}$$
17 $$1 + (-99 + 99i)T - 4.91e3iT^{2}$$
19 $$1 + 6.85e3T^{2}$$
23 $$1 - 1.21e4iT^{2}$$
29 $$1 - 284iT - 2.43e4T^{2}$$
31 $$1 - 2.97e4T^{2}$$
37 $$1 + (91 - 91i)T - 5.06e4iT^{2}$$
41 $$1 - 472T + 6.89e4T^{2}$$
43 $$1 - 7.95e4iT^{2}$$
47 $$1 + 1.03e5iT^{2}$$
53 $$1 + (27 + 27i)T + 1.48e5iT^{2}$$
59 $$1 + 2.05e5T^{2}$$
61 $$1 + 468T + 2.26e5T^{2}$$
67 $$1 + 3.00e5iT^{2}$$
71 $$1 - 3.57e5T^{2}$$
73 $$1 + (-253 - 253i)T + 3.89e5iT^{2}$$
79 $$1 + 4.93e5T^{2}$$
83 $$1 - 5.71e5iT^{2}$$
89 $$1 + 176iT - 7.04e5T^{2}$$
97 $$1 + (611 - 611i)T - 9.12e5iT^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}