Properties

 Degree 2 Conductor $2^{2} \cdot 5$ Sign $0.890 + 0.455i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (2.70 − 0.813i)2-s + (−2.61 − 2.61i)3-s + (6.67 − 4.40i)4-s + (−0.435 + 11.1i)5-s + (−9.22 − 4.96i)6-s + (−17.7 + 17.7i)7-s + (14.4 − 17.3i)8-s − 13.2i·9-s + (7.91 + 30.6i)10-s − 7.37i·11-s + (−29.0 − 5.93i)12-s + (−2.68 + 2.68i)13-s + (−33.6 + 62.6i)14-s + (30.3 − 28.1i)15-s + (25.1 − 58.8i)16-s + (−20.2 − 20.2i)17-s + ⋯
 L(s)  = 1 + (0.957 − 0.287i)2-s + (−0.503 − 0.503i)3-s + (0.834 − 0.551i)4-s + (−0.0389 + 0.999i)5-s + (−0.627 − 0.337i)6-s + (−0.959 + 0.959i)7-s + (0.640 − 0.767i)8-s − 0.492i·9-s + (0.250 + 0.968i)10-s − 0.202i·11-s + (−0.698 − 0.142i)12-s + (−0.0572 + 0.0572i)13-s + (−0.643 + 1.19i)14-s + (0.523 − 0.483i)15-s + (0.392 − 0.919i)16-s + (−0.288 − 0.288i)17-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$20$$    =    $$2^{2} \cdot 5$$ $$\varepsilon$$ = $0.890 + 0.455i$ motivic weight = $$3$$ character : $\chi_{20} (7, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 20,\ (\ :3/2),\ 0.890 + 0.455i)$ $L(2)$ $\approx$ $1.41729 - 0.341235i$ $L(\frac12)$ $\approx$ $1.41729 - 0.341235i$ $L(\frac{5}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5\}$, $$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-2.70 + 0.813i)T$$
5 $$1 + (0.435 - 11.1i)T$$
good3 $$1 + (2.61 + 2.61i)T + 27iT^{2}$$
7 $$1 + (17.7 - 17.7i)T - 343iT^{2}$$
11 $$1 + 7.37iT - 1.33e3T^{2}$$
13 $$1 + (2.68 - 2.68i)T - 2.19e3iT^{2}$$
17 $$1 + (20.2 + 20.2i)T + 4.91e3iT^{2}$$
19 $$1 - 135.T + 6.85e3T^{2}$$
23 $$1 + (71.0 + 71.0i)T + 1.21e4iT^{2}$$
29 $$1 + 34.2iT - 2.43e4T^{2}$$
31 $$1 - 187. iT - 2.97e4T^{2}$$
37 $$1 + (-250. - 250. i)T + 5.06e4iT^{2}$$
41 $$1 + 211.T + 6.89e4T^{2}$$
43 $$1 + (-46.7 - 46.7i)T + 7.95e4iT^{2}$$
47 $$1 + (-189. + 189. i)T - 1.03e5iT^{2}$$
53 $$1 + (74.5 - 74.5i)T - 1.48e5iT^{2}$$
59 $$1 - 101.T + 2.05e5T^{2}$$
61 $$1 - 232.T + 2.26e5T^{2}$$
67 $$1 + (34.7 - 34.7i)T - 3.00e5iT^{2}$$
71 $$1 + 614. iT - 3.57e5T^{2}$$
73 $$1 + (37.4 - 37.4i)T - 3.89e5iT^{2}$$
79 $$1 + 1.00e3T + 4.93e5T^{2}$$
83 $$1 + (423. + 423. i)T + 5.71e5iT^{2}$$
89 $$1 - 1.04e3iT - 7.04e5T^{2}$$
97 $$1 + (536. + 536. i)T + 9.12e5iT^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}