Properties

 Degree 2 Conductor $2^{2} \cdot 5$ Sign $0.309 + 0.951i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (−1.61 − 1.17i)2-s − 3.80i·3-s + (1.23 + 3.80i)4-s + 2.23·5-s + (−4.47 + 6.15i)6-s + 8.50i·7-s + (2.47 − 7.60i)8-s − 5.47·9-s + (−3.61 − 2.62i)10-s − 1.79i·11-s + (14.4 − 4.70i)12-s + 0.472·13-s + (10 − 13.7i)14-s − 8.50i·15-s + (−12.9 + 9.40i)16-s − 23.8·17-s + ⋯
 L(s)  = 1 + (−0.809 − 0.587i)2-s − 1.26i·3-s + (0.309 + 0.951i)4-s + 0.447·5-s + (−0.745 + 1.02i)6-s + 1.21i·7-s + (0.309 − 0.951i)8-s − 0.608·9-s + (−0.361 − 0.262i)10-s − 0.163i·11-s + (1.20 − 0.391i)12-s + 0.0363·13-s + (0.714 − 0.983i)14-s − 0.567i·15-s + (−0.809 + 0.587i)16-s − 1.40·17-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.951i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.309 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$20$$    =    $$2^{2} \cdot 5$$ $$\varepsilon$$ = $0.309 + 0.951i$ motivic weight = $$2$$ character : $\chi_{20} (11, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 20,\ (\ :1),\ 0.309 + 0.951i)$ $L(\frac{3}{2})$ $\approx$ $0.529541 - 0.384734i$ $L(\frac12)$ $\approx$ $0.529541 - 0.384734i$ $L(2)$ 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 + (1.61 + 1.17i)T$$
5 $$1 - 2.23T$$
good3 $$1 + 3.80iT - 9T^{2}$$
7 $$1 - 8.50iT - 49T^{2}$$
11 $$1 + 1.79iT - 121T^{2}$$
13 $$1 - 0.472T + 169T^{2}$$
17 $$1 + 23.8T + 289T^{2}$$
19 $$1 - 9.40iT - 361T^{2}$$
23 $$1 - 16.1iT - 529T^{2}$$
29 $$1 - 6.94T + 841T^{2}$$
31 $$1 + 47.4iT - 961T^{2}$$
37 $$1 - 26.3T + 1.36e3T^{2}$$
41 $$1 + 41.4T + 1.68e3T^{2}$$
43 $$1 - 2.00iT - 1.84e3T^{2}$$
47 $$1 - 35.3iT - 2.20e3T^{2}$$
53 $$1 + 21.6T + 2.80e3T^{2}$$
59 $$1 + 73.8iT - 3.48e3T^{2}$$
61 $$1 + 26.1T + 3.72e3T^{2}$$
67 $$1 + 88.8iT - 4.48e3T^{2}$$
71 $$1 - 39.4iT - 5.04e3T^{2}$$
73 $$1 - 137.T + 5.32e3T^{2}$$
79 $$1 - 113. iT - 6.24e3T^{2}$$
83 $$1 + 21.2iT - 6.88e3T^{2}$$
89 $$1 - 67.4T + 7.92e3T^{2}$$
97 $$1 + 39.1T + 9.40e3T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}