Properties

 Degree $2$ Conductor $810$ Sign $-0.803 - 0.595i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.133 − 2.23i)5-s + (−1.73 − i)7-s − 0.999i·8-s + (−1 + 1.99i)10-s + (1 − 1.73i)11-s + (−5.19 + 3i)13-s + (0.999 + 1.73i)14-s + (−0.5 + 0.866i)16-s − 2i·17-s + (1.86 − 1.23i)20-s + (−1.73 + 0.999i)22-s + (−3.46 + 2i)23-s + (−4.96 + 0.598i)25-s + 6·26-s + ⋯
 L(s)  = 1 + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.0599 − 0.998i)5-s + (−0.654 − 0.377i)7-s − 0.353i·8-s + (−0.316 + 0.632i)10-s + (0.301 − 0.522i)11-s + (−1.44 + 0.832i)13-s + (0.267 + 0.462i)14-s + (−0.125 + 0.216i)16-s − 0.485i·17-s + (0.417 − 0.275i)20-s + (−0.369 + 0.213i)22-s + (−0.722 + 0.417i)23-s + (−0.992 + 0.119i)25-s + 1.17·26-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$810$$    =    $$2 \cdot 3^{4} \cdot 5$$ Sign: $-0.803 - 0.595i$ Motivic weight: $$1$$ Character: $\chi_{810} (379, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 810,\ (\ :1/2),\ -0.803 - 0.595i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.0495725 + 0.150047i$$ $$L(\frac12)$$ $$\approx$$ $$0.0495725 + 0.150047i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.866 + 0.5i)T$$
3 $$1$$
5 $$1 + (0.133 + 2.23i)T$$
good7 $$1 + (1.73 + i)T + (3.5 + 6.06i)T^{2}$$
11 $$1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + (5.19 - 3i)T + (6.5 - 11.2i)T^{2}$$
17 $$1 + 2iT - 17T^{2}$$
19 $$1 + 19T^{2}$$
23 $$1 + (3.46 - 2i)T + (11.5 - 19.9i)T^{2}$$
29 $$1 + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 - 2iT - 37T^{2}$$
41 $$1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (3.46 + 2i)T + (21.5 + 37.2i)T^{2}$$
47 $$1 + (6.92 + 4i)T + (23.5 + 40.7i)T^{2}$$
53 $$1 - 6iT - 53T^{2}$$
59 $$1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (6.92 - 4i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 + 12T + 71T^{2}$$
73 $$1 - 4iT - 73T^{2}$$
79 $$1 + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (-3.46 - 2i)T + (41.5 + 71.8i)T^{2}$$
89 $$1 + 10T + 89T^{2}$$
97 $$1 + (-6.92 - 4i)T + (48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$