Properties

 Label 2-24e2-3.2-c2-0-15 Degree $2$ Conductor $576$ Sign $-0.577 + 0.816i$ Analytic cond. $15.6948$ Root an. cond. $3.96167$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 4.24i·5-s + 4·7-s − 16.9i·11-s − 8·13-s + 12.7i·17-s − 16·19-s − 16.9i·23-s + 7.00·25-s + 4.24i·29-s − 44·31-s − 16.9i·35-s + 34·37-s − 46.6i·41-s − 40·43-s − 84.8i·47-s + ⋯
 L(s)  = 1 − 0.848i·5-s + 0.571·7-s − 1.54i·11-s − 0.615·13-s + 0.748i·17-s − 0.842·19-s − 0.737i·23-s + 0.280·25-s + 0.146i·29-s − 1.41·31-s − 0.484i·35-s + 0.918·37-s − 1.13i·41-s − 0.930·43-s − 1.80i·47-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$576$$    =    $$2^{6} \cdot 3^{2}$$ Sign: $-0.577 + 0.816i$ Analytic conductor: $$15.6948$$ Root analytic conductor: $$3.96167$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{576} (449, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 576,\ (\ :1),\ -0.577 + 0.816i)$$

Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.284658515$$ $$L(\frac12)$$ $$\approx$$ $$1.284658515$$ $$L(2)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5 $$1 + 4.24iT - 25T^{2}$$
7 $$1 - 4T + 49T^{2}$$
11 $$1 + 16.9iT - 121T^{2}$$
13 $$1 + 8T + 169T^{2}$$
17 $$1 - 12.7iT - 289T^{2}$$
19 $$1 + 16T + 361T^{2}$$
23 $$1 + 16.9iT - 529T^{2}$$
29 $$1 - 4.24iT - 841T^{2}$$
31 $$1 + 44T + 961T^{2}$$
37 $$1 - 34T + 1.36e3T^{2}$$
41 $$1 + 46.6iT - 1.68e3T^{2}$$
43 $$1 + 40T + 1.84e3T^{2}$$
47 $$1 + 84.8iT - 2.20e3T^{2}$$
53 $$1 - 38.1iT - 2.80e3T^{2}$$
59 $$1 + 33.9iT - 3.48e3T^{2}$$
61 $$1 + 50T + 3.72e3T^{2}$$
67 $$1 - 8T + 4.48e3T^{2}$$
71 $$1 + 50.9iT - 5.04e3T^{2}$$
73 $$1 + 16T + 5.32e3T^{2}$$
79 $$1 - 76T + 6.24e3T^{2}$$
83 $$1 + 118. iT - 6.88e3T^{2}$$
89 $$1 + 12.7iT - 7.92e3T^{2}$$
97 $$1 - 176T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$