Properties

 Label 2-24e2-8.5-c3-0-24 Degree $2$ Conductor $576$ Sign $-0.707 + 0.707i$ Analytic cond. $33.9851$ Root an. cond. $5.82967$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + 18i·11-s − 90·17-s − 106i·19-s + 125·25-s − 522·41-s − 290i·43-s − 343·49-s − 846i·59-s + 70i·67-s − 430·73-s − 1.35e3i·83-s − 1.02e3·89-s − 1.91e3·97-s − 1.71e3i·107-s + 270·113-s + ⋯
 L(s)  = 1 + 0.493i·11-s − 1.28·17-s − 1.27i·19-s + 25-s − 1.98·41-s − 1.02i·43-s − 49-s − 1.86i·59-s + 0.127i·67-s − 0.689·73-s − 1.78i·83-s − 1.22·89-s − 1.99·97-s − 1.54i·107-s + 0.224·113-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$576$$    =    $$2^{6} \cdot 3^{2}$$ Sign: $-0.707 + 0.707i$ Analytic conductor: $$33.9851$$ Root analytic conductor: $$5.82967$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{576} (289, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 576,\ (\ :3/2),\ -0.707 + 0.707i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$0.6923090619$$ $$L(\frac12)$$ $$\approx$$ $$0.6923090619$$ $$L(\frac{5}{2})$$ 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 - 125T^{2}$$
7 $$1 + 343T^{2}$$
11 $$1 - 18iT - 1.33e3T^{2}$$
13 $$1 - 2.19e3T^{2}$$
17 $$1 + 90T + 4.91e3T^{2}$$
19 $$1 + 106iT - 6.85e3T^{2}$$
23 $$1 + 1.21e4T^{2}$$
29 $$1 - 2.43e4T^{2}$$
31 $$1 + 2.97e4T^{2}$$
37 $$1 - 5.06e4T^{2}$$
41 $$1 + 522T + 6.89e4T^{2}$$
43 $$1 + 290iT - 7.95e4T^{2}$$
47 $$1 + 1.03e5T^{2}$$
53 $$1 - 1.48e5T^{2}$$
59 $$1 + 846iT - 2.05e5T^{2}$$
61 $$1 - 2.26e5T^{2}$$
67 $$1 - 70iT - 3.00e5T^{2}$$
71 $$1 + 3.57e5T^{2}$$
73 $$1 + 430T + 3.89e5T^{2}$$
79 $$1 + 4.93e5T^{2}$$
83 $$1 + 1.35e3iT - 5.71e5T^{2}$$
89 $$1 + 1.02e3T + 7.04e5T^{2}$$
97 $$1 + 1.91e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$