# Properties

 Label 2-648-9.4-c3-0-4 Degree $2$ Conductor $648$ Sign $-0.173 - 0.984i$ Analytic cond. $38.2332$ Root an. cond. $6.18330$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (8 − 13.8i)5-s + (6 + 10.3i)7-s + (32 + 55.4i)11-s + (−29 + 50.2i)13-s − 32·17-s − 136·19-s + (−64 + 110. i)23-s + (−65.4 − 113. i)25-s + (−72 − 124. i)29-s + (−10 + 17.3i)31-s + 192·35-s − 18·37-s + (−144 + 249. i)41-s + (100 + 173. i)43-s + (192 + 332. i)47-s + ⋯
 L(s)  = 1 + (0.715 − 1.23i)5-s + (0.323 + 0.561i)7-s + (0.877 + 1.51i)11-s + (−0.618 + 1.07i)13-s − 0.456·17-s − 1.64·19-s + (−0.580 + 1.00i)23-s + (−0.523 − 0.907i)25-s + (−0.461 − 0.798i)29-s + (−0.0579 + 0.100i)31-s + 0.927·35-s − 0.0799·37-s + (−0.548 + 0.950i)41-s + (0.354 + 0.614i)43-s + (0.595 + 1.03i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$648$$    =    $$2^{3} \cdot 3^{4}$$ Sign: $-0.173 - 0.984i$ Analytic conductor: $$38.2332$$ Root analytic conductor: $$6.18330$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{648} (433, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 648,\ (\ :3/2),\ -0.173 - 0.984i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.457042844$$ $$L(\frac12)$$ $$\approx$$ $$1.457042844$$ $$L(\frac{5}{2})$$ 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 + (-8 + 13.8i)T + (-62.5 - 108. i)T^{2}$$
7 $$1 + (-6 - 10.3i)T + (-171.5 + 297. i)T^{2}$$
11 $$1 + (-32 - 55.4i)T + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 + (29 - 50.2i)T + (-1.09e3 - 1.90e3i)T^{2}$$
17 $$1 + 32T + 4.91e3T^{2}$$
19 $$1 + 136T + 6.85e3T^{2}$$
23 $$1 + (64 - 110. i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + (72 + 124. i)T + (-1.21e4 + 2.11e4i)T^{2}$$
31 $$1 + (10 - 17.3i)T + (-1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + 18T + 5.06e4T^{2}$$
41 $$1 + (144 - 249. i)T + (-3.44e4 - 5.96e4i)T^{2}$$
43 $$1 + (-100 - 173. i)T + (-3.97e4 + 6.88e4i)T^{2}$$
47 $$1 + (-192 - 332. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + 496T + 1.48e5T^{2}$$
59 $$1 + (64 - 110. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (-229 - 396. i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (-248 + 429. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + 512T + 3.57e5T^{2}$$
73 $$1 + 602T + 3.89e5T^{2}$$
79 $$1 + (554 + 959. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 + (-352 - 609. i)T + (-2.85e5 + 4.95e5i)T^{2}$$
89 $$1 - 960T + 7.04e5T^{2}$$
97 $$1 + (103 + 178. i)T + (-4.56e5 + 7.90e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$