# Properties

 Degree $2$ Conductor $648$ Sign $-0.173 + 0.984i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1 − 1.73i)5-s + (−12 + 20.7i)7-s + (−22 + 38.1i)11-s + (−11 − 19.0i)13-s − 50·17-s + 44·19-s + (−28 − 48.4i)23-s + (60.5 − 104. i)25-s + (99 − 171. i)29-s + (80 + 138. i)31-s + 48·35-s − 162·37-s + (−99 − 171. i)41-s + (−26 + 45.0i)43-s + (264 − 457. i)47-s + ⋯
 L(s)  = 1 + (−0.0894 − 0.154i)5-s + (−0.647 + 1.12i)7-s + (−0.603 + 1.04i)11-s + (−0.234 − 0.406i)13-s − 0.713·17-s + 0.531·19-s + (−0.253 − 0.439i)23-s + (0.483 − 0.838i)25-s + (0.633 − 1.09i)29-s + (0.463 + 0.802i)31-s + 0.231·35-s − 0.719·37-s + (−0.377 − 0.653i)41-s + (−0.0922 + 0.159i)43-s + (0.819 − 1.41i)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: $-0.173 + 0.984i$ Motivic weight: $$3$$ Character: $\chi_{648} (217, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 648,\ (\ :3/2),\ -0.173 + 0.984i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.6599684407$$ $$L(\frac12)$$ $$\approx$$ $$0.6599684407$$ $$L(\frac{5}{2})$$ 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 + (1 + 1.73i)T + (-62.5 + 108. i)T^{2}$$
7 $$1 + (12 - 20.7i)T + (-171.5 - 297. i)T^{2}$$
11 $$1 + (22 - 38.1i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 + (11 + 19.0i)T + (-1.09e3 + 1.90e3i)T^{2}$$
17 $$1 + 50T + 4.91e3T^{2}$$
19 $$1 - 44T + 6.85e3T^{2}$$
23 $$1 + (28 + 48.4i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + (-99 + 171. i)T + (-1.21e4 - 2.11e4i)T^{2}$$
31 $$1 + (-80 - 138. i)T + (-1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + 162T + 5.06e4T^{2}$$
41 $$1 + (99 + 171. i)T + (-3.44e4 + 5.96e4i)T^{2}$$
43 $$1 + (26 - 45.0i)T + (-3.97e4 - 6.88e4i)T^{2}$$
47 $$1 + (-264 + 457. i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 - 242T + 1.48e5T^{2}$$
59 $$1 + (334 + 578. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (275 - 476. i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (94 + 162. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 + 728T + 3.57e5T^{2}$$
73 $$1 - 154T + 3.89e5T^{2}$$
79 $$1 + (-328 + 568. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + (-118 + 204. i)T + (-2.85e5 - 4.95e5i)T^{2}$$
89 $$1 + 714T + 7.04e5T^{2}$$
97 $$1 + (-239 + 413. i)T + (-4.56e5 - 7.90e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$