# Properties

 Degree $2$ Conductor $324$ Sign $-0.529 - 0.848i$ Motivic weight $4$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (3.49 + 1.94i)2-s + (8.46 + 13.5i)4-s + 33.2·5-s + 46.1i·7-s + (3.25 + 63.9i)8-s + (116. + 64.4i)10-s + 73.4i·11-s − 303.·13-s + (−89.5 + 161. i)14-s + (−112. + 229. i)16-s + 182.·17-s + 314. i·19-s + (281. + 451. i)20-s + (−142. + 256. i)22-s − 335. i·23-s + ⋯
 L(s)  = 1 + (0.874 + 0.485i)2-s + (0.529 + 0.848i)4-s + 1.32·5-s + 0.942i·7-s + (0.0508 + 0.998i)8-s + (1.16 + 0.644i)10-s + 0.607i·11-s − 1.79·13-s + (−0.457 + 0.823i)14-s + (−0.440 + 0.897i)16-s + 0.629·17-s + 0.870i·19-s + (0.703 + 1.12i)20-s + (−0.294 + 0.530i)22-s − 0.635i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 - 0.848i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.529 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$324$$    =    $$2^{2} \cdot 3^{4}$$ Sign: $-0.529 - 0.848i$ Motivic weight: $$4$$ Character: $\chi_{324} (163, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 324,\ (\ :2),\ -0.529 - 0.848i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$3.863815025$$ $$L(\frac12)$$ $$\approx$$ $$3.863815025$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-3.49 - 1.94i)T$$
3 $$1$$
good5 $$1 - 33.2T + 625T^{2}$$
7 $$1 - 46.1iT - 2.40e3T^{2}$$
11 $$1 - 73.4iT - 1.46e4T^{2}$$
13 $$1 + 303.T + 2.85e4T^{2}$$
17 $$1 - 182.T + 8.35e4T^{2}$$
19 $$1 - 314. iT - 1.30e5T^{2}$$
23 $$1 + 335. iT - 2.79e5T^{2}$$
29 $$1 - 714.T + 7.07e5T^{2}$$
31 $$1 + 1.13e3iT - 9.23e5T^{2}$$
37 $$1 - 1.00e3T + 1.87e6T^{2}$$
41 $$1 + 1.11e3T + 2.82e6T^{2}$$
43 $$1 - 2.51e3iT - 3.41e6T^{2}$$
47 $$1 + 1.13e3iT - 4.87e6T^{2}$$
53 $$1 - 1.05e3T + 7.89e6T^{2}$$
59 $$1 - 1.01e3iT - 1.21e7T^{2}$$
61 $$1 - 860.T + 1.38e7T^{2}$$
67 $$1 + 645. iT - 2.01e7T^{2}$$
71 $$1 - 9.56e3iT - 2.54e7T^{2}$$
73 $$1 - 1.89e3T + 2.83e7T^{2}$$
79 $$1 + 7.81e3iT - 3.89e7T^{2}$$
83 $$1 - 8.14e3iT - 4.74e7T^{2}$$
89 $$1 + 7.65e3T + 6.27e7T^{2}$$
97 $$1 - 1.27e4T + 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$