# Properties

 Degree $2$ Conductor $324$ Sign $0.996 - 0.0820i$ Motivic weight $4$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (3.99 − 0.164i)2-s + (15.9 − 1.31i)4-s + 29.7·5-s + 59.8i·7-s + (63.5 − 7.86i)8-s + (118. − 4.89i)10-s − 225. i·11-s + 171.·13-s + (9.84 + 239. i)14-s + (252. − 41.8i)16-s − 99.0·17-s + 169. i·19-s + (474. − 39.0i)20-s + (−37.0 − 901. i)22-s + 358. i·23-s + ⋯
 L(s)  = 1 + (0.999 − 0.0410i)2-s + (0.996 − 0.0820i)4-s + 1.19·5-s + 1.22i·7-s + (0.992 − 0.122i)8-s + (1.18 − 0.0489i)10-s − 1.86i·11-s + 1.01·13-s + (0.0502 + 1.22i)14-s + (0.986 − 0.163i)16-s − 0.342·17-s + 0.468i·19-s + (1.18 − 0.0977i)20-s + (−0.0765 − 1.86i)22-s + 0.678i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0820i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.996 - 0.0820i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$324$$    =    $$2^{2} \cdot 3^{4}$$ Sign: $0.996 - 0.0820i$ Motivic weight: $$4$$ Character: $\chi_{324} (163, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 324,\ (\ :2),\ 0.996 - 0.0820i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$5.168872825$$ $$L(\frac12)$$ $$\approx$$ $$5.168872825$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-3.99 + 0.164i)T$$
3 $$1$$
good5 $$1 - 29.7T + 625T^{2}$$
7 $$1 - 59.8iT - 2.40e3T^{2}$$
11 $$1 + 225. iT - 1.46e4T^{2}$$
13 $$1 - 171.T + 2.85e4T^{2}$$
17 $$1 + 99.0T + 8.35e4T^{2}$$
19 $$1 - 169. iT - 1.30e5T^{2}$$
23 $$1 - 358. iT - 2.79e5T^{2}$$
29 $$1 + 18.0T + 7.07e5T^{2}$$
31 $$1 - 775. iT - 9.23e5T^{2}$$
37 $$1 - 609.T + 1.87e6T^{2}$$
41 $$1 + 413.T + 2.82e6T^{2}$$
43 $$1 + 306. iT - 3.41e6T^{2}$$
47 $$1 + 2.62e3iT - 4.87e6T^{2}$$
53 $$1 - 2.03e3T + 7.89e6T^{2}$$
59 $$1 + 2.59e3iT - 1.21e7T^{2}$$
61 $$1 + 1.41e3T + 1.38e7T^{2}$$
67 $$1 - 5.99e3iT - 2.01e7T^{2}$$
71 $$1 - 1.23e3iT - 2.54e7T^{2}$$
73 $$1 + 5.06e3T + 2.83e7T^{2}$$
79 $$1 + 1.88e3iT - 3.89e7T^{2}$$
83 $$1 - 6.73e3iT - 4.74e7T^{2}$$
89 $$1 - 9.43e3T + 6.27e7T^{2}$$
97 $$1 + 1.45e4T + 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$