# Properties

 Degree $2$ Conductor $324$ Sign $-0.546 - 0.837i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (4.80 − 1.74i)5-s + (5.74 + 32.5i)7-s + (−28.2 − 10.2i)11-s + (7.59 + 6.37i)13-s + (−44.7 + 77.4i)17-s + (−29.6 − 51.3i)19-s + (17.0 − 96.7i)23-s + (−75.7 + 63.5i)25-s + (−108. + 90.6i)29-s + (4.10 − 23.2i)31-s + (84.5 + 146. i)35-s + (−114. + 197. i)37-s + (357. + 300. i)41-s + (−10.1 − 3.68i)43-s + (66.0 + 374. i)47-s + ⋯
 L(s)  = 1 + (0.429 − 0.156i)5-s + (0.310 + 1.75i)7-s + (−0.773 − 0.281i)11-s + (0.162 + 0.136i)13-s + (−0.638 + 1.10i)17-s + (−0.358 − 0.620i)19-s + (0.154 − 0.876i)23-s + (−0.605 + 0.508i)25-s + (−0.692 + 0.580i)29-s + (0.0237 − 0.134i)31-s + (0.408 + 0.707i)35-s + (−0.506 + 0.877i)37-s + (1.36 + 1.14i)41-s + (−0.0358 − 0.0130i)43-s + (0.205 + 1.16i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$324$$    =    $$2^{2} \cdot 3^{4}$$ Sign: $-0.546 - 0.837i$ Motivic weight: $$3$$ Character: $\chi_{324} (37, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 324,\ (\ :3/2),\ -0.546 - 0.837i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.287689047$$ $$L(\frac12)$$ $$\approx$$ $$1.287689047$$ $$L(\frac{5}{2})$$ 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 + (-4.80 + 1.74i)T + (95.7 - 80.3i)T^{2}$$
7 $$1 + (-5.74 - 32.5i)T + (-322. + 117. i)T^{2}$$
11 $$1 + (28.2 + 10.2i)T + (1.01e3 + 855. i)T^{2}$$
13 $$1 + (-7.59 - 6.37i)T + (381. + 2.16e3i)T^{2}$$
17 $$1 + (44.7 - 77.4i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (29.6 + 51.3i)T + (-3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (-17.0 + 96.7i)T + (-1.14e4 - 4.16e3i)T^{2}$$
29 $$1 + (108. - 90.6i)T + (4.23e3 - 2.40e4i)T^{2}$$
31 $$1 + (-4.10 + 23.2i)T + (-2.79e4 - 1.01e4i)T^{2}$$
37 $$1 + (114. - 197. i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 + (-357. - 300. i)T + (1.19e4 + 6.78e4i)T^{2}$$
43 $$1 + (10.1 + 3.68i)T + (6.09e4 + 5.11e4i)T^{2}$$
47 $$1 + (-66.0 - 374. i)T + (-9.75e4 + 3.55e4i)T^{2}$$
53 $$1 - 202.T + 1.48e5T^{2}$$
59 $$1 + (766. - 279. i)T + (1.57e5 - 1.32e5i)T^{2}$$
61 $$1 + (109. + 622. i)T + (-2.13e5 + 7.76e4i)T^{2}$$
67 $$1 + (466. + 391. i)T + (5.22e4 + 2.96e5i)T^{2}$$
71 $$1 + (140. - 243. i)T + (-1.78e5 - 3.09e5i)T^{2}$$
73 $$1 + (-608. - 1.05e3i)T + (-1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (-278. + 233. i)T + (8.56e4 - 4.85e5i)T^{2}$$
83 $$1 + (453. - 380. i)T + (9.92e4 - 5.63e5i)T^{2}$$
89 $$1 + (-359. - 622. i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 + (-776. - 282. i)T + (6.99e5 + 5.86e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$