# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{4}$ Sign $-0.766 + 0.642i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (3.5 − 6.06i)5-s + (7.5 − 4.33i)7-s − 7.99·8-s + (−7 − 12.1i)10-s + (7.5 − 4.33i)11-s + (−10 + 17.3i)13-s − 17.3i·14-s + (−8 + 13.8i)16-s + 8·17-s − 10.3i·19-s − 28·20-s − 17.3i·22-s + (3 + 1.73i)23-s + ⋯
 L(s)  = 1 + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.700 − 1.21i)5-s + (1.07 − 0.618i)7-s − 0.999·8-s + (−0.700 − 1.21i)10-s + (0.681 − 0.393i)11-s + (−0.769 + 1.33i)13-s − 1.23i·14-s + (−0.5 + 0.866i)16-s + 0.470·17-s − 0.546i·19-s − 1.40·20-s − 0.787i·22-s + (0.130 + 0.0753i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$324$$    =    $$2^{2} \cdot 3^{4}$$ $$\varepsilon$$ = $-0.766 + 0.642i$ motivic weight = $$2$$ character : $\chi_{324} (271, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 324,\ (\ :1),\ -0.766 + 0.642i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.818797 - 2.24962i$$ $$L(\frac12)$$ $$\approx$$ $$0.818797 - 2.24962i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-1 + 1.73i)T$$
3 $$1$$
good5 $$1 + (-3.5 + 6.06i)T + (-12.5 - 21.6i)T^{2}$$
7 $$1 + (-7.5 + 4.33i)T + (24.5 - 42.4i)T^{2}$$
11 $$1 + (-7.5 + 4.33i)T + (60.5 - 104. i)T^{2}$$
13 $$1 + (10 - 17.3i)T + (-84.5 - 146. i)T^{2}$$
17 $$1 - 8T + 289T^{2}$$
19 $$1 + 10.3iT - 361T^{2}$$
23 $$1 + (-3 - 1.73i)T + (264.5 + 458. i)T^{2}$$
29 $$1 + (-5 - 8.66i)T + (-420.5 + 728. i)T^{2}$$
31 $$1 + (46.5 + 26.8i)T + (480.5 + 832. i)T^{2}$$
37 $$1 + 10T + 1.36e3T^{2}$$
41 $$1 + (25 - 43.3i)T + (-840.5 - 1.45e3i)T^{2}$$
43 $$1 + (15 - 8.66i)T + (924.5 - 1.60e3i)T^{2}$$
47 $$1 + (-75 + 43.3i)T + (1.10e3 - 1.91e3i)T^{2}$$
53 $$1 - 47T + 2.80e3T^{2}$$
59 $$1 + (-30 - 17.3i)T + (1.74e3 + 3.01e3i)T^{2}$$
61 $$1 + (-32 - 55.4i)T + (-1.86e3 + 3.22e3i)T^{2}$$
67 $$1 + (-75 - 43.3i)T + (2.24e3 + 3.88e3i)T^{2}$$
71 $$1 - 5.04e3T^{2}$$
73 $$1 + 55T + 5.32e3T^{2}$$
79 $$1 + (6 - 3.46i)T + (3.12e3 - 5.40e3i)T^{2}$$
83 $$1 + (-25.5 + 14.7i)T + (3.44e3 - 5.96e3i)T^{2}$$
89 $$1 + 10T + 7.92e3T^{2}$$
97 $$1 + (-12.5 - 21.6i)T + (-4.70e3 + 8.14e3i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}