# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{3}$ Sign $-0.948 + 0.316i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (55.1 + 95.6i)5-s + (50.8 − 88.1i)7-s + (75.1 − 130. i)11-s + (−317. − 550. i)13-s − 1.49e3·17-s − 1.43e3·19-s + (632. + 1.09e3i)23-s + (−4.53e3 + 7.84e3i)25-s + (−1.38e3 + 2.40e3i)29-s + (−3.48e3 − 6.03e3i)31-s + 1.12e4·35-s − 7.95e3·37-s + (−1.01e3 − 1.75e3i)41-s + (−6.26e3 + 1.08e4i)43-s + (3.24e3 − 5.61e3i)47-s + ⋯
 L(s)  = 1 + (0.987 + 1.71i)5-s + (0.392 − 0.679i)7-s + (0.187 − 0.324i)11-s + (−0.521 − 0.903i)13-s − 1.25·17-s − 0.913·19-s + (0.249 + 0.431i)23-s + (−1.45 + 2.51i)25-s + (−0.306 + 0.531i)29-s + (−0.651 − 1.12i)31-s + 1.54·35-s − 0.954·37-s + (−0.0941 − 0.163i)41-s + (−0.516 + 0.894i)43-s + (0.214 − 0.370i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$432$$    =    $$2^{4} \cdot 3^{3}$$ $$\varepsilon$$ = $-0.948 + 0.316i$ motivic weight = $$5$$ character : $\chi_{432} (289, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 432,\ (\ :5/2),\ -0.948 + 0.316i)$$ $$L(3)$$ $$\approx$$ $$0.2816215007$$ $$L(\frac12)$$ $$\approx$$ $$0.2816215007$$ $$L(\frac{7}{2})$$ 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$$
3 $$1$$
good5 $$1 + (-55.1 - 95.6i)T + (-1.56e3 + 2.70e3i)T^{2}$$
7 $$1 + (-50.8 + 88.1i)T + (-8.40e3 - 1.45e4i)T^{2}$$
11 $$1 + (-75.1 + 130. i)T + (-8.05e4 - 1.39e5i)T^{2}$$
13 $$1 + (317. + 550. i)T + (-1.85e5 + 3.21e5i)T^{2}$$
17 $$1 + 1.49e3T + 1.41e6T^{2}$$
19 $$1 + 1.43e3T + 2.47e6T^{2}$$
23 $$1 + (-632. - 1.09e3i)T + (-3.21e6 + 5.57e6i)T^{2}$$
29 $$1 + (1.38e3 - 2.40e3i)T + (-1.02e7 - 1.77e7i)T^{2}$$
31 $$1 + (3.48e3 + 6.03e3i)T + (-1.43e7 + 2.47e7i)T^{2}$$
37 $$1 + 7.95e3T + 6.93e7T^{2}$$
41 $$1 + (1.01e3 + 1.75e3i)T + (-5.79e7 + 1.00e8i)T^{2}$$
43 $$1 + (6.26e3 - 1.08e4i)T + (-7.35e7 - 1.27e8i)T^{2}$$
47 $$1 + (-3.24e3 + 5.61e3i)T + (-1.14e8 - 1.98e8i)T^{2}$$
53 $$1 + 9.82e3T + 4.18e8T^{2}$$
59 $$1 + (2.35e4 + 4.07e4i)T + (-3.57e8 + 6.19e8i)T^{2}$$
61 $$1 + (4.16e3 - 7.22e3i)T + (-4.22e8 - 7.31e8i)T^{2}$$
67 $$1 + (3.63e3 + 6.28e3i)T + (-6.75e8 + 1.16e9i)T^{2}$$
71 $$1 - 3.58e3T + 1.80e9T^{2}$$
73 $$1 - 5.80e4T + 2.07e9T^{2}$$
79 $$1 + (3.18e4 - 5.52e4i)T + (-1.53e9 - 2.66e9i)T^{2}$$
83 $$1 + (4.14e4 - 7.17e4i)T + (-1.96e9 - 3.41e9i)T^{2}$$
89 $$1 - 3.86e3T + 5.58e9T^{2}$$
97 $$1 + (3.46e4 - 5.99e4i)T + (-4.29e9 - 7.43e9i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}