# Properties

 Degree 2 Conductor $3^{3}$ Sign $0.710 + 0.703i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.404 − 2.29i)2-s + (4.52 + 2.54i)3-s + (2.41 − 0.878i)4-s + (−4.78 − 4.01i)5-s + (4.01 − 11.4i)6-s + (2.53 + 0.924i)7-s + (−12.3 − 21.3i)8-s + (14.0 + 23.0i)9-s + (−7.27 + 12.5i)10-s + (−30.2 + 25.4i)11-s + (13.1 + 2.16i)12-s + (−12.8 + 73.0i)13-s + (1.09 − 6.20i)14-s + (−11.4 − 30.3i)15-s + (−28.2 + 23.6i)16-s + (29.1 − 50.4i)17-s + ⋯
 L(s)  = 1 + (−0.143 − 0.811i)2-s + (0.871 + 0.490i)3-s + (0.301 − 0.109i)4-s + (−0.427 − 0.358i)5-s + (0.273 − 0.777i)6-s + (0.137 + 0.0499i)7-s + (−0.544 − 0.942i)8-s + (0.519 + 0.854i)9-s + (−0.230 + 0.398i)10-s + (−0.830 + 0.696i)11-s + (0.316 + 0.0521i)12-s + (−0.274 + 1.55i)13-s + (0.0208 − 0.118i)14-s + (−0.196 − 0.522i)15-s + (−0.441 + 0.370i)16-s + (0.415 − 0.719i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$27$$    =    $$3^{3}$$ $$\varepsilon$$ = $0.710 + 0.703i$ motivic weight = $$3$$ character : $\chi_{27} (13, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 27,\ (\ :3/2),\ 0.710 + 0.703i)$ $L(2)$ $\approx$ $1.29197 - 0.531602i$ $L(\frac12)$ $\approx$ $1.29197 - 0.531602i$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 3$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 + (-4.52 - 2.54i)T$$
good2 $$1 + (0.404 + 2.29i)T + (-7.51 + 2.73i)T^{2}$$
5 $$1 + (4.78 + 4.01i)T + (21.7 + 123. i)T^{2}$$
7 $$1 + (-2.53 - 0.924i)T + (262. + 220. i)T^{2}$$
11 $$1 + (30.2 - 25.4i)T + (231. - 1.31e3i)T^{2}$$
13 $$1 + (12.8 - 73.0i)T + (-2.06e3 - 751. i)T^{2}$$
17 $$1 + (-29.1 + 50.4i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (41.2 + 71.3i)T + (-3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (25.8 - 9.40i)T + (9.32e3 - 7.82e3i)T^{2}$$
29 $$1 + (-30.1 - 170. i)T + (-2.29e4 + 8.34e3i)T^{2}$$
31 $$1 + (-149. + 54.3i)T + (2.28e4 - 1.91e4i)T^{2}$$
37 $$1 + (-220. + 382. i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 + (-14.7 + 83.6i)T + (-6.47e4 - 2.35e4i)T^{2}$$
43 $$1 + (150. - 126. i)T + (1.38e4 - 7.82e4i)T^{2}$$
47 $$1 + (-402. - 146. i)T + (7.95e4 + 6.67e4i)T^{2}$$
53 $$1 - 448.T + 1.48e5T^{2}$$
59 $$1 + (269. + 225. i)T + (3.56e4 + 2.02e5i)T^{2}$$
61 $$1 + (346. + 126. i)T + (1.73e5 + 1.45e5i)T^{2}$$
67 $$1 + (104. - 594. i)T + (-2.82e5 - 1.02e5i)T^{2}$$
71 $$1 + (423. - 733. i)T + (-1.78e5 - 3.09e5i)T^{2}$$
73 $$1 + (-21.1 - 36.7i)T + (-1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (-63.6 - 361. i)T + (-4.63e5 + 1.68e5i)T^{2}$$
83 $$1 + (10.0 + 57.0i)T + (-5.37e5 + 1.95e5i)T^{2}$$
89 $$1 + (713. + 1.23e3i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 + (-875. + 735. i)T + (1.58e5 - 8.98e5i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}