# Properties

 Degree 2 Conductor $2 \cdot 3^{4}$ Sign $-0.939 - 0.342i$ Motivic weight 3 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 + 5.19i)5-s + (8 + 13.8i)7-s − 7.99·8-s − 12·10-s + (−6 − 10.3i)11-s + (−19 + 32.9i)13-s + (−15.9 + 27.7i)14-s + (−8 − 13.8i)16-s − 126·17-s + 20·19-s + (−12.0 − 20.7i)20-s + (12 − 20.7i)22-s + (−84 + 145. i)23-s + ⋯
 L(s)  = 1 + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.268 + 0.464i)5-s + (0.431 + 0.748i)7-s − 0.353·8-s − 0.379·10-s + (−0.164 − 0.284i)11-s + (−0.405 + 0.702i)13-s + (−0.305 + 0.529i)14-s + (−0.125 − 0.216i)16-s − 1.79·17-s + 0.241·19-s + (−0.134 − 0.232i)20-s + (0.116 − 0.201i)22-s + (−0.761 + 1.31i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$162$$    =    $$2 \cdot 3^{4}$$ $$\varepsilon$$ = $-0.939 - 0.342i$ motivic weight = $$3$$ character : $\chi_{162} (109, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 162,\ (\ :3/2),\ -0.939 - 0.342i)$$ $$L(2)$$ $$\approx$$ $$0.236027 + 1.33857i$$ $$L(\frac12)$$ $$\approx$$ $$0.236027 + 1.33857i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ 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 - 5.19i)T + (-62.5 - 108. i)T^{2}$$
7 $$1 + (-8 - 13.8i)T + (-171.5 + 297. i)T^{2}$$
11 $$1 + (6 + 10.3i)T + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 + (19 - 32.9i)T + (-1.09e3 - 1.90e3i)T^{2}$$
17 $$1 + 126T + 4.91e3T^{2}$$
19 $$1 - 20T + 6.85e3T^{2}$$
23 $$1 + (84 - 145. i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + (15 + 25.9i)T + (-1.21e4 + 2.11e4i)T^{2}$$
31 $$1 + (-44 + 76.2i)T + (-1.48e4 - 2.57e4i)T^{2}$$
37 $$1 - 254T + 5.06e4T^{2}$$
41 $$1 + (21 - 36.3i)T + (-3.44e4 - 5.96e4i)T^{2}$$
43 $$1 + (-26 - 45.0i)T + (-3.97e4 + 6.88e4i)T^{2}$$
47 $$1 + (-48 - 83.1i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 - 198T + 1.48e5T^{2}$$
59 $$1 + (-330 + 571. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (-269 - 465. i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (442 - 765. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 - 792T + 3.57e5T^{2}$$
73 $$1 - 218T + 3.89e5T^{2}$$
79 $$1 + (-260 - 450. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 + (-246 - 426. i)T + (-2.85e5 + 4.95e5i)T^{2}$$
89 $$1 - 810T + 7.04e5T^{2}$$
97 $$1 + (577 + 999. i)T + (-4.56e5 + 7.90e5i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}