# Properties

 Degree 2 Conductor 3 Sign $1$ Motivic weight 5 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 6·2-s + 9·3-s + 4·4-s + 6·5-s − 54·6-s − 40·7-s + 168·8-s + 81·9-s − 36·10-s − 564·11-s + 36·12-s + 638·13-s + 240·14-s + 54·15-s − 1.13e3·16-s + 882·17-s − 486·18-s − 556·19-s + 24·20-s − 360·21-s + 3.38e3·22-s − 840·23-s + 1.51e3·24-s − 3.08e3·25-s − 3.82e3·26-s + 729·27-s − 160·28-s + ⋯
 L(s)  = 1 − 1.06·2-s + 0.577·3-s + 1/8·4-s + 0.107·5-s − 0.612·6-s − 0.308·7-s + 0.928·8-s + 1/3·9-s − 0.113·10-s − 1.40·11-s + 0.0721·12-s + 1.04·13-s + 0.327·14-s + 0.0619·15-s − 1.10·16-s + 0.740·17-s − 0.353·18-s − 0.353·19-s + 0.0134·20-s − 0.178·21-s + 1.49·22-s − 0.331·23-s + 0.535·24-s − 0.988·25-s − 1.11·26-s + 0.192·27-s − 0.0385·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3$$ $$\varepsilon$$ = $1$ motivic weight = $$5$$ character : $\chi_{3} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 3,\ (\ :5/2),\ 1)$ $L(3)$ $\approx$ $0.560038$ $L(\frac12)$ $\approx$ $0.560038$ $L(\frac{7}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 3$, $$F_p$$ is a polynomial of degree 2. If $p = 3$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 $$1 - p^{2} T$$
good2 $$1 + 3 p T + p^{5} T^{2}$$
5 $$1 - 6 T + p^{5} T^{2}$$
7 $$1 + 40 T + p^{5} T^{2}$$
11 $$1 + 564 T + p^{5} T^{2}$$
13 $$1 - 638 T + p^{5} T^{2}$$
17 $$1 - 882 T + p^{5} T^{2}$$
19 $$1 + 556 T + p^{5} T^{2}$$
23 $$1 + 840 T + p^{5} T^{2}$$
29 $$1 - 4638 T + p^{5} T^{2}$$
31 $$1 - 4400 T + p^{5} T^{2}$$
37 $$1 + 2410 T + p^{5} T^{2}$$
41 $$1 + 6870 T + p^{5} T^{2}$$
43 $$1 - 9644 T + p^{5} T^{2}$$
47 $$1 + 18672 T + p^{5} T^{2}$$
53 $$1 - 33750 T + p^{5} T^{2}$$
59 $$1 + 18084 T + p^{5} T^{2}$$
61 $$1 - 39758 T + p^{5} T^{2}$$
67 $$1 + 23068 T + p^{5} T^{2}$$
71 $$1 + 4248 T + p^{5} T^{2}$$
73 $$1 + 41110 T + p^{5} T^{2}$$
79 $$1 - 21920 T + p^{5} T^{2}$$
83 $$1 - 82452 T + p^{5} T^{2}$$
89 $$1 + 94086 T + p^{5} T^{2}$$
97 $$1 - 49442 T + p^{5} T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}