# Properties

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

# Origins

## Dirichlet series

 L(s)  = 1 + 4·2-s − 9·3-s + 16·4-s − 66·5-s − 36·6-s + 176·7-s + 64·8-s + 81·9-s − 264·10-s − 60·11-s − 144·12-s − 658·13-s + 704·14-s + 594·15-s + 256·16-s − 414·17-s + 324·18-s + 956·19-s − 1.05e3·20-s − 1.58e3·21-s − 240·22-s + 600·23-s − 576·24-s + 1.23e3·25-s − 2.63e3·26-s − 729·27-s + 2.81e3·28-s + ⋯
 L(s)  = 1 + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.18·5-s − 0.408·6-s + 1.35·7-s + 0.353·8-s + 1/3·9-s − 0.834·10-s − 0.149·11-s − 0.288·12-s − 1.07·13-s + 0.959·14-s + 0.681·15-s + 1/4·16-s − 0.347·17-s + 0.235·18-s + 0.607·19-s − 0.590·20-s − 0.783·21-s − 0.105·22-s + 0.236·23-s − 0.204·24-s + 0.393·25-s − 0.763·26-s − 0.192·27-s + 0.678·28-s + ⋯

## Functional equation

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

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$6$$    =    $$2 \cdot 3$$ $$\varepsilon$$ = $1$ motivic weight = $$5$$ character : $\chi_{6} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 6,\ (\ :5/2),\ 1)$$ $$L(3)$$ $$\approx$$ $$1.15777$$ $$L(\frac12)$$ $$\approx$$ $$1.15777$$ $$L(\frac{7}{2})$$ 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 - p^{2} T$$
3 $$1 + p^{2} T$$
good5 $$1 + 66 T + p^{5} T^{2}$$
7 $$1 - 176 T + p^{5} T^{2}$$
11 $$1 + 60 T + p^{5} T^{2}$$
13 $$1 + 658 T + p^{5} T^{2}$$
17 $$1 + 414 T + p^{5} T^{2}$$
19 $$1 - 956 T + p^{5} T^{2}$$
23 $$1 - 600 T + p^{5} T^{2}$$
29 $$1 - 5574 T + p^{5} T^{2}$$
31 $$1 + 3592 T + p^{5} T^{2}$$
37 $$1 + 8458 T + p^{5} T^{2}$$
41 $$1 - 19194 T + p^{5} T^{2}$$
43 $$1 - 13316 T + p^{5} T^{2}$$
47 $$1 + 19680 T + p^{5} T^{2}$$
53 $$1 + 31266 T + p^{5} T^{2}$$
59 $$1 - 26340 T + p^{5} T^{2}$$
61 $$1 + 31090 T + p^{5} T^{2}$$
67 $$1 + 16804 T + p^{5} T^{2}$$
71 $$1 - 6120 T + p^{5} T^{2}$$
73 $$1 + 25558 T + p^{5} T^{2}$$
79 $$1 - 74408 T + p^{5} T^{2}$$
83 $$1 + 6468 T + p^{5} T^{2}$$
89 $$1 + 32742 T + p^{5} T^{2}$$
97 $$1 - 166082 T + p^{5} T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}