# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 5 \cdot 7$ Sign $-1$ Motivic weight 3 Primitive yes Self-dual yes Analytic rank 1

# Origins

## Dirichlet series

 L(s)  = 1 − 2·2-s − 3·3-s + 4·4-s − 5·5-s + 6·6-s + 7·7-s − 8·8-s + 9·9-s + 10·10-s + 12·11-s − 12·12-s + 2·13-s − 14·14-s + 15·15-s + 16·16-s − 18·17-s − 18·18-s + 56·19-s − 20·20-s − 21·21-s − 24·22-s − 156·23-s + 24·24-s + 25·25-s − 4·26-s − 27·27-s + 28·28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.328·11-s − 0.288·12-s + 0.0426·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.256·17-s − 0.235·18-s + 0.676·19-s − 0.223·20-s − 0.218·21-s − 0.232·22-s − 1.41·23-s + 0.204·24-s + 1/5·25-s − 0.0301·26-s − 0.192·27-s + 0.188·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$210$$    =    $$2 \cdot 3 \cdot 5 \cdot 7$$ $$\varepsilon$$ = $-1$ motivic weight = $$3$$ character : $\chi_{210} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$1$$ Selberg data = $$(2,\ 210,\ (\ :3/2),\ -1)$$ $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + p T$$
3 $$1 + p T$$
5 $$1 + p T$$
7 $$1 - p T$$
good11 $$1 - 12 T + p^{3} T^{2}$$
13 $$1 - 2 T + p^{3} T^{2}$$
17 $$1 + 18 T + p^{3} T^{2}$$
19 $$1 - 56 T + p^{3} T^{2}$$
23 $$1 + 156 T + p^{3} T^{2}$$
29 $$1 + 186 T + p^{3} T^{2}$$
31 $$1 + 52 T + p^{3} T^{2}$$
37 $$1 + 178 T + p^{3} T^{2}$$
41 $$1 + 138 T + p^{3} T^{2}$$
43 $$1 + 412 T + p^{3} T^{2}$$
47 $$1 + 456 T + p^{3} T^{2}$$
53 $$1 + 198 T + p^{3} T^{2}$$
59 $$1 - 348 T + p^{3} T^{2}$$
61 $$1 - 110 T + p^{3} T^{2}$$
67 $$1 + 196 T + p^{3} T^{2}$$
71 $$1 + 936 T + p^{3} T^{2}$$
73 $$1 - 542 T + p^{3} T^{2}$$
79 $$1 - 992 T + p^{3} T^{2}$$
83 $$1 + 276 T + p^{3} T^{2}$$
89 $$1 - 630 T + p^{3} T^{2}$$
97 $$1 - 110 T + p^{3} T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}