# Properties

 Degree 2 Conductor $2^{5} \cdot 7$ Sign $1$ Motivic weight 2 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.82·3-s − 8.48·5-s + 7·7-s − 0.999·9-s + 25.4·13-s + 24·15-s + 8.48·19-s − 19.7·21-s + 10·23-s + 46.9·25-s + 28.2·27-s − 59.3·35-s − 72·39-s + 8.48·45-s + 49·49-s − 24·57-s + 76.3·59-s − 8.48·61-s − 6.99·63-s − 215.·65-s − 28.2·69-s + 110·71-s − 132.·75-s − 130·79-s − 71.0·81-s − 25.4·83-s + 178.·91-s + ⋯
 L(s)  = 1 − 0.942·3-s − 1.69·5-s + 7-s − 0.111·9-s + 1.95·13-s + 1.60·15-s + 0.446·19-s − 0.942·21-s + 0.434·23-s + 1.87·25-s + 1.04·27-s − 1.69·35-s − 1.84·39-s + 0.188·45-s + 0.999·49-s − 0.421·57-s + 1.29·59-s − 0.139·61-s − 0.111·63-s − 3.32·65-s − 0.409·69-s + 1.54·71-s − 1.77·75-s − 1.64·79-s − 0.876·81-s − 0.306·83-s + 1.95·91-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$224$$    =    $$2^{5} \cdot 7$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : $\chi_{224} (209, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 224,\ (\ :1),\ 1)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.903497$$ $$L(\frac12)$$ $$\approx$$ $$0.903497$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
7 $$1 - 7T$$
good3 $$1 + 2.82T + 9T^{2}$$
5 $$1 + 8.48T + 25T^{2}$$
11 $$1 - 121T^{2}$$
13 $$1 - 25.4T + 169T^{2}$$
17 $$1 - 289T^{2}$$
19 $$1 - 8.48T + 361T^{2}$$
23 $$1 - 10T + 529T^{2}$$
29 $$1 - 841T^{2}$$
31 $$1 - 961T^{2}$$
37 $$1 - 1.36e3T^{2}$$
41 $$1 - 1.68e3T^{2}$$
43 $$1 - 1.84e3T^{2}$$
47 $$1 - 2.20e3T^{2}$$
53 $$1 - 2.80e3T^{2}$$
59 $$1 - 76.3T + 3.48e3T^{2}$$
61 $$1 + 8.48T + 3.72e3T^{2}$$
67 $$1 - 4.48e3T^{2}$$
71 $$1 - 110T + 5.04e3T^{2}$$
73 $$1 - 5.32e3T^{2}$$
79 $$1 + 130T + 6.24e3T^{2}$$
83 $$1 + 25.4T + 6.88e3T^{2}$$
89 $$1 - 7.92e3T^{2}$$
97 $$1 - 9.40e3T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}