# Properties

 Degree 2 Conductor $2^{4} \cdot 7$ Sign $0.382 - 0.923i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + i·2-s − 4-s + i·7-s − i·8-s − i·9-s + (−1 − i)11-s − 14-s + 16-s + 18-s + (1 − i)22-s + i·25-s − i·28-s + (−1 + i)29-s + i·32-s + i·36-s + (−1 − i)37-s + ⋯
 L(s)  = 1 + i·2-s − 4-s + i·7-s − i·8-s − i·9-s + (−1 − i)11-s − 14-s + 16-s + 18-s + (1 − i)22-s + i·25-s − i·28-s + (−1 + i)29-s + i·32-s + i·36-s + (−1 − i)37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$112$$    =    $$2^{4} \cdot 7$$ $$\varepsilon$$ = $0.382 - 0.923i$ motivic weight = $$0$$ character : $\chi_{112} (69, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 112,\ (\ :0),\ 0.382 - 0.923i)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.5314225250$$ $$L(\frac12)$$ $$\approx$$ $$0.5314225250$$ $$L(1)$$ 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 - iT$$
7 $$1 - iT$$
good3 $$1 + iT^{2}$$
5 $$1 - iT^{2}$$
11 $$1 + (1 + i)T + iT^{2}$$
13 $$1 + iT^{2}$$
17 $$1 - T^{2}$$
19 $$1 + iT^{2}$$
23 $$1 - T^{2}$$
29 $$1 + (1 - i)T - iT^{2}$$
31 $$1 - T^{2}$$
37 $$1 + (1 + i)T + iT^{2}$$
41 $$1 + T^{2}$$
43 $$1 + (-1 - i)T + iT^{2}$$
47 $$1 - T^{2}$$
53 $$1 + (-1 - i)T + iT^{2}$$
59 $$1 - iT^{2}$$
61 $$1 + iT^{2}$$
67 $$1 + (-1 + i)T - iT^{2}$$
71 $$1 + 2iT - T^{2}$$
73 $$1 + T^{2}$$
79 $$1 + T^{2}$$
83 $$1 + iT^{2}$$
89 $$1 + T^{2}$$
97 $$1 - T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}