# Properties

 Degree 2 Conductor $2^{6} \cdot 7$ Sign $0.755 - 0.654i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2·3-s + 3.46i·5-s + (2 − 1.73i)7-s + 9-s + 3.46i·11-s − 3.46i·13-s + 6.92i·15-s + 2·19-s + (4 − 3.46i)21-s + 3.46i·23-s − 6.99·25-s − 4·27-s − 6·29-s + 8·31-s + 6.92i·33-s + ⋯
 L(s)  = 1 + 1.15·3-s + 1.54i·5-s + (0.755 − 0.654i)7-s + 0.333·9-s + 1.04i·11-s − 0.960i·13-s + 1.78i·15-s + 0.458·19-s + (0.872 − 0.755i)21-s + 0.722i·23-s − 1.39·25-s − 0.769·27-s − 1.11·29-s + 1.43·31-s + 1.20i·33-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$448$$    =    $$2^{6} \cdot 7$$ $$\varepsilon$$ = $0.755 - 0.654i$ motivic weight = $$1$$ character : $\chi_{448} (447, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 448,\ (\ :1/2),\ 0.755 - 0.654i)$ $L(1)$ $\approx$ $1.95274 + 0.728030i$ $L(\frac12)$ $\approx$ $1.95274 + 0.728030i$ $L(\frac{3}{2})$ 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 + (-2 + 1.73i)T$$
good3 $$1 - 2T + 3T^{2}$$
5 $$1 - 3.46iT - 5T^{2}$$
11 $$1 - 3.46iT - 11T^{2}$$
13 $$1 + 3.46iT - 13T^{2}$$
17 $$1 - 17T^{2}$$
19 $$1 - 2T + 19T^{2}$$
23 $$1 - 3.46iT - 23T^{2}$$
29 $$1 + 6T + 29T^{2}$$
31 $$1 - 8T + 31T^{2}$$
37 $$1 - 2T + 37T^{2}$$
41 $$1 + 6.92iT - 41T^{2}$$
43 $$1 + 10.3iT - 43T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 + 6T + 53T^{2}$$
59 $$1 - 6T + 59T^{2}$$
61 $$1 + 3.46iT - 61T^{2}$$
67 $$1 - 3.46iT - 67T^{2}$$
71 $$1 + 3.46iT - 71T^{2}$$
73 $$1 - 6.92iT - 73T^{2}$$
79 $$1 + 3.46iT - 79T^{2}$$
83 $$1 + 6T + 83T^{2}$$
89 $$1 + 6.92iT - 89T^{2}$$
97 $$1 - 13.8iT - 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}