# Properties

 Degree 2 Conductor $2^{3} \cdot 11^{2}$ Sign $0.426 + 0.904i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 3-s + 5-s − 1.41i·7-s − 15-s − 1.41i·17-s + 1.41i·21-s − 23-s + 27-s + 31-s − 1.41i·35-s + 37-s − 1.41i·43-s − 1.00·49-s + 1.41i·51-s + 59-s + ⋯
 L(s)  = 1 − 3-s + 5-s − 1.41i·7-s − 15-s − 1.41i·17-s + 1.41i·21-s − 23-s + 27-s + 31-s − 1.41i·35-s + 37-s − 1.41i·43-s − 1.00·49-s + 1.41i·51-s + 59-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$968$$    =    $$2^{3} \cdot 11^{2}$$ $$\varepsilon$$ = $0.426 + 0.904i$ motivic weight = $$0$$ character : $\chi_{968} (241, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 968,\ (\ :0),\ 0.426 + 0.904i)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.7886763449$$ $$L(\frac12)$$ $$\approx$$ $$0.7886763449$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;11\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
11 $$1$$
good3 $$1 + T + T^{2}$$
5 $$1 - T + T^{2}$$
7 $$1 + 1.41iT - T^{2}$$
13 $$1 - T^{2}$$
17 $$1 + 1.41iT - T^{2}$$
19 $$1 - T^{2}$$
23 $$1 + T + T^{2}$$
29 $$1 - T^{2}$$
31 $$1 - T + T^{2}$$
37 $$1 - T + T^{2}$$
41 $$1 - T^{2}$$
43 $$1 + 1.41iT - T^{2}$$
47 $$1 + T^{2}$$
53 $$1 + T^{2}$$
59 $$1 - T + T^{2}$$
61 $$1 - T^{2}$$
67 $$1 - T + T^{2}$$
71 $$1 + T + T^{2}$$
73 $$1 - T^{2}$$
79 $$1 - 1.41iT - T^{2}$$
83 $$1 - 1.41iT - T^{2}$$
89 $$1 + T + T^{2}$$
97 $$1 + T + T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}