# Properties

 Degree 2 Conductor $3^{2} \cdot 239$ Sign $1$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + 5-s − 1.41i·7-s + 8-s − 10-s + 11-s + 1.41i·13-s + 1.41i·14-s − 16-s + 17-s + 1.41i·19-s − 22-s + 1.41i·23-s − 1.41i·26-s + 29-s + ⋯
 L(s)  = 1 − 2-s + 5-s − 1.41i·7-s + 8-s − 10-s + 11-s + 1.41i·13-s + 1.41i·14-s − 16-s + 17-s + 1.41i·19-s − 22-s + 1.41i·23-s − 1.41i·26-s + 29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$2151$$    =    $$3^{2} \cdot 239$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : $\chi_{2151} (955, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 2151,\ (\ :0),\ 1)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.8490923334$$ $$L(\frac12)$$ $$\approx$$ $$0.8490923334$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;239\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{3,\;239\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1$$
239 $$1 - T$$
good2 $$1 + T + T^{2}$$
5 $$1 - T + T^{2}$$
7 $$1 + 1.41iT - T^{2}$$
11 $$1 - T + T^{2}$$
13 $$1 - 1.41iT - T^{2}$$
17 $$1 - T + T^{2}$$
19 $$1 - 1.41iT - T^{2}$$
23 $$1 - 1.41iT - T^{2}$$
29 $$1 - T + T^{2}$$
31 $$1 + T + T^{2}$$
37 $$1 + 1.41iT - T^{2}$$
41 $$1 + 1.41iT - T^{2}$$
43 $$1 - T^{2}$$
47 $$1 - 1.41iT - T^{2}$$
53 $$1 - T^{2}$$
59 $$1 - T^{2}$$
61 $$1 - T + T^{2}$$
67 $$1 + T + T^{2}$$
71 $$1 + T^{2}$$
73 $$1 - T^{2}$$
79 $$1 - T^{2}$$
83 $$1 + T + T^{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}