Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 167$ Sign $-0.959 + 0.281i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (0.841 − 0.540i)2-s + (−0.654 − 0.755i)3-s + (0.415 − 0.909i)4-s + (−0.959 − 0.281i)6-s − 0.563i·7-s + (−0.142 − 0.989i)8-s + (−0.142 + 0.989i)9-s − 0.830·11-s + (−0.959 + 0.281i)12-s + (−0.304 − 0.474i)14-s + (−0.654 − 0.755i)16-s + (0.415 + 0.909i)18-s − 1.97i·19-s + (−0.425 + 0.368i)21-s + (−0.698 + 0.449i)22-s + ⋯
 L(s)  = 1 + (0.841 − 0.540i)2-s + (−0.654 − 0.755i)3-s + (0.415 − 0.909i)4-s + (−0.959 − 0.281i)6-s − 0.563i·7-s + (−0.142 − 0.989i)8-s + (−0.142 + 0.989i)9-s − 0.830·11-s + (−0.959 + 0.281i)12-s + (−0.304 − 0.474i)14-s + (−0.654 − 0.755i)16-s + (0.415 + 0.909i)18-s − 1.97i·19-s + (−0.425 + 0.368i)21-s + (−0.698 + 0.449i)22-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$2004$$    =    $$2^{2} \cdot 3 \cdot 167$$ $$\varepsilon$$ = $-0.959 + 0.281i$ motivic weight = $$0$$ character : $\chi_{2004} (2003, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 2004,\ (\ :0),\ -0.959 + 0.281i)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$1.246580327$$ $$L(\frac12)$$ $$\approx$$ $$1.246580327$$ $$L(1)$$ not available $$L(1)$$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;167\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;167\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-0.841 + 0.540i)T$$
3 $$1 + (0.654 + 0.755i)T$$
167 $$1 - T$$
good5 $$1 + T^{2}$$
7 $$1 + 0.563iT - T^{2}$$
11 $$1 + 0.830T + T^{2}$$
13 $$1 - T^{2}$$
17 $$1 + T^{2}$$
19 $$1 + 1.97iT - T^{2}$$
23 $$1 - T^{2}$$
29 $$1 + 0.563iT - T^{2}$$
31 $$1 - 1.08iT - T^{2}$$
37 $$1 - T^{2}$$
41 $$1 + T^{2}$$
43 $$1 + T^{2}$$
47 $$1 - 0.284T + T^{2}$$
53 $$1 + T^{2}$$
59 $$1 - T^{2}$$
61 $$1 - 1.30T + T^{2}$$
67 $$1 + T^{2}$$
71 $$1 - T^{2}$$
73 $$1 - T^{2}$$
79 $$1 + T^{2}$$
83 $$1 - T^{2}$$
89 $$1 - 1.51iT - T^{2}$$
97 $$1 - 0.830T + T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}