# Properties

 Degree 2 Conductor $3 \cdot 23 \cdot 29$ Sign $0.981 - 0.189i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.366 − 0.366i)2-s + (−0.866 − 0.5i)3-s + 0.732i·4-s + (−0.5 + 0.133i)6-s + (0.633 + 0.633i)8-s + (0.499 + 0.866i)9-s + (0.366 − 0.633i)12-s − i·13-s − 0.267·16-s + (0.499 + 0.133i)18-s + i·23-s + (−0.232 − 0.866i)24-s + 25-s + (−0.366 − 0.366i)26-s − 0.999i·27-s + ⋯
 L(s)  = 1 + (0.366 − 0.366i)2-s + (−0.866 − 0.5i)3-s + 0.732i·4-s + (−0.5 + 0.133i)6-s + (0.633 + 0.633i)8-s + (0.499 + 0.866i)9-s + (0.366 − 0.633i)12-s − i·13-s − 0.267·16-s + (0.499 + 0.133i)18-s + i·23-s + (−0.232 − 0.866i)24-s + 25-s + (−0.366 − 0.366i)26-s − 0.999i·27-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$2001$$    =    $$3 \cdot 23 \cdot 29$$ $$\varepsilon$$ = $0.981 - 0.189i$ motivic weight = $$0$$ character : $\chi_{2001} (1172, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 2001,\ (\ :0),\ 0.981 - 0.189i)$ $L(\frac{1}{2})$ $\approx$ $1.084475138$ $L(\frac12)$ $\approx$ $1.084475138$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;23,\;29\}$, $$F_p(T)$$ is a polynomial of degree 2. If $p \in \{3,\;23,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 + (0.866 + 0.5i)T$$
23 $$1 - iT$$
29 $$1 + (0.866 - 0.5i)T$$
good2 $$1 + (-0.366 + 0.366i)T - iT^{2}$$
5 $$1 - T^{2}$$
7 $$1 - T^{2}$$
11 $$1 + iT^{2}$$
13 $$1 + iT - T^{2}$$
17 $$1 + iT^{2}$$
19 $$1 - iT^{2}$$
31 $$1 + (-1.36 - 1.36i)T + iT^{2}$$
37 $$1 + iT^{2}$$
41 $$1 + (-1.36 - 1.36i)T + iT^{2}$$
43 $$1 - iT^{2}$$
47 $$1 + (-0.366 - 0.366i)T + iT^{2}$$
53 $$1 + T^{2}$$
59 $$1 + 2iT - T^{2}$$
61 $$1 - iT^{2}$$
67 $$1 + T^{2}$$
71 $$1 + 1.73T + T^{2}$$
73 $$1 + (-1.36 + 1.36i)T - iT^{2}$$
79 $$1 - iT^{2}$$
83 $$1 + T^{2}$$
89 $$1 + iT^{2}$$
97 $$1 + iT^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}