# Properties

 Label 2-2001-2001.1931-c0-0-5 Degree $2$ Conductor $2001$ Sign $0.981 + 0.189i$ Analytic cond. $0.998629$ Root an. cond. $0.999314$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1 + i)2-s − i·3-s + i·4-s + (1 − i)6-s − 9-s + 12-s − 2i·13-s + 16-s + (−1 − i)18-s − i·23-s + 25-s + (2 − 2i)26-s + i·27-s + i·29-s + (−1 + i)31-s + (1 + i)32-s + ⋯
 L(s)  = 1 + (1 + i)2-s − i·3-s + i·4-s + (1 − i)6-s − 9-s + 12-s − 2i·13-s + 16-s + (−1 − i)18-s − i·23-s + 25-s + (2 − 2i)26-s + i·27-s + i·29-s + (−1 + i)31-s + (1 + i)32-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

 Degree: $$2$$ Conductor: $$2001$$    =    $$3 \cdot 23 \cdot 29$$ Sign: $0.981 + 0.189i$ Analytic conductor: $$0.998629$$ Root analytic conductor: $$0.999314$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{2001} (1931, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2001,\ (\ :0),\ 0.981 + 0.189i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.933768823$$ $$L(\frac12)$$ $$\approx$$ $$1.933768823$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + iT$$
23 $$1 + iT$$
29 $$1 - iT$$
good2 $$1 + (-1 - i)T + iT^{2}$$
5 $$1 - T^{2}$$
7 $$1 - T^{2}$$
11 $$1 - iT^{2}$$
13 $$1 + 2iT - T^{2}$$
17 $$1 - iT^{2}$$
19 $$1 + iT^{2}$$
31 $$1 + (1 - i)T - iT^{2}$$
37 $$1 - iT^{2}$$
41 $$1 + (1 - i)T - iT^{2}$$
43 $$1 + iT^{2}$$
47 $$1 + (-1 + i)T - iT^{2}$$
53 $$1 + T^{2}$$
59 $$1 - 2iT - T^{2}$$
61 $$1 + iT^{2}$$
67 $$1 + T^{2}$$
71 $$1 + T^{2}$$
73 $$1 + (1 + i)T + iT^{2}$$
79 $$1 + iT^{2}$$
83 $$1 + T^{2}$$
89 $$1 - iT^{2}$$
97 $$1 - iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$