Properties

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

Related objects

Dirichlet series

 L(s)  = 1 + (−1 − i)2-s + 3-s + i·4-s + (−1 − i)6-s + 9-s + i·12-s − 2i·13-s + 16-s + (−1 − i)18-s + i·23-s + 25-s + (−2 + 2i)26-s + 27-s − i·29-s + (−1 + i)31-s + (−1 − i)32-s + ⋯
 L(s)  = 1 + (−1 − i)2-s + 3-s + i·4-s + (−1 − i)6-s + 9-s + i·12-s − 2i·13-s + 16-s + (−1 − i)18-s + i·23-s + 25-s + (−2 + 2i)26-s + 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 & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$2001$$    =    $$3 \cdot 23 \cdot 29$$ $$\varepsilon$$ = $-0.189 + 0.981i$ motivic weight = $$0$$ character : $\chi_{2001} (1931, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 2001,\ (\ :0),\ -0.189 + 0.981i)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.9781764484$$ $$L(\frac12)$$ $$\approx$$ $$0.9781764484$$ $$L(1)$$ 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 - T$$
23 $$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}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}