# 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 + (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

 $$d$$ = $$2$$ $$N$$ = $$2001$$    =    $$3 \cdot 23 \cdot 29$$ $$\varepsilon$$ = $0.981 + 0.189i$ motivic weight = $$0$$ character : $\chi_{2001} (1931, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 2001,\ (\ :0),\ 0.981 + 0.189i)$ $L(\frac{1}{2})$ $\approx$ $1.933768823$ $L(\frac12)$ $\approx$ $1.933768823$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;23,\;29\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{3,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 $$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}$$
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\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

## Imaginary part of the first few zeros on the critical line

−8.818386907003194553262343213001, −8.311054210908338613961222226780, −7.38519636203980971489244978722, −7.01173845027374617510831424490, −6.08930402049363758613191143349, −5.46915395071510417297680507734, −4.84606035679535512913514622259, −3.48871735581238695571082916185, −2.76570478491314705304799496869, −1.11946198236062556720060919945, 1.80863970995744292176853033784, 2.71534346064884293610064227614, 3.85620080643315639556510783474, 4.14873997327109448197734214665, 5.06090099359527545749738494172, 5.77487939191565230315492397537, 6.82301674379719791647076645185, 7.909930291327359386067163393486, 9.013583199878180017521722825469, 9.448714402438412153078856361613