# Properties

 Degree $2$ Conductor $1000$ Sign $1$ Motivic weight $0$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·2-s − 4-s + 1.61i·7-s + i·8-s − 9-s + 0.618·11-s + 0.618i·13-s + 1.61·14-s + 16-s + i·18-s + 1.61·19-s − 0.618i·22-s + 0.618i·23-s + 0.618·26-s − 1.61i·28-s + ⋯
 L(s)  = 1 − i·2-s − 4-s + 1.61i·7-s + i·8-s − 9-s + 0.618·11-s + 0.618i·13-s + 1.61·14-s + 16-s + i·18-s + 1.61·19-s − 0.618i·22-s + 0.618i·23-s + 0.618·26-s − 1.61i·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1000$$    =    $$2^{3} \cdot 5^{3}$$ Sign: $1$ Motivic weight: $$0$$ Character: $\chi_{1000} (251, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1000,\ (\ :0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8505397702$$ $$L(\frac12)$$ $$\approx$$ $$0.8505397702$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + iT$$
5 $$1$$
good3 $$1 + T^{2}$$
7 $$1 - 1.61iT - T^{2}$$
11 $$1 - 0.618T + T^{2}$$
13 $$1 - 0.618iT - T^{2}$$
17 $$1 + T^{2}$$
19 $$1 - 1.61T + T^{2}$$
23 $$1 - 0.618iT - T^{2}$$
29 $$1 - T^{2}$$
31 $$1 - T^{2}$$
37 $$1 - 1.61iT - T^{2}$$
41 $$1 + 1.61T + T^{2}$$
43 $$1 + T^{2}$$
47 $$1 + 0.618iT - T^{2}$$
53 $$1 + 1.61iT - T^{2}$$
59 $$1 - 1.61T + T^{2}$$
61 $$1 - T^{2}$$
67 $$1 + T^{2}$$
71 $$1 - T^{2}$$
73 $$1 + T^{2}$$
79 $$1 - T^{2}$$
83 $$1 + T^{2}$$
89 $$1 + 0.618T + T^{2}$$
97 $$1 + T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

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

−10.02137939848365779774768363054, −9.397382726591370986549255208385, −8.725658628376628377253661881080, −8.142710175677256728199506049288, −6.64651576831471675882014119021, −5.52315502847623226942703368889, −5.08697837931114319036111818488, −3.57312699061533403071699906158, −2.80119684492478915363972911188, −1.69926707956014038299708096864, 0.886336849121244270002008400291, 3.23438279179993723560700966856, 4.05044386761420648592790285583, 5.11089164664135601252674248407, 5.95425057796081598403791478785, 6.96030708153170355150733223390, 7.52763351210575740728574484470, 8.339414635898990353612887641657, 9.230219139527150670788931860659, 10.05140006647898626354693581657