# Properties

 Label 1-2001-2001.800-r0-0-0 Degree $1$ Conductor $2001$ Sign $-0.258 + 0.966i$ Analytic cond. $9.29260$ Root an. cond. $9.29260$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.540 + 0.841i)2-s + (−0.415 − 0.909i)4-s + (−0.959 − 0.281i)5-s + (−0.654 + 0.755i)7-s + (0.989 + 0.142i)8-s + (0.755 − 0.654i)10-s + (0.540 + 0.841i)11-s + (0.654 + 0.755i)13-s + (−0.281 − 0.959i)14-s + (−0.654 + 0.755i)16-s + (0.909 + 0.415i)17-s + (0.909 − 0.415i)19-s + (0.142 + 0.989i)20-s − 22-s + (0.841 + 0.540i)25-s + (−0.989 + 0.142i)26-s + ⋯
 L(s)  = 1 + (−0.540 + 0.841i)2-s + (−0.415 − 0.909i)4-s + (−0.959 − 0.281i)5-s + (−0.654 + 0.755i)7-s + (0.989 + 0.142i)8-s + (0.755 − 0.654i)10-s + (0.540 + 0.841i)11-s + (0.654 + 0.755i)13-s + (−0.281 − 0.959i)14-s + (−0.654 + 0.755i)16-s + (0.909 + 0.415i)17-s + (0.909 − 0.415i)19-s + (0.142 + 0.989i)20-s − 22-s + (0.841 + 0.540i)25-s + (−0.989 + 0.142i)26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$2001$$    =    $$3 \cdot 23 \cdot 29$$ Sign: $-0.258 + 0.966i$ Analytic conductor: $$9.29260$$ Root analytic conductor: $$9.29260$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{2001} (800, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 2001,\ (0:\ ),\ -0.258 + 0.966i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.5874571230 + 0.7651272626i$$ $$L(\frac12)$$ $$\approx$$ $$0.5874571230 + 0.7651272626i$$ $$L(1)$$ $$\approx$$ $$0.6408295049 + 0.3497745806i$$ $$L(1)$$ $$\approx$$ $$0.6408295049 + 0.3497745806i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
23 $$1$$
29 $$1$$
good2 $$1 + (-0.540 + 0.841i)T$$
5 $$1 + (-0.959 - 0.281i)T$$
7 $$1 + (-0.654 + 0.755i)T$$
11 $$1 + (0.540 + 0.841i)T$$
13 $$1 + (0.654 + 0.755i)T$$
17 $$1 + (0.909 + 0.415i)T$$
19 $$1 + (0.909 - 0.415i)T$$
31 $$1 + (0.989 + 0.142i)T$$
37 $$1 + (-0.281 - 0.959i)T$$
41 $$1 + (-0.281 + 0.959i)T$$
43 $$1 + (-0.989 + 0.142i)T$$
47 $$1 - iT$$
53 $$1 + (0.654 - 0.755i)T$$
59 $$1 + (0.654 + 0.755i)T$$
61 $$1 + (0.989 + 0.142i)T$$
67 $$1 + (-0.841 - 0.540i)T$$
71 $$1 + (0.841 + 0.540i)T$$
73 $$1 + (-0.909 + 0.415i)T$$
79 $$1 + (-0.755 + 0.654i)T$$
83 $$1 + (0.959 - 0.281i)T$$
89 $$1 + (0.989 - 0.142i)T$$
97 $$1 + (0.281 - 0.959i)T$$
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$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$

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

−19.69577349852606197222483734027, −18.94720098700154072683353215672, −18.696544435015553254966005888837, −17.65043855358243058787999511417, −16.83877241288302586356552994879, −16.197949711726464759294907955946, −15.68907335075253048104826685718, −14.42228222464994375262582725324, −13.69423025443139910089314087191, −13.07405733415550532301281340127, −11.95552354706698247156467878262, −11.75308532502535232447219405473, −10.701663429364387670645541940062, −10.269194303243351551435101529541, −9.4060056347351743577501971309, −8.4458939807130354664475911104, −7.8636459517558625707037176483, −7.1482125085884322954890690807, −6.217125298538465991843481674546, −4.98658895418549271687241857597, −3.8423206150354688760271495403, −3.4300175368330962195786849624, −2.86615710895773537229957075866, −1.19372175099730137145917601466, −0.636524781180682298464293726305, 0.87632092098673468307087101902, 1.87303453209081306378938235749, 3.27991063359869965320909293033, 4.13345691327510954317987922768, 5.0031361528463134151146712149, 5.82837804871033432370690191400, 6.767120038782547875153278782530, 7.23424481151028586874372931916, 8.28952514737803535343029384744, 8.76555770309242229574410312982, 9.59718651269779752045489165493, 10.17304887080294486174486530647, 11.45928438432201255426267990280, 11.90807887684974750652863991746, 12.830078464548967521247760297154, 13.6768465059440253356352409943, 14.69239986534860987241076300458, 15.13243892618404232568614598851, 15.979403712256853934225998811865, 16.32354941590502748671258016565, 17.10169340674175707804359006876, 18.08666780564741142057399529523, 18.622029090580110240873926765585, 19.44414489873775278615092919225, 19.7382551596377887281330583709