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Properties

Label 1-2001-2001.500-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.380 + 0.924i$
Analytic cond. $9.29260$
Root an. cond. $9.29260$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.917 − 0.396i)2-s + (0.685 + 0.728i)4-s + (0.0203 + 0.999i)5-s + (0.377 + 0.925i)7-s + (−0.339 − 0.940i)8-s + (0.377 − 0.925i)10-s + (0.882 + 0.470i)11-s + (0.488 + 0.872i)13-s + (0.0203 − 0.999i)14-s + (−0.0611 + 0.998i)16-s + (−0.142 − 0.989i)17-s + (−0.685 − 0.728i)19-s + (−0.714 + 0.699i)20-s + (−0.623 − 0.781i)22-s + (−0.999 + 0.0407i)25-s + (−0.101 − 0.994i)26-s + ⋯
L(s)  = 1  + (−0.917 − 0.396i)2-s + (0.685 + 0.728i)4-s + (0.0203 + 0.999i)5-s + (0.377 + 0.925i)7-s + (−0.339 − 0.940i)8-s + (0.377 − 0.925i)10-s + (0.882 + 0.470i)11-s + (0.488 + 0.872i)13-s + (0.0203 − 0.999i)14-s + (−0.0611 + 0.998i)16-s + (−0.142 − 0.989i)17-s + (−0.685 − 0.728i)19-s + (−0.714 + 0.699i)20-s + (−0.623 − 0.781i)22-s + (−0.999 + 0.0407i)25-s + (−0.101 − 0.994i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.380 + 0.924i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (500, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.380 + 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5654089613 + 0.8444899273i\)
\(L(\frac12)\) \(\approx\) \(0.5654089613 + 0.8444899273i\)
\(L(1)\) \(\approx\) \(0.7451641512 + 0.2311965771i\)
\(L(1)\) \(\approx\) \(0.7451641512 + 0.2311965771i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.917 - 0.396i)T \)
5 \( 1 + (0.0203 + 0.999i)T \)
7 \( 1 + (0.377 + 0.925i)T \)
11 \( 1 + (0.882 + 0.470i)T \)
13 \( 1 + (0.488 + 0.872i)T \)
17 \( 1 + (-0.142 - 0.989i)T \)
19 \( 1 + (-0.685 - 0.728i)T \)
31 \( 1 + (-0.523 + 0.852i)T \)
37 \( 1 + (0.979 - 0.202i)T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (0.523 + 0.852i)T \)
47 \( 1 + (0.222 + 0.974i)T \)
53 \( 1 + (0.986 - 0.162i)T \)
59 \( 1 + (0.959 - 0.281i)T \)
61 \( 1 + (-0.947 - 0.320i)T \)
67 \( 1 + (-0.882 + 0.470i)T \)
71 \( 1 + (0.591 - 0.806i)T \)
73 \( 1 + (0.557 - 0.830i)T \)
79 \( 1 + (0.0611 + 0.998i)T \)
83 \( 1 + (0.970 + 0.242i)T \)
89 \( 1 + (-0.992 - 0.122i)T \)
97 \( 1 + (0.452 + 0.891i)T \)
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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

−19.732110048721321201564249266812, −19.06152619717843923311915358362, −18.10808018953431338313831491147, −17.34400885466401466865557708747, −16.8000150333729245003289295938, −16.51520573871409371514374602806, −15.407107149094863634795035115285, −14.827687320117586267558108558962, −13.90961655137506810735250162262, −13.17987484340208829872474515788, −12.25887026031037639354611866091, −11.39732753072997501542899694509, −10.62296453761444152930684238021, −10.060348140237286529886926751672, −9.04533689365880736489968215609, −8.43930070651487454163342524144, −7.92371248318817702590161669962, −7.01578393454309020423491917794, −6.00683294249437672744587646073, −5.54033187073595668277387587895, −4.28038874996092874510898614001, −3.66847373260155103345020922723, −2.061543840675293374936857198382, −1.2815682951807998960396679072, −0.519638387368779095762126734897, 1.28137337282506230845734340462, 2.20047369518612541924991087194, 2.768195954636572904538098918652, 3.79047161526324317642545536156, 4.73186932938013441872821728809, 6.15324452890302306689751747110, 6.65788481516380114221349290056, 7.38616752267224625567556803855, 8.309347067327662261673918687787, 9.28697032721168885617746242598, 9.41933802029129366336460308855, 10.660313491359530145538203264220, 11.29996829927424999666678094723, 11.7262201245836656824404378440, 12.52176105686246574648332778794, 13.58274716253576457034917365582, 14.53931541978514123922106473090, 15.0774769750721242276324834096, 15.91436976922871693523002648766, 16.611882381792579862299952283899, 17.6257045911156057035512303219, 18.07717750080442333860085166001, 18.62003676587365874482443274969, 19.39444413281955866218440672757, 19.886836390360602675682140466919

Graph of the $Z$-function along the critical line