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Properties

Label 1-2001-2001.1169-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.998 - 0.0622i$
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  + (0.182 + 0.983i)2-s + (−0.933 + 0.359i)4-s + (0.970 + 0.242i)5-s + (0.0611 + 0.998i)7-s + (−0.523 − 0.852i)8-s + (−0.0611 + 0.998i)10-s + (−0.917 − 0.396i)11-s + (0.986 − 0.162i)13-s + (−0.970 + 0.242i)14-s + (0.742 − 0.670i)16-s + (0.142 − 0.989i)17-s + (−0.933 + 0.359i)19-s + (−0.992 + 0.122i)20-s + (0.222 − 0.974i)22-s + (0.882 + 0.470i)25-s + (0.339 + 0.940i)26-s + ⋯
L(s)  = 1  + (0.182 + 0.983i)2-s + (−0.933 + 0.359i)4-s + (0.970 + 0.242i)5-s + (0.0611 + 0.998i)7-s + (−0.523 − 0.852i)8-s + (−0.0611 + 0.998i)10-s + (−0.917 − 0.396i)11-s + (0.986 − 0.162i)13-s + (−0.970 + 0.242i)14-s + (0.742 − 0.670i)16-s + (0.142 − 0.989i)17-s + (−0.933 + 0.359i)19-s + (−0.992 + 0.122i)20-s + (0.222 − 0.974i)22-s + (0.882 + 0.470i)25-s + (0.339 + 0.940i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.998 - 0.0622i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.998 - 0.0622i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04170951895 + 1.339085320i\)
\(L(\frac12)\) \(\approx\) \(-0.04170951895 + 1.339085320i\)
\(L(1)\) \(\approx\) \(0.8100848249 + 0.7443126373i\)
\(L(1)\) \(\approx\) \(0.8100848249 + 0.7443126373i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.182 + 0.983i)T \)
5 \( 1 + (0.970 + 0.242i)T \)
7 \( 1 + (0.0611 + 0.998i)T \)
11 \( 1 + (-0.917 - 0.396i)T \)
13 \( 1 + (0.986 - 0.162i)T \)
17 \( 1 + (0.142 - 0.989i)T \)
19 \( 1 + (-0.933 + 0.359i)T \)
31 \( 1 + (-0.947 + 0.320i)T \)
37 \( 1 + (-0.768 + 0.639i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (0.947 + 0.320i)T \)
47 \( 1 + (-0.900 + 0.433i)T \)
53 \( 1 + (-0.377 + 0.925i)T \)
59 \( 1 + (0.959 + 0.281i)T \)
61 \( 1 + (-0.714 + 0.699i)T \)
67 \( 1 + (-0.917 + 0.396i)T \)
71 \( 1 + (-0.262 + 0.965i)T \)
73 \( 1 + (-0.685 + 0.728i)T \)
79 \( 1 + (0.742 + 0.670i)T \)
83 \( 1 + (-0.979 - 0.202i)T \)
89 \( 1 + (-0.101 + 0.994i)T \)
97 \( 1 + (0.794 - 0.607i)T \)
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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

−19.66450470289520591414947489981, −18.97977332892230795143778807990, −18.07637313381941994753264616818, −17.58804217745837423744299449849, −16.9433447817038749479440646594, −16.016740893096578462057070360485, −14.903180577644506941591653786110, −14.22411009677972537164922890184, −13.437977112908426304498257454977, −13.01882076542526963316956575954, −12.45676481020614520723192919508, −11.16446175503265752204786943451, −10.56948884889482107657027819254, −10.25229809231407425560210177526, −9.22705889595558907136342384119, −8.59097937014418330761327931935, −7.65528213262790634197782766591, −6.479905467991811248573850853050, −5.71339839169195032788226397169, −4.8782309116132526410356346863, −4.07135689612895833846436509229, −3.314708343355456292409982464710, −2.07108811046283050043817958620, −1.6646706147215683110295424468, −0.42354385066992363021440292807, 1.320398882434471627540960818631, 2.59008168926660708174711006400, 3.21370638119729034112827587092, 4.50005121525435770526218979699, 5.41175215511884611004084512262, 5.830484268418879639171081470792, 6.4986725993767636688039452638, 7.469533835253012268737378126853, 8.41012186817202076414713185422, 8.90371057805269378444225398248, 9.70724802985273673817453549022, 10.55143738227502166589140788585, 11.45609916764684536973358325110, 12.638007242990580043305477974401, 13.06524805098568597457985457509, 13.84617761336786363870845243164, 14.49868329224138757247684480064, 15.23578875699952855354390169, 15.99442635463082787099033500890, 16.47811337489665000391844393222, 17.49142322339536605795439495326, 18.20352358548971379117320902041, 18.44765928348624815356922575927, 19.21660313417472659087566733965, 20.799691307058965260641983298604

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