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Properties

Label 1-2001-2001.1022-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.959 + 0.281i$
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  + (0.301 + 0.953i)2-s + (−0.818 + 0.574i)4-s + (0.917 + 0.396i)5-s + (−0.101 − 0.994i)7-s + (−0.794 − 0.607i)8-s + (−0.101 + 0.994i)10-s + (−0.933 + 0.359i)11-s + (−0.714 − 0.699i)13-s + (0.917 − 0.396i)14-s + (0.339 − 0.940i)16-s + (−0.959 + 0.281i)17-s + (0.818 − 0.574i)19-s + (−0.979 + 0.202i)20-s + (−0.623 − 0.781i)22-s + (0.685 + 0.728i)25-s + (0.452 − 0.891i)26-s + ⋯
L(s)  = 1  + (0.301 + 0.953i)2-s + (−0.818 + 0.574i)4-s + (0.917 + 0.396i)5-s + (−0.101 − 0.994i)7-s + (−0.794 − 0.607i)8-s + (−0.101 + 0.994i)10-s + (−0.933 + 0.359i)11-s + (−0.714 − 0.699i)13-s + (0.917 − 0.396i)14-s + (0.339 − 0.940i)16-s + (−0.959 + 0.281i)17-s + (0.818 − 0.574i)19-s + (−0.979 + 0.202i)20-s + (−0.623 − 0.781i)22-s + (0.685 + 0.728i)25-s + (0.452 − 0.891i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.959 + 0.281i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1022, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.959 + 0.281i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1639299868 + 1.143077228i\)
\(L(\frac12)\) \(\approx\) \(0.1639299868 + 1.143077228i\)
\(L(1)\) \(\approx\) \(0.8747847399 + 0.5909224153i\)
\(L(1)\) \(\approx\) \(0.8747847399 + 0.5909224153i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.301 + 0.953i)T \)
5 \( 1 + (0.917 + 0.396i)T \)
7 \( 1 + (-0.101 - 0.994i)T \)
11 \( 1 + (-0.933 + 0.359i)T \)
13 \( 1 + (-0.714 - 0.699i)T \)
17 \( 1 + (-0.959 + 0.281i)T \)
19 \( 1 + (0.818 - 0.574i)T \)
31 \( 1 + (0.0203 + 0.999i)T \)
37 \( 1 + (0.591 + 0.806i)T \)
41 \( 1 + (0.654 + 0.755i)T \)
43 \( 1 + (-0.0203 + 0.999i)T \)
47 \( 1 + (0.222 + 0.974i)T \)
53 \( 1 + (-0.992 - 0.122i)T \)
59 \( 1 + (-0.841 + 0.540i)T \)
61 \( 1 + (-0.970 + 0.242i)T \)
67 \( 1 + (0.933 + 0.359i)T \)
71 \( 1 + (-0.996 + 0.0815i)T \)
73 \( 1 + (0.742 + 0.670i)T \)
79 \( 1 + (-0.339 - 0.940i)T \)
83 \( 1 + (0.182 + 0.983i)T \)
89 \( 1 + (-0.768 - 0.639i)T \)
97 \( 1 + (0.999 - 0.0407i)T \)
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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

−19.72994989205302344263705906909, −18.76430769239104547889865248971, −18.40445634478070072383326544515, −17.69393322697988097086278677765, −16.84265568142951271273239657278, −15.88516881013294724430832168233, −15.14429422455954194561092978642, −14.15815971748542804259619120253, −13.70425299411189592122738361628, −12.855747075363063239761467501485, −12.34647423976934464788736421478, −11.54628381819476671870326900808, −10.7711061516562122054990080517, −9.86845208215521531314303230955, −9.30249221054384914568544521502, −8.77356835621412839865504917023, −7.72910364819080950537840140928, −6.389919266725366872073915122136, −5.61962683523592941906599320720, −5.12596584817726832162901757929, −4.26857963959620045065393836305, −3.0435510037604115815967460054, −2.28642371985950031935074828667, −1.83722425107507379918359767989, −0.36113199013220442490176652990, 1.147727133030979508971122880605, 2.64119692202024672321595389168, 3.191901702583891409534915866843, 4.56642195080493208711615237576, 4.94505696055992304489237502848, 5.9630842206142508691487392037, 6.659473477518122226781904436100, 7.42354960728984132857188729350, 7.932623661340927718577720544, 9.10758906675165357922755779042, 9.826362674016885289489700446940, 10.426719437971797234460372808319, 11.312888905611022437316581429107, 12.75181636943758545096401158910, 13.0075657560065422692632806600, 13.80135253354391463842544224724, 14.3657189537803044533957265685, 15.194410504529457253084605207103, 15.834819343790116840538070212149, 16.67955229780847862124167713015, 17.49036435516889828713655571255, 17.78623448126385401552318785584, 18.43766164005841580955594150549, 19.61956458677948885495986559286, 20.333780416527853674733503879118

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