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Properties

Label 1-2001-2001.1268-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.501 + 0.865i$
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  + (−0.925 − 0.377i)2-s + (0.714 + 0.699i)4-s + (−0.862 − 0.505i)5-s + (0.794 + 0.607i)7-s + (−0.396 − 0.917i)8-s + (0.607 + 0.794i)10-s + (0.162 + 0.986i)11-s + (0.768 + 0.639i)13-s + (−0.505 − 0.862i)14-s + (0.0203 + 0.999i)16-s + (−0.540 + 0.841i)17-s + (−0.699 + 0.714i)19-s + (−0.262 − 0.965i)20-s + (0.222 − 0.974i)22-s + (0.488 + 0.872i)25-s + (−0.470 − 0.882i)26-s + ⋯
L(s)  = 1  + (−0.925 − 0.377i)2-s + (0.714 + 0.699i)4-s + (−0.862 − 0.505i)5-s + (0.794 + 0.607i)7-s + (−0.396 − 0.917i)8-s + (0.607 + 0.794i)10-s + (0.162 + 0.986i)11-s + (0.768 + 0.639i)13-s + (−0.505 − 0.862i)14-s + (0.0203 + 0.999i)16-s + (−0.540 + 0.841i)17-s + (−0.699 + 0.714i)19-s + (−0.262 − 0.965i)20-s + (0.222 − 0.974i)22-s + (0.488 + 0.872i)25-s + (−0.470 − 0.882i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.501 + 0.865i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.501 + 0.865i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3130098375 + 0.5431307540i\)
\(L(\frac12)\) \(\approx\) \(0.3130098375 + 0.5431307540i\)
\(L(1)\) \(\approx\) \(0.6279469773 + 0.08547157272i\)
\(L(1)\) \(\approx\) \(0.6279469773 + 0.08547157272i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.925 - 0.377i)T \)
5 \( 1 + (-0.862 - 0.505i)T \)
7 \( 1 + (0.794 + 0.607i)T \)
11 \( 1 + (0.162 + 0.986i)T \)
13 \( 1 + (0.768 + 0.639i)T \)
17 \( 1 + (-0.540 + 0.841i)T \)
19 \( 1 + (-0.699 + 0.714i)T \)
31 \( 1 + (0.983 - 0.182i)T \)
37 \( 1 + (0.830 + 0.557i)T \)
41 \( 1 + (-0.989 + 0.142i)T \)
43 \( 1 + (-0.983 - 0.182i)T \)
47 \( 1 + (0.433 + 0.900i)T \)
53 \( 1 + (0.452 + 0.891i)T \)
59 \( 1 + (-0.415 - 0.909i)T \)
61 \( 1 + (-0.806 + 0.591i)T \)
67 \( 1 + (-0.986 - 0.162i)T \)
71 \( 1 + (0.742 - 0.670i)T \)
73 \( 1 + (-0.320 + 0.947i)T \)
79 \( 1 + (-0.999 - 0.0203i)T \)
83 \( 1 + (-0.996 + 0.0815i)T \)
89 \( 1 + (-0.0407 - 0.999i)T \)
97 \( 1 + (-0.359 - 0.933i)T \)
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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

−19.76346652115490571728532210141, −18.81282089217411655306308821881, −18.26955849324396022443041664888, −17.6393181727873689365564943206, −16.78389348954298028821302426431, −16.13665445627165150229062540417, −15.383292621599718353777454346908, −14.88812889550859207630267080123, −13.95406019399615552785246194605, −13.34402923683752489796409039591, −11.844543712875309730568781435010, −11.36852064935949737641338837761, −10.777817970955520631623631740805, −10.22381049785413016601924850862, −8.92701577906316334105581501297, −8.395871877073941956897708738221, −7.79764002633841862117847423505, −6.95050881810031000605471418363, −6.371076466139362092428588325518, −5.28720661802077923947974877371, −4.35985759224198736504022920834, −3.321441745145809328538322634316, −2.45008151095690759296492009337, −1.15434127581059360921375751892, −0.330740885054602140300671688219, 1.384007339614935070682301647510, 1.80461916827968873758269144206, 2.970383402456449262962459146231, 4.1756767414679241614954628377, 4.49048070852570961979603788889, 5.93916679752064689073856000641, 6.761777534766942489814471086978, 7.712705661825026505068772529383, 8.386347987555409536439315802488, 8.76235700228854802348789175794, 9.6921987246278164880376196222, 10.61975843610467207143943169764, 11.34349329319833237963471549233, 11.95868029555433394749366706768, 12.48369777618684392722907680360, 13.33396081882469898107562459888, 14.64807324221415246095232158942, 15.34086680869714926382165598792, 15.75199422667976083113994967316, 16.952734172289296032191879944901, 17.09325053808867402433646861644, 18.254622913474036385910336863196, 18.64233415722099962599932829870, 19.46506455911461126162866697605, 20.12735581747376575368474660647

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