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Properties

Label 1-2001-2001.50-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.923 + 0.383i$
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  + (−0.983 + 0.182i)2-s + (0.933 − 0.359i)4-s + (0.970 + 0.242i)5-s + (−0.0611 − 0.998i)7-s + (−0.852 + 0.523i)8-s + (−0.998 − 0.0611i)10-s + (−0.396 + 0.917i)11-s + (−0.986 + 0.162i)13-s + (0.242 + 0.970i)14-s + (0.742 − 0.670i)16-s + (−0.989 − 0.142i)17-s + (0.359 + 0.933i)19-s + (0.992 − 0.122i)20-s + (0.222 − 0.974i)22-s + (0.882 + 0.470i)25-s + (0.940 − 0.339i)26-s + ⋯
L(s)  = 1  + (−0.983 + 0.182i)2-s + (0.933 − 0.359i)4-s + (0.970 + 0.242i)5-s + (−0.0611 − 0.998i)7-s + (−0.852 + 0.523i)8-s + (−0.998 − 0.0611i)10-s + (−0.396 + 0.917i)11-s + (−0.986 + 0.162i)13-s + (0.242 + 0.970i)14-s + (0.742 − 0.670i)16-s + (−0.989 − 0.142i)17-s + (0.359 + 0.933i)19-s + (0.992 − 0.122i)20-s + (0.222 − 0.974i)22-s + (0.882 + 0.470i)25-s + (0.940 − 0.339i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.923 + 0.383i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.923 + 0.383i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.021539035 + 0.2039313157i\)
\(L(\frac12)\) \(\approx\) \(1.021539035 + 0.2039313157i\)
\(L(1)\) \(\approx\) \(0.7729401746 + 0.06311572107i\)
\(L(1)\) \(\approx\) \(0.7729401746 + 0.06311572107i\)

Euler product

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

−19.71384471061178946744133884174, −19.19972735474140181826836095691, −18.226969593883572540330829614186, −17.890563426143233841455904569803, −17.169384685840243675425889027652, −16.3887221542247443683616672920, −15.69326417887845077481422395567, −15.021382566801680270910866443009, −14.03021857057201366569187745972, −13.09149722995130941052245287709, −12.508423033745435475143097896539, −11.61940470329888802355948592422, −10.90193396847817672916106488442, −10.118691480702198929962293757534, −9.31476880462754641687961909267, −8.8595435003604940528746964004, −8.16989296781518636913650887668, −7.07242663916735646140112770255, −6.33706977254892228323373286037, −5.556751562596544255436682423343, −4.80503666978062330653106552135, −3.165787575126623295524894592135, −2.54658040243356872386385381203, −1.88670807314882352471970994131, −0.65767410900141819130958563642, 0.77476436776709034705241146754, 2.061481571898046731172555390156, 2.31591490345833986758215408151, 3.70737062493687979330730337570, 4.84402025856845746916440500183, 5.699481926118053580333397237034, 6.620212502374815337407010114528, 7.28634397745487489007218620623, 7.74535438593072569892705979744, 9.03837159338459896602729847796, 9.56559549504708334763844116170, 10.32425502370397055869785791007, 10.646177891655303814139668253032, 11.717047582952001066598480377475, 12.619525466361408219409625806134, 13.43425532966052785906124249721, 14.37287620129218110195995245194, 14.80547152845359017882469908732, 15.838948164531894985014724347953, 16.55166679135106828153675957724, 17.381473374129459054813885341305, 17.59903060398059966748460846550, 18.36802289455151016772841918084, 19.24922599701051087520731282425, 19.91263504646261753968230117523

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