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Properties

Label 1-2001-2001.269-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.575 - 0.818i$
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  + (−0.505 + 0.862i)2-s + (−0.488 − 0.872i)4-s + (0.182 + 0.983i)5-s + (0.339 + 0.940i)7-s + (0.999 + 0.0203i)8-s + (−0.940 − 0.339i)10-s + (−0.953 + 0.301i)11-s + (0.992 + 0.122i)13-s + (−0.983 − 0.182i)14-s + (−0.523 + 0.852i)16-s + (−0.281 + 0.959i)17-s + (−0.872 + 0.488i)19-s + (0.768 − 0.639i)20-s + (0.222 − 0.974i)22-s + (−0.933 + 0.359i)25-s + (−0.607 + 0.794i)26-s + ⋯
L(s)  = 1  + (−0.505 + 0.862i)2-s + (−0.488 − 0.872i)4-s + (0.182 + 0.983i)5-s + (0.339 + 0.940i)7-s + (0.999 + 0.0203i)8-s + (−0.940 − 0.339i)10-s + (−0.953 + 0.301i)11-s + (0.992 + 0.122i)13-s + (−0.983 − 0.182i)14-s + (−0.523 + 0.852i)16-s + (−0.281 + 0.959i)17-s + (−0.872 + 0.488i)19-s + (0.768 − 0.639i)20-s + (0.222 − 0.974i)22-s + (−0.933 + 0.359i)25-s + (−0.607 + 0.794i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.575 - 0.818i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.575 - 0.818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3272676292 + 0.6300433378i\)
\(L(\frac12)\) \(\approx\) \(-0.3272676292 + 0.6300433378i\)
\(L(1)\) \(\approx\) \(0.5017978184 + 0.5394938333i\)
\(L(1)\) \(\approx\) \(0.5017978184 + 0.5394938333i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.505 + 0.862i)T \)
5 \( 1 + (0.182 + 0.983i)T \)
7 \( 1 + (0.339 + 0.940i)T \)
11 \( 1 + (-0.953 + 0.301i)T \)
13 \( 1 + (0.992 + 0.122i)T \)
17 \( 1 + (-0.281 + 0.959i)T \)
19 \( 1 + (-0.872 + 0.488i)T \)
31 \( 1 + (-0.242 + 0.970i)T \)
37 \( 1 + (0.965 - 0.262i)T \)
41 \( 1 + (-0.755 - 0.654i)T \)
43 \( 1 + (0.242 + 0.970i)T \)
47 \( 1 + (-0.433 - 0.900i)T \)
53 \( 1 + (-0.101 + 0.994i)T \)
59 \( 1 + (-0.841 - 0.540i)T \)
61 \( 1 + (-0.202 - 0.979i)T \)
67 \( 1 + (0.301 - 0.953i)T \)
71 \( 1 + (0.557 + 0.830i)T \)
73 \( 1 + (-0.574 + 0.818i)T \)
79 \( 1 + (0.852 - 0.523i)T \)
83 \( 1 + (0.591 - 0.806i)T \)
89 \( 1 + (-0.891 + 0.452i)T \)
97 \( 1 + (0.470 - 0.882i)T \)
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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

−19.65278920806235429225951144800, −18.73522654104966198176317020104, −18.02671412145973813522757693440, −17.46209855890118875686761560009, −16.568494706995380782555697721165, −16.25744231031841202091335024062, −15.21805983360506941911844042567, −13.888908210021180130846583905805, −13.289146168795328359830811699912, −13.07521836479293452273737405470, −11.96787840676032447564325928734, −11.153419888713266491543969509723, −10.68944456875285532426884680048, −9.78595387652102051019442325054, −9.083139081235991123878286963156, −8.2285812072447954409542314917, −7.83603993705079003622364238643, −6.745047719353241168188752517533, −5.52087958340508462725333078173, −4.62650180237881398245149782801, −4.09638058198128070712186667034, −3.027255799864339183069462314115, −2.06186253306445504933725364420, −1.102520343205318118758131061327, −0.31268537353878142516590866340, 1.60478912543921263812019377259, 2.25290308965723214440889240172, 3.44286832672093975274533676568, 4.52609522859632381323583246580, 5.50607336318950525186439949163, 6.14191113880970572718687137505, 6.7000424715084260179671496706, 7.79258521998111346475404811197, 8.30120882489184040469891992322, 9.06263666932794882540960438201, 10.007007132670365741998427310106, 10.73475062914500303978472231240, 11.1583195477183110447354486010, 12.472894017659303985169650137684, 13.228795610262446485348712935687, 14.11911017924057028761151685167, 14.78226926013950457104153921790, 15.37994746210608166909548036259, 15.832392542806516116597032953666, 16.8506021832802547280305076334, 17.6887738566000202346193254078, 18.26294812603255867884774709707, 18.66682497515303566920983107002, 19.32196194355799987913410297357, 20.315548674279596407814855300021

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