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Properties

Label 1-2001-2001.1235-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.489 - 0.871i$
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  + (0.989 − 0.142i)2-s + (0.959 − 0.281i)4-s + (−0.654 + 0.755i)5-s + (0.841 − 0.540i)7-s + (0.909 − 0.415i)8-s + (−0.540 + 0.841i)10-s + (−0.989 − 0.142i)11-s + (−0.841 − 0.540i)13-s + (0.755 − 0.654i)14-s + (0.841 − 0.540i)16-s + (0.281 − 0.959i)17-s + (0.281 + 0.959i)19-s + (−0.415 + 0.909i)20-s − 22-s + (−0.142 − 0.989i)25-s + (−0.909 − 0.415i)26-s + ⋯
L(s)  = 1  + (0.989 − 0.142i)2-s + (0.959 − 0.281i)4-s + (−0.654 + 0.755i)5-s + (0.841 − 0.540i)7-s + (0.909 − 0.415i)8-s + (−0.540 + 0.841i)10-s + (−0.989 − 0.142i)11-s + (−0.841 − 0.540i)13-s + (0.755 − 0.654i)14-s + (0.841 − 0.540i)16-s + (0.281 − 0.959i)17-s + (0.281 + 0.959i)19-s + (−0.415 + 0.909i)20-s − 22-s + (−0.142 − 0.989i)25-s + (−0.909 − 0.415i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.489 - 0.871i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.489 - 0.871i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.404255780 - 1.406801838i\)
\(L(\frac12)\) \(\approx\) \(2.404255780 - 1.406801838i\)
\(L(1)\) \(\approx\) \(1.773809338 - 0.3452450869i\)
\(L(1)\) \(\approx\) \(1.773809338 - 0.3452450869i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.989 - 0.142i)T \)
5 \( 1 + (-0.654 + 0.755i)T \)
7 \( 1 + (0.841 - 0.540i)T \)
11 \( 1 + (-0.989 - 0.142i)T \)
13 \( 1 + (-0.841 - 0.540i)T \)
17 \( 1 + (0.281 - 0.959i)T \)
19 \( 1 + (0.281 + 0.959i)T \)
31 \( 1 + (0.909 - 0.415i)T \)
37 \( 1 + (0.755 - 0.654i)T \)
41 \( 1 + (0.755 + 0.654i)T \)
43 \( 1 + (-0.909 - 0.415i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.841 + 0.540i)T \)
59 \( 1 + (-0.841 - 0.540i)T \)
61 \( 1 + (0.909 - 0.415i)T \)
67 \( 1 + (0.142 + 0.989i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (-0.281 - 0.959i)T \)
79 \( 1 + (0.540 - 0.841i)T \)
83 \( 1 + (0.654 + 0.755i)T \)
89 \( 1 + (0.909 + 0.415i)T \)
97 \( 1 + (-0.755 - 0.654i)T \)
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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

−20.231880148948888836706266732424, −19.52675796930860235048783654568, −18.82788703983958789988629106729, −17.629395983814962207147228734562, −17.10014059031990594688763000635, −16.16947010354494165180071400085, −15.57462229120968030508742913941, −14.99245464723251518876983726689, −14.32067800664380153325306870025, −13.35588662197783802656185024888, −12.69465462774404204341872689358, −12.08022891866514737650048103068, −11.45124363611977561767192870438, −10.75123368233250232074847191118, −9.65791150937062779344451011677, −8.544112801608788066396593462154, −7.94153216549961617090438446479, −7.33400186877740364892430544145, −6.261083178509886072689672979541, −5.259912031307925038694360834884, −4.81620299835358738408436546197, −4.20781933844359254112978556952, −3.0192190603788446191611143160, −2.244744259398548618391777537010, −1.247039142137229780725140831488, 0.68314993955848650597268591992, 2.06858968852810046254397657466, 2.846192258532533535851299438279, 3.557608884530547644659580788301, 4.54614693790946218899369037309, 5.10906224669851186347396330554, 6.013834844353127423186675924454, 7.04661870642928111868492272713, 7.75672593809113442909544585252, 7.994801672042518964066632830090, 9.79442843213864257509202739926, 10.405117298800673873413838435476, 11.059488260351209157809965216349, 11.760134020003242997601884626869, 12.356280804725324886709811492261, 13.35588871711756126258158107937, 14.0270040471972930409157380188, 14.68515888394045924154702539701, 15.19605001249713535225725613736, 16.02246003191322997298437229016, 16.665087144358090420753502593553, 17.75131316784773353570526440852, 18.480868890726124371707428923540, 19.19611995780285519683774830193, 20.10120260074440473464339389759

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