↑

Properties

Label 1-2001-2001.1124-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.974 - 0.226i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.742 − 0.670i)2-s + (0.101 − 0.994i)4-s + (0.557 + 0.830i)5-s + (−0.970 + 0.242i)7-s + (−0.591 − 0.806i)8-s + (0.970 + 0.242i)10-s + (0.0611 − 0.998i)11-s + (0.794 − 0.607i)13-s + (−0.557 + 0.830i)14-s + (−0.979 − 0.202i)16-s + (−0.841 − 0.540i)17-s + (0.101 − 0.994i)19-s + (0.882 − 0.470i)20-s + (−0.623 − 0.781i)22-s + (−0.377 + 0.925i)25-s + (0.182 − 0.983i)26-s + ⋯
L(s)  = 1  + (0.742 − 0.670i)2-s + (0.101 − 0.994i)4-s + (0.557 + 0.830i)5-s + (−0.970 + 0.242i)7-s + (−0.591 − 0.806i)8-s + (0.970 + 0.242i)10-s + (0.0611 − 0.998i)11-s + (0.794 − 0.607i)13-s + (−0.557 + 0.830i)14-s + (−0.979 − 0.202i)16-s + (−0.841 − 0.540i)17-s + (0.101 − 0.994i)19-s + (0.882 − 0.470i)20-s + (−0.623 − 0.781i)22-s + (−0.377 + 0.925i)25-s + (0.182 − 0.983i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.974 - 0.226i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.974 - 0.226i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1698454212 - 1.483152584i\)
\(L(\frac12)\) \(\approx\) \(0.1698454212 - 1.483152584i\)
\(L(1)\) \(\approx\) \(1.137022033 - 0.6844052600i\)
\(L(1)\) \(\approx\) \(1.137022033 - 0.6844052600i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.742 - 0.670i)T \)
5 \( 1 + (0.557 + 0.830i)T \)
7 \( 1 + (-0.970 + 0.242i)T \)
11 \( 1 + (0.0611 - 0.998i)T \)
13 \( 1 + (0.794 - 0.607i)T \)
17 \( 1 + (-0.841 - 0.540i)T \)
19 \( 1 + (0.101 - 0.994i)T \)
31 \( 1 + (-0.262 + 0.965i)T \)
37 \( 1 + (-0.933 - 0.359i)T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (0.262 + 0.965i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (0.0203 + 0.999i)T \)
59 \( 1 + (-0.415 - 0.909i)T \)
61 \( 1 + (-0.999 - 0.0407i)T \)
67 \( 1 + (0.0611 + 0.998i)T \)
71 \( 1 + (-0.488 - 0.872i)T \)
73 \( 1 + (0.992 - 0.122i)T \)
79 \( 1 + (-0.979 + 0.202i)T \)
83 \( 1 + (0.685 + 0.728i)T \)
89 \( 1 + (-0.917 - 0.396i)T \)
97 \( 1 + (-0.862 - 0.505i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.45962937343939110086285385053, −19.8395907134664450580655044900, −18.72652188889628973998200425469, −17.86118002249707525672859499124, −17.09852894516334291811546815273, −16.61524206272621939359438796264, −15.910872217036032310203831537591, −15.29305245532738845722649448825, −14.349480854591810958844731810524, −13.56266108287377072411192499700, −13.08277017273541642702096280232, −12.45676895660632070284723491102, −11.78619356700224778962807205999, −10.617958519658948535900996105611, −9.66630539508244686986083914348, −9.033480570582616970110060187812, −8.24263494111960769284266933990, −7.305864442422448739655901367379, −6.38887912253889288504292284678, −6.044875011736665378734579415415, −5.00092554241651204694378606811, −4.19992759339151453855632953559, −3.63464573006288826565063478159, −2.37420526724294221317610516971, −1.51497236016066416498915369214, 0.359156078050378119429033386692, 1.64863697727062241293195760068, 2.799865074258279591223986141038, 3.08239207478056773736674667878, 3.93552794616089958494517663093, 5.19593560991998127190291578410, 5.85109692595593543934348721987, 6.53615673853368476912715190830, 7.1192203828754604658951312501, 8.71863687721997957300462188323, 9.261471794956271107632919590414, 10.172719501545101373551862893341, 10.876471203184491807711602714294, 11.27857822182064611518172015247, 12.35036898793486107927280807370, 13.12850870133924146531658305863, 13.71951223650751815201354654195, 14.10879832125517906708044185073, 15.39246492611293067846915532226, 15.5757925855955257572067898414, 16.5196428094030232788462888967, 17.78527795735020839921222350004, 18.27532688092630111556847776542, 19.06218719166289955066553816674, 19.59174648476146532265261314563

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