↑

Properties

Label 1-2001-2001.215-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.0585 - 0.998i$
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  + (−0.281 + 0.959i)2-s + (−0.841 − 0.540i)4-s + (−0.142 + 0.989i)5-s + (0.415 + 0.909i)7-s + (0.755 − 0.654i)8-s + (−0.909 − 0.415i)10-s + (0.281 + 0.959i)11-s + (−0.415 + 0.909i)13-s + (−0.989 + 0.142i)14-s + (0.415 + 0.909i)16-s + (−0.540 − 0.841i)17-s + (−0.540 + 0.841i)19-s + (0.654 − 0.755i)20-s − 22-s + (−0.959 − 0.281i)25-s + (−0.755 − 0.654i)26-s + ⋯
L(s)  = 1  + (−0.281 + 0.959i)2-s + (−0.841 − 0.540i)4-s + (−0.142 + 0.989i)5-s + (0.415 + 0.909i)7-s + (0.755 − 0.654i)8-s + (−0.909 − 0.415i)10-s + (0.281 + 0.959i)11-s + (−0.415 + 0.909i)13-s + (−0.989 + 0.142i)14-s + (0.415 + 0.909i)16-s + (−0.540 − 0.841i)17-s + (−0.540 + 0.841i)19-s + (0.654 − 0.755i)20-s − 22-s + (−0.959 − 0.281i)25-s + (−0.755 − 0.654i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0585 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0585 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.0585 - 0.998i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.0585 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3802392001 + 0.4031883480i\)
\(L(\frac12)\) \(\approx\) \(-0.3802392001 + 0.4031883480i\)
\(L(1)\) \(\approx\) \(0.4616508670 + 0.5638674209i\)
\(L(1)\) \(\approx\) \(0.4616508670 + 0.5638674209i\)

Euler product

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

−19.632275966105438396545075274015, −18.966380162648365428900768851914, −17.79614256547553330255039569356, −17.27547900897948048634731924721, −16.88136008080734054274540074001, −15.93615735318512076423401782118, −14.96892273031206598428438549647, −13.92811391934175187704542895012, −13.3507981229221398595845387253, −12.79955801463771324901535379116, −11.9545307653098369937815156802, −11.21459307482345337285442533138, −10.53978184006580698812708153437, −9.89091557748271197486331818165, −8.74074551889215461186839399104, −8.461734828766694543522738352, −7.68278294124190514776267912399, −6.56673286231930805682010751173, −5.284393570252068785622252755202, −4.69791929493498619038348955533, −3.8792666928743502567954166175, −3.14130403379730398061910966883, −1.91219464724482065729439308801, −1.07718357298735734770995765543, −0.23007613384215921806916290653, 1.71922373316414659942498231176, 2.380077412895852444751072499595, 3.72136991839510348856688574338, 4.60692757586462747020563487007, 5.3020313743206035178138600554, 6.447958453746925329102431746745, 6.75167257452372381970796280038, 7.64792064277359610337520744214, 8.37729779229742026431520470236, 9.301849522584637802743687347419, 9.83857762895169821436227296917, 10.71664752564360262877920941588, 11.7239094879675148270827078597, 12.24813188442383823633892234950, 13.46632647069838506413269778930, 14.219140774314464003195757112139, 14.77794598959807250190879297704, 15.31077635659330143633067020184, 15.95016009589612559698069925667, 16.95658736464933148487826854761, 17.60599042941758181604626141863, 18.25059282138836619990458696156, 18.89951517878380208144593832095, 19.31377401530608113516710912819, 20.4641328473525992970556405699

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