↑

Properties

Label 1-2001-2001.962-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.913 + 0.407i$
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  + (−0.591 − 0.806i)2-s + (−0.301 + 0.953i)4-s + (−0.979 + 0.202i)5-s + (−0.742 + 0.670i)7-s + (0.947 − 0.320i)8-s + (0.742 + 0.670i)10-s + (−0.182 + 0.983i)11-s + (−0.377 − 0.925i)13-s + (0.979 + 0.202i)14-s + (−0.818 − 0.574i)16-s + (0.142 − 0.989i)17-s + (−0.301 + 0.953i)19-s + (0.101 − 0.994i)20-s + (0.900 − 0.433i)22-s + (0.917 − 0.396i)25-s + (−0.523 + 0.852i)26-s + ⋯
L(s)  = 1  + (−0.591 − 0.806i)2-s + (−0.301 + 0.953i)4-s + (−0.979 + 0.202i)5-s + (−0.742 + 0.670i)7-s + (0.947 − 0.320i)8-s + (0.742 + 0.670i)10-s + (−0.182 + 0.983i)11-s + (−0.377 − 0.925i)13-s + (0.979 + 0.202i)14-s + (−0.818 − 0.574i)16-s + (0.142 − 0.989i)17-s + (−0.301 + 0.953i)19-s + (0.101 − 0.994i)20-s + (0.900 − 0.433i)22-s + (0.917 − 0.396i)25-s + (−0.523 + 0.852i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.913 + 0.407i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (962, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.913 + 0.407i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5251186204 + 0.1117628683i\)
\(L(\frac12)\) \(\approx\) \(0.5251186204 + 0.1117628683i\)
\(L(1)\) \(\approx\) \(0.5451913261 - 0.08892472125i\)
\(L(1)\) \(\approx\) \(0.5451913261 - 0.08892472125i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.591 - 0.806i)T \)
5 \( 1 + (-0.979 + 0.202i)T \)
7 \( 1 + (-0.742 + 0.670i)T \)
11 \( 1 + (-0.182 + 0.983i)T \)
13 \( 1 + (-0.377 - 0.925i)T \)
17 \( 1 + (0.142 - 0.989i)T \)
19 \( 1 + (-0.301 + 0.953i)T \)
31 \( 1 + (0.714 - 0.699i)T \)
37 \( 1 + (-0.452 - 0.891i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (-0.714 - 0.699i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + (-0.0611 - 0.998i)T \)
59 \( 1 + (0.959 + 0.281i)T \)
61 \( 1 + (-0.992 - 0.122i)T \)
67 \( 1 + (-0.182 - 0.983i)T \)
71 \( 1 + (0.999 + 0.0407i)T \)
73 \( 1 + (0.933 - 0.359i)T \)
79 \( 1 + (-0.818 + 0.574i)T \)
83 \( 1 + (-0.768 + 0.639i)T \)
89 \( 1 + (-0.339 - 0.940i)T \)
97 \( 1 + (0.0203 - 0.999i)T \)
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

−19.59296334331356032210241843632, −19.15295146559308931331905526642, −18.68497781842458598242978795646, −17.446837214316965948514555793026, −16.86595349973306867454819675384, −16.33121829272344612815888864953, −15.66766816988524019265658439435, −15.0590120979760403518009121094, −14.0842269874246396405002374955, −13.50961954737379302046441668884, −12.622124345857093963196563243371, −11.61981823394021760442140358210, −10.82759503286795172335455642451, −10.22537016802726941626938963997, −9.21003177917342555756394041989, −8.57660386622446303298168417561, −7.92518637662963173232461100072, −6.94600269980510059209517110337, −6.61845587084738065096037567212, −5.55173013332280611291808594143, −4.54251113664448459730488446506, −3.91165784456639091681677806000, −2.85773704405767833904522548786, −1.3774451071525605272362413630, −0.39015546686745909070706187930, 0.66175190536312904984383753521, 2.10854004083599366012453786334, 2.831113878966965845212540075043, 3.54312020866308974966052797497, 4.43429116165463880359880179953, 5.33954858481735217839793156995, 6.60784907243514945941152003095, 7.47280587294441510029001979678, 7.9835497618096177388251379414, 8.85894595889559252290854161732, 9.77286360469039136886697041413, 10.17766160429680540269049320333, 11.162076078362129702036970431353, 11.93401686602747091321594815328, 12.48089741637413471644117453298, 12.915554573419943620106350315205, 14.09369584327121981545941737707, 15.12077411262080893444728471972, 15.657417139294739011391777869173, 16.40367278770559643436273128677, 17.18990166699937813409696781057, 18.22008196154876724035429319450, 18.48366082821463029230357705081, 19.40412733137016672260122386164, 19.83080205888170629431334545838

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