↑

Properties

Label 1-2001-2001.1244-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.152 - 0.988i$
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  + (−0.639 + 0.768i)2-s + (−0.182 − 0.983i)4-s + (−0.992 − 0.122i)5-s + (0.685 − 0.728i)7-s + (0.872 + 0.488i)8-s + (0.728 − 0.685i)10-s + (−0.202 + 0.979i)11-s + (−0.996 + 0.0815i)13-s + (0.122 + 0.992i)14-s + (−0.933 + 0.359i)16-s + (−0.755 + 0.654i)17-s + (−0.983 + 0.182i)19-s + (0.0611 + 0.998i)20-s + (−0.623 − 0.781i)22-s + (0.970 + 0.242i)25-s + (0.574 − 0.818i)26-s + ⋯
L(s)  = 1  + (−0.639 + 0.768i)2-s + (−0.182 − 0.983i)4-s + (−0.992 − 0.122i)5-s + (0.685 − 0.728i)7-s + (0.872 + 0.488i)8-s + (0.728 − 0.685i)10-s + (−0.202 + 0.979i)11-s + (−0.996 + 0.0815i)13-s + (0.122 + 0.992i)14-s + (−0.933 + 0.359i)16-s + (−0.755 + 0.654i)17-s + (−0.983 + 0.182i)19-s + (0.0611 + 0.998i)20-s + (−0.623 − 0.781i)22-s + (0.970 + 0.242i)25-s + (0.574 − 0.818i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.152 - 0.988i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.152 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2177857848 - 0.1868155611i\)
\(L(\frac12)\) \(\approx\) \(0.2177857848 - 0.1868155611i\)
\(L(1)\) \(\approx\) \(0.5354983482 + 0.1325994916i\)
\(L(1)\) \(\approx\) \(0.5354983482 + 0.1325994916i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.639 + 0.768i)T \)
5 \( 1 + (-0.992 - 0.122i)T \)
7 \( 1 + (0.685 - 0.728i)T \)
11 \( 1 + (-0.202 + 0.979i)T \)
13 \( 1 + (-0.996 + 0.0815i)T \)
17 \( 1 + (-0.755 + 0.654i)T \)
19 \( 1 + (-0.983 + 0.182i)T \)
31 \( 1 + (0.162 + 0.986i)T \)
37 \( 1 + (0.940 - 0.339i)T \)
41 \( 1 + (-0.540 + 0.841i)T \)
43 \( 1 + (-0.162 + 0.986i)T \)
47 \( 1 + (0.974 - 0.222i)T \)
53 \( 1 + (-0.557 - 0.830i)T \)
59 \( 1 + (0.142 - 0.989i)T \)
61 \( 1 + (0.925 - 0.377i)T \)
67 \( 1 + (0.979 - 0.202i)T \)
71 \( 1 + (0.794 - 0.607i)T \)
73 \( 1 + (-0.396 - 0.917i)T \)
79 \( 1 + (0.359 - 0.933i)T \)
83 \( 1 + (-0.101 + 0.994i)T \)
89 \( 1 + (-0.670 - 0.742i)T \)
97 \( 1 + (-0.320 - 0.947i)T \)
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

−20.17372756482242914372623714584, −19.06269484991789209659477262048, −18.97798045476183241666544726535, −18.152541188291655931521497523291, −17.27299224886232367359279171065, −16.68429983959916521421799003733, −15.70197354525207812249378492845, −15.21125386904265550195972238432, −14.238230836156637224249146203514, −13.31498387007636946635233172913, −12.439898535575959715438475317242, −11.84348897703445641111955758714, −11.20861678745670268745796730872, −10.73446579532818082527306713749, −9.65473039863322686065741934202, −8.75121833024570345497509218543, −8.34021916269436094774072524004, −7.55893512394470620593556677269, −6.790456233569722810458252822665, −5.48367075916010513965598677871, −4.531970369249564344070427367786, −3.89231267569699704838755929558, −2.647950919806774810464035912712, −2.38303898820884256041183061062, −0.89060792390340445980176820782, 0.157449036451098553854973868831, 1.440882213398598256544153078763, 2.32781971756892687130329010148, 3.910146883586520154240705213505, 4.63185002013616408970566402189, 5.05302343849055598306364265704, 6.53395176944633258699814151355, 6.99821724956729188139768456677, 7.91781968632850139188317408881, 8.17169827277806908738301754002, 9.21412442550461780588404187180, 10.09295158033859734546309766281, 10.766745484277641695327986224351, 11.432481450667039169060574769330, 12.46565748236986020306810611070, 13.17051128171125252014121462906, 14.38768496455777510628099016572, 14.789319671287949763991612340523, 15.36996809651545396437135653257, 16.201519071742249561386073758981, 16.99454635169116648252088427938, 17.45830387810828140003022540544, 18.16825319503868904947556654088, 19.08446674781539815468378205354, 19.86866244385119962108275540388

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