↑

Properties

Label 1-1001-1001.20-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.999 - 0.0331i$
Analytic cond. $4.64862$
Root an. cond. $4.64862$
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  + (−0.994 − 0.104i)2-s + (0.978 − 0.207i)3-s + (0.978 + 0.207i)4-s + (0.587 + 0.809i)5-s + (−0.994 + 0.104i)6-s + (−0.951 − 0.309i)8-s + (0.913 − 0.406i)9-s + (−0.5 − 0.866i)10-s + 12-s + (0.743 + 0.669i)15-s + (0.913 + 0.406i)16-s + (−0.104 − 0.994i)17-s + (−0.951 + 0.309i)18-s + (0.743 − 0.669i)19-s + (0.406 + 0.913i)20-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)2-s + (0.978 − 0.207i)3-s + (0.978 + 0.207i)4-s + (0.587 + 0.809i)5-s + (−0.994 + 0.104i)6-s + (−0.951 − 0.309i)8-s + (0.913 − 0.406i)9-s + (−0.5 − 0.866i)10-s + 12-s + (0.743 + 0.669i)15-s + (0.913 + 0.406i)16-s + (−0.104 − 0.994i)17-s + (−0.951 + 0.309i)18-s + (0.743 − 0.669i)19-s + (0.406 + 0.913i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.999 - 0.0331i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.999 - 0.0331i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.662631607 + 0.02757732173i\)
\(L(\frac12)\) \(\approx\) \(1.662631607 + 0.02757732173i\)
\(L(1)\) \(\approx\) \(1.152474321 + 0.01588260587i\)
\(L(1)\) \(\approx\) \(1.152474321 + 0.01588260587i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.994 - 0.104i)T \)
3 \( 1 + (0.978 - 0.207i)T \)
5 \( 1 + (0.587 + 0.809i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
19 \( 1 + (0.743 - 0.669i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.669 - 0.743i)T \)
31 \( 1 + (-0.587 + 0.809i)T \)
37 \( 1 + (-0.743 - 0.669i)T \)
41 \( 1 + (0.207 + 0.978i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.207 - 0.978i)T \)
61 \( 1 + (0.104 + 0.994i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.994 - 0.104i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.587 + 0.809i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.406 - 0.913i)T \)
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

−21.35310032361173242454511424720, −20.44507859787183175730310866896, −20.34064277610158632050969363502, −19.25595221460911965208736774713, −18.66664940747932121211988586027, −17.742644684761636910188427406389, −16.92800598817053281558660617039, −16.251480926698659353540671594168, −15.51866860088197953233442002384, −14.61776006312099309302212046068, −13.91320385623433100700147177660, −12.81032010199380247781672436105, −12.228924865057558503047188219, −10.836715453598629489626819110709, −10.15028885624382680037304593335, −9.404330724570336131191490247622, −8.70729866713126346738200095796, −8.1514813627257811174229990601, −7.219541233490831650582723538675, −6.17829407603772242364913187010, −5.19424454249867125227140367901, −4.00804621592017499785386737334, −2.86255744458829473724686870840, −1.91254025441919489164701249441, −1.12485331323437293085425370604, 1.07321477231833789221884745837, 2.15739719336111652293711624409, 2.846224889963636141742798251, 3.59484296326096704967402073734, 5.23901275415695760324835233534, 6.486566851794836735345926424056, 7.17824504386547809872562968157, 7.72998941004603166422033380460, 8.92925368737219024559227181747, 9.41541757798717783720874395950, 10.14824492975140358970043598220, 11.048099217446669959540294135473, 11.88665266308101571877628271197, 12.99627839758487958253248504726, 13.868931261939588326022620368478, 14.50623310268093337406418530212, 15.547895179779578134300375856874, 15.93257386274642824618121546640, 17.32635233562041775588703515539, 17.87192531446870593851694663270, 18.56499743760874641349620715234, 19.238112137089032169367973272600, 19.92052209671036924359896050944, 20.73334435716541709260268854773, 21.3869118747988504442408473430

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