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Properties

Label 1-1001-1001.16-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.380 - 0.924i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.669 − 0.743i)3-s + (0.309 − 0.951i)4-s + (−0.104 + 0.994i)5-s + (−0.104 + 0.994i)6-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s + (0.669 + 0.743i)15-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (0.669 + 0.743i)18-s + (−0.978 − 0.207i)19-s + (0.913 + 0.406i)20-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.669 − 0.743i)3-s + (0.309 − 0.951i)4-s + (−0.104 + 0.994i)5-s + (−0.104 + 0.994i)6-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s + (0.669 + 0.743i)15-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (0.669 + 0.743i)18-s + (−0.978 − 0.207i)19-s + (0.913 + 0.406i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.380 - 0.924i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.380 - 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7615173627 - 0.5100159957i\)
\(L(\frac12)\) \(\approx\) \(0.7615173627 - 0.5100159957i\)
\(L(1)\) \(\approx\) \(0.8122008598 - 0.04741523099i\)
\(L(1)\) \(\approx\) \(0.8122008598 - 0.04741523099i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−21.438947924662902298252883320525, −20.91498357068828121809860127821, −20.23081676997238905318281222958, −19.51077741984796362465036628456, −19.073574170332146565204964528950, −17.80619417141356676165699462744, −16.99811118466878911207183441027, −16.50166651977230028954163745276, −15.59087484720349706824003325286, −14.99630428390186094527861487284, −13.60890545496769423023035944258, −13.03777025185719480810490653476, −12.165119831205093271603537321, −11.17973460377553191990027874564, −10.41350404290625505460058466203, −9.65673046770298396025009076644, −8.66425579312217737890637468006, −8.56257756922266554856157082705, −7.52882443804725356211416490080, −6.29709088785944216230963057211, −4.802140598418404703472118330259, −4.253921685118851993993415370375, −3.2321873281237988920495240702, −2.24030195537782892581110477663, −1.24637365634705558870659122981, 0.48170875163216392246332667591, 1.976224839310848408117287771957, 2.562944241290642721924321697, 3.76167503873045253925328535463, 5.188900571840407759160204435368, 6.4669928354541123007709225124, 6.81402689140189583347908711104, 7.566475870333981486685128120853, 8.4879863019900827300624321345, 9.129272068617105258630910268398, 10.10497636620304414943858530664, 10.98680046331467880655261077634, 11.69880793943882090645691173952, 12.975234007316971918583940590292, 13.79769159157748423124427247715, 14.57390095912689616223661115064, 15.13927126379019623383911798376, 15.82998329408063121929957882643, 17.0883219872611280130978128972, 17.698626913297750929546876772179, 18.54984966238383354801036475195, 18.890825843079481331018339046025, 19.73111420161854527385184174565, 20.32891663511448515626158930119, 21.404882385691136604271373319213

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