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Properties

Label 1-1001-1001.739-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.549 - 0.835i$
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  + (0.994 + 0.104i)2-s + (0.309 − 0.951i)3-s + (0.978 + 0.207i)4-s + (0.994 − 0.104i)5-s + (0.406 − 0.913i)6-s + (0.951 + 0.309i)8-s + (−0.809 − 0.587i)9-s + 10-s + (0.5 − 0.866i)12-s + (0.207 − 0.978i)15-s + (0.913 + 0.406i)16-s + (−0.104 − 0.994i)17-s + (−0.743 − 0.669i)18-s + (−0.951 − 0.309i)19-s + (0.994 + 0.104i)20-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s + (0.309 − 0.951i)3-s + (0.978 + 0.207i)4-s + (0.994 − 0.104i)5-s + (0.406 − 0.913i)6-s + (0.951 + 0.309i)8-s + (−0.809 − 0.587i)9-s + 10-s + (0.5 − 0.866i)12-s + (0.207 − 0.978i)15-s + (0.913 + 0.406i)16-s + (−0.104 − 0.994i)17-s + (−0.743 − 0.669i)18-s + (−0.951 − 0.309i)19-s + (0.994 + 0.104i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.549 - 0.835i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.549 - 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.358361862 - 1.811257026i\)
\(L(\frac12)\) \(\approx\) \(3.358361862 - 1.811257026i\)
\(L(1)\) \(\approx\) \(2.336178369 - 0.7014298496i\)
\(L(1)\) \(\approx\) \(2.336178369 - 0.7014298496i\)

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 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.994 - 0.104i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
19 \( 1 + (-0.951 - 0.309i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.978 + 0.207i)T \)
31 \( 1 + (-0.994 - 0.104i)T \)
37 \( 1 + (-0.743 - 0.669i)T \)
41 \( 1 + (0.743 - 0.669i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.743 - 0.669i)T \)
53 \( 1 + (-0.104 + 0.994i)T \)
59 \( 1 + (-0.207 + 0.978i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + iT \)
71 \( 1 + (-0.406 + 0.913i)T \)
73 \( 1 + (0.743 + 0.669i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (0.587 + 0.809i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.994 - 0.104i)T \)
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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

−21.68284974310930173548697938665, −21.169071352273881249036472399376, −20.60063748362542693242581093276, −19.69092896390806362505705308270, −18.94332666597753361710826590903, −17.60166955852113947333573962892, −16.82312792045449718686521198767, −16.21420559897240015199262696988, −15.146209487285227796773057742418, −14.65457165399941589433429763857, −14.01692883086744300422956869190, −13.11333123067165897710081599528, −12.47314655586612955523423115330, −11.18882579716809400958636008016, −10.55278109344743066444341789662, −9.995425770209739935419543487546, −8.92780032799379973056868150451, −8.0265246917584282830162809876, −6.59545585677129050371257210699, −6.035813631380719055317159668852, −5.05493526727613974672222805379, −4.38113943375001362901489730229, −3.375239571341518877002890213743, −2.52256267405440556019666650815, −1.649837262558309595519005032145, 1.17950845912255391028016043122, 2.20333591036992942170791875000, 2.77246910344024856860493677447, 3.9461892201024505204427994520, 5.24304980005269724152458495031, 5.75842668052515254938560405406, 6.8633622861089625801588962895, 7.18169589910774999597177957659, 8.47549344675013434464767259719, 9.26215752653271629237689618625, 10.490462525456841996444977954708, 11.34989417108893135617483853330, 12.31254850049991650635281429949, 12.891580596150331830359765968540, 13.691078614335810920788224986644, 14.078298490215646666152789846425, 14.94279565180487452237534595782, 15.866600893256983756972467604785, 16.960943524475626475173935032263, 17.496915409097670673131871514389, 18.406388199973506862479893109858, 19.33321563002938067168268466335, 20.12463154566141449717802399971, 20.83196433114747258273838962075, 21.549954517519819127955044346335

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