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Properties

Label 1-1001-1001.123-r0-0-0
Degree $1$
Conductor $1001$
Sign $-0.483 + 0.875i$
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  + (0.587 − 0.809i)2-s + (0.669 + 0.743i)3-s + (−0.309 − 0.951i)4-s + (−0.994 + 0.104i)5-s + (0.994 − 0.104i)6-s + (−0.951 − 0.309i)8-s + (−0.104 + 0.994i)9-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)12-s + (−0.743 − 0.669i)15-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (0.743 + 0.669i)18-s + (−0.207 − 0.978i)19-s + (0.406 + 0.913i)20-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (0.669 + 0.743i)3-s + (−0.309 − 0.951i)4-s + (−0.994 + 0.104i)5-s + (0.994 − 0.104i)6-s + (−0.951 − 0.309i)8-s + (−0.104 + 0.994i)9-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)12-s + (−0.743 − 0.669i)15-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (0.743 + 0.669i)18-s + (−0.207 − 0.978i)19-s + (0.406 + 0.913i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.483 + 0.875i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ -0.483 + 0.875i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2502842809 + 0.4240244926i\)
\(L(\frac12)\) \(\approx\) \(0.2502842809 + 0.4240244926i\)
\(L(1)\) \(\approx\) \(1.027258909 - 0.1337525306i\)
\(L(1)\) \(\approx\) \(1.027258909 - 0.1337525306i\)

Euler product

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

−21.38439482718930028659840402876, −20.52970507262914528300099587635, −19.95097713479765788038635999934, −18.94407150609468654939134240651, −18.3219678091281810083640653472, −17.43769403974776168301045782102, −16.50567069129190777894180450886, −15.65544254717535538758085922337, −15.10461974542114686169259208393, −14.27088777659131690287535062244, −13.56984521123625371105051093556, −12.78811734425595134650478658519, −11.995695119425575862645530021098, −11.48831089198785774626903104431, −9.8863380857407259894380613604, −8.74143487519301967670762994266, −8.20863268888291674926447329420, −7.5085751873046087892881541656, −6.75772917232132406159042339664, −5.9281571410810620109600640809, −4.66406580403052526186989099705, −3.84981867250303740721043436476, −3.09511779715742901210756125634, −1.930588465991292096041895203636, −0.14575239901706483707542648798, 1.67528814255584139330568982472, 2.77146941951960167358796607001, 3.47898955028428087857022790021, 4.36309496824465983996729339437, 4.825184871763111902912256904355, 6.130384243073832944525792435602, 7.27483111782235473310039718734, 8.43362564579006902089143394745, 8.99096447557880171668398416902, 10.04995543637446952208063853181, 10.79333792850955195455140442335, 11.3917712980707770201576640831, 12.32964748203552755228419142027, 13.20910736916296607468848818836, 14.00977671527537424216121366676, 14.79782522586471004045259412454, 15.508228468186306833008707144595, 15.9120568102175587955969002509, 17.200574050947607358509032788186, 18.34175234809421445458064989542, 19.22752758926921696892833288481, 19.75628995771074854735243306751, 20.27020107185271154490177942660, 21.08436197441425762949576844387, 21.976579196398713845537527113443

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