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Properties

Label 1-1001-1001.324-r0-0-0
Degree $1$
Conductor $1001$
Sign $-0.798 - 0.601i$
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  + (−0.913 − 0.406i)2-s + (−0.978 − 0.207i)3-s + (0.669 + 0.743i)4-s + (0.104 − 0.994i)5-s + (0.809 + 0.587i)6-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)12-s + (−0.309 + 0.951i)15-s + (−0.104 + 0.994i)16-s + (0.913 − 0.406i)17-s + (−0.669 − 0.743i)18-s + (−0.669 + 0.743i)19-s + (0.809 − 0.587i)20-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (−0.978 − 0.207i)3-s + (0.669 + 0.743i)4-s + (0.104 − 0.994i)5-s + (0.809 + 0.587i)6-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)12-s + (−0.309 + 0.951i)15-s + (−0.104 + 0.994i)16-s + (0.913 − 0.406i)17-s + (−0.669 − 0.743i)18-s + (−0.669 + 0.743i)19-s + (0.809 − 0.587i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.798 - 0.601i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ -0.798 - 0.601i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1706582593 - 0.5099062668i\)
\(L(\frac12)\) \(\approx\) \(0.1706582593 - 0.5099062668i\)
\(L(1)\) \(\approx\) \(0.4716508062 - 0.2522234140i\)
\(L(1)\) \(\approx\) \(0.4716508062 - 0.2522234140i\)

Euler product

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

−21.80861949989894707620990401845, −21.46546837616690914857054163272, −20.23542444773531382243567720565, −19.30663842888576785083472393781, −18.6127050965436266740913038909, −18.00814770261400248801376495516, −17.29886546203188252002908744470, −16.66626393538622518203186351358, −15.76388903440183421999914721256, −15.09791445803259319918001950929, −14.41805517093512081022321610352, −13.23093016022263754844037795266, −12.02147068780015367519308341477, −11.29872918243227535837436099233, −10.65764905698509513857700136886, −9.97587217583848537365832934612, −9.27122851622794862528248871808, −7.93726581707585669544588392553, −7.25772318059030394824881186054, −6.33640774121758540238140463792, −5.89008968676179241854696938847, −4.804202304790299773804983392045, −3.49696277548431819888903988427, −2.26622276055720189226304514570, −1.10518372039718251805058449510, 0.43027687130231498263114267238, 1.352180294929907895164979561934, 2.30153098584245347711196591008, 3.812615046775281590398484104533, 4.72966402336098839669860014045, 5.81541358439276935260132573856, 6.55489986027848570482457128429, 7.71936991188022481611003166175, 8.27424122963569391682530515486, 9.38591212413690198339108364955, 10.08066197142785112263727604046, 10.83530825964069444469533602317, 11.833083516189487926917538934224, 12.36572084673623362431196546876, 12.897342351752013740402108424787, 14.068986051472635259972171864639, 15.48470530614008490250400148490, 16.344953472365983448475120447276, 16.63782799757556297137631927779, 17.475134495421308051948221855, 18.09381987177868120451069003896, 18.972655681141570069262945920720, 19.583424217027338221663057676255, 20.77353274070184891631512153020, 21.03061697192568231389350440954

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