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Properties

Label 1-1001-1001.822-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.447 - 0.894i$
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  + (0.978 − 0.207i)2-s + (0.809 − 0.587i)3-s + (0.913 − 0.406i)4-s + (0.978 + 0.207i)5-s + (0.669 − 0.743i)6-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + 10-s + (0.5 − 0.866i)12-s + (0.913 − 0.406i)15-s + (0.669 − 0.743i)16-s + (−0.978 − 0.207i)17-s + (0.104 − 0.994i)18-s + (−0.809 + 0.587i)19-s + (0.978 − 0.207i)20-s + ⋯
L(s)  = 1  + (0.978 − 0.207i)2-s + (0.809 − 0.587i)3-s + (0.913 − 0.406i)4-s + (0.978 + 0.207i)5-s + (0.669 − 0.743i)6-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + 10-s + (0.5 − 0.866i)12-s + (0.913 − 0.406i)15-s + (0.669 − 0.743i)16-s + (−0.978 − 0.207i)17-s + (0.104 − 0.994i)18-s + (−0.809 + 0.587i)19-s + (0.978 − 0.207i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (822, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.757320877 - 2.319952107i\)
\(L(\frac12)\) \(\approx\) \(3.757320877 - 2.319952107i\)
\(L(1)\) \(\approx\) \(2.587667973 - 0.9680493453i\)
\(L(1)\) \(\approx\) \(2.587667973 - 0.9680493453i\)

Euler product

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

−21.63189590125057869700740371073, −21.31217743216454327810384947130, −20.34954604068367089033260876361, −19.87775202595427244542765309092, −18.88398326137610647124053350785, −17.53181294578604267652315789349, −17.01814322188085063587404741722, −15.95717913017180923724954634375, −15.402701871694278450233284570853, −14.59517115763836396594508031503, −13.836401402814716795722459887913, −13.29674906043227545952791781356, −12.6414957093309310397269679297, −11.33358959403088731213611809206, −10.6069415524001249441801615084, −9.687166632408187738789105868402, −8.837398669398546784625405742175, −7.97494798536633464470736722884, −6.898321937730119842799495884223, −6.028992973122456546390447351907, −5.09102109855564308325050055301, −4.36205191172193611204252217443, −3.452727490264099240651314203155, −2.35579393440939899085644568172, −1.85321509431633233305417807955, 1.31952027613740828914433570921, 2.2082594530728120776191402452, 2.7692022001005753107107443090, 3.89397328457385034956076795498, 4.810661619544450189449524266842, 6.11591905981539785635510516747, 6.458632725487756691493273219658, 7.44194212637263212304806841840, 8.502583744364494786459991707860, 9.484460656670799113595577163751, 10.319006074235433183915201055274, 11.1823193360074861926326826003, 12.3408074988460130576431551398, 12.88799420267001129862683723424, 13.6499929712059041441762004596, 14.203734524451637842020479251773, 14.88794921632296080864006689370, 15.66601740726913043646198454687, 16.790224067183009170190796820985, 17.68918438925593687829226752981, 18.64456664629757179568638587671, 19.20887943888497268656647515488, 20.2968419823344316912520112231, 20.67873841733734285400577642812, 21.45578461418936068989119405118

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