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Properties

Label 1-1300-1300.447-r1-0-0
Degree $1$
Conductor $1300$
Sign $0.922 - 0.386i$
Analytic cond. $139.704$
Root an. cond. $139.704$
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  + (−0.951 + 0.309i)3-s + 7-s + (0.809 − 0.587i)9-s + (−0.587 + 0.809i)11-s + (−0.951 − 0.309i)17-s + (−0.951 − 0.309i)19-s + (−0.951 + 0.309i)21-s + (−0.587 + 0.809i)23-s + (−0.587 + 0.809i)27-s + (−0.309 − 0.951i)29-s + (0.951 + 0.309i)31-s + (0.309 − 0.951i)33-s + (0.809 − 0.587i)37-s + (0.587 + 0.809i)41-s − i·43-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)3-s + 7-s + (0.809 − 0.587i)9-s + (−0.587 + 0.809i)11-s + (−0.951 − 0.309i)17-s + (−0.951 − 0.309i)19-s + (−0.951 + 0.309i)21-s + (−0.587 + 0.809i)23-s + (−0.587 + 0.809i)27-s + (−0.309 − 0.951i)29-s + (0.951 + 0.309i)31-s + (0.309 − 0.951i)33-s + (0.809 − 0.587i)37-s + (0.587 + 0.809i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.922 - 0.386i$
Analytic conductor: \(139.704\)
Root analytic conductor: \(139.704\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1300,\ (1:\ ),\ 0.922 - 0.386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.137825198 - 0.2289605028i\)
\(L(\frac12)\) \(\approx\) \(1.137825198 - 0.2289605028i\)
\(L(1)\) \(\approx\) \(0.7938807279 + 0.06479877272i\)
\(L(1)\) \(\approx\) \(0.7938807279 + 0.06479877272i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.951 + 0.309i)T \)
7 \( 1 + T \)
11 \( 1 + (-0.587 + 0.809i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (-0.951 - 0.309i)T \)
23 \( 1 + (-0.587 + 0.809i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (0.951 + 0.309i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (0.587 + 0.809i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (-0.587 - 0.809i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (-0.951 + 0.309i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (-0.587 + 0.809i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−21.0986879936278911810393522384, −20.07862105657466743844328813719, −19.141282348601060013273272698, −18.31712764873636384208535221996, −17.94687415573345752436411206814, −16.985427709740938296075324440130, −16.49557148690667634386823438426, −15.5337050377641799162206701104, −14.7801490412014960606555044572, −13.75525523322083293880969916222, −13.101625613837792923367871553221, −12.22629168903749267523186812525, −11.48082164684490327755678926875, −10.720087437765789671105089086082, −10.35598923140955071567540685590, −8.85565284491452353572594714948, −8.17981732941290307582876292971, −7.369662002437737968510550613177, −6.31184942643453031081560041412, −5.76080523288514882937112298417, −4.71969158268182371422269154586, −4.17712233105278790792563073218, −2.604363734344102460669899388183, −1.707650061585354190162459123696, −0.64557740237672694755017027781, 0.39305629600736525648687825779, 1.64284123142588490852798776724, 2.523988852357073845436910828337, 4.196835849202683599660880210644, 4.530301075634317665249364317, 5.45192864113074380170103792848, 6.2863031274871479440849588503, 7.2485040191647030358971394571, 7.98560452314090361498335225104, 9.08308004147605893387790862039, 9.9471879672807041599158358797, 10.74660701755955549007009176338, 11.36025842511666532293949783399, 12.06711937464706100097029276596, 12.93113708706323830325664981199, 13.73267425636474738103994451209, 14.90276914310340188797894827847, 15.38857618552859938047994782251, 16.10628721269428551039639084058, 17.20268035598367087029423737700, 17.636314425939697879794012872763, 18.13941582155460098943221529246, 19.12301304423222561292964964805, 20.17499630299173248589849205888, 20.88971403353721604748517549520

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