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Properties

Label 1-5200-5200.1477-r0-0-0
Degree $1$
Conductor $5200$
Sign $0.0672 - 0.997i$
Analytic cond. $24.1486$
Root an. cond. $24.1486$
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  + (0.809 + 0.587i)3-s − 7-s + (0.309 + 0.951i)9-s + (0.309 − 0.951i)11-s + (0.587 + 0.809i)17-s + (0.809 − 0.587i)19-s + (−0.809 − 0.587i)21-s + (−0.951 − 0.309i)23-s + (−0.309 + 0.951i)27-s + (0.587 − 0.809i)29-s + (−0.587 − 0.809i)31-s + (0.809 − 0.587i)33-s + (−0.951 + 0.309i)37-s + (0.951 − 0.309i)41-s − 43-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)3-s − 7-s + (0.309 + 0.951i)9-s + (0.309 − 0.951i)11-s + (0.587 + 0.809i)17-s + (0.809 − 0.587i)19-s + (−0.809 − 0.587i)21-s + (−0.951 − 0.309i)23-s + (−0.309 + 0.951i)27-s + (0.587 − 0.809i)29-s + (−0.587 − 0.809i)31-s + (0.809 − 0.587i)33-s + (−0.951 + 0.309i)37-s + (0.951 − 0.309i)41-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0672 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0672 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $0.0672 - 0.997i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (1477, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ 0.0672 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8919543163 - 0.8338739426i\)
\(L(\frac12)\) \(\approx\) \(0.8919543163 - 0.8338739426i\)
\(L(1)\) \(\approx\) \(1.131852154 + 0.05473801077i\)
\(L(1)\) \(\approx\) \(1.131852154 + 0.05473801077i\)

Euler product

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

−18.00668869522852572947300719427, −17.8957619747477649864067875750, −16.68095869168597551466818782659, −16.07205035772469032799716587750, −15.545657655024712440902146954942, −14.60433303538053053360743887268, −14.17858400372578570964185263171, −13.542792693441173292350489206585, −12.77049166915495896954822170035, −12.17436191006064889731249360467, −11.87449148232032941586780601260, −10.52201637249866540710443522491, −9.862330051046186673642996814158, −9.37516064965050902172370363300, −8.77223926374322548767150872736, −7.72366125118327477035875105776, −7.34901653588561037103601476204, −6.64471006252222918529748729716, −5.953637380823716362323837096057, −5.01092899083119826616356627575, −4.04329061272871481453531681814, −3.2789547259586393827980739432, −2.84233498889160402136330131723, −1.76960062409782488099349177048, −1.16746622051740175084517950125, 0.27389403922699679809101402883, 1.54559549119191198942569271193, 2.46549545821005976805832867091, 3.345594431435580347953210489719, 3.59025115861334870305886328135, 4.491144828083819623970482332251, 5.43528073930353728016511828000, 6.139712921944840946682259821427, 6.844307943589323589749429189262, 7.89042984666174529279126722246, 8.27128872798564195682216721386, 9.18669735656174116045240236861, 9.628870408706754848304107711344, 10.28238036134532847093668214304, 10.95666147462025286057058438713, 11.81500207766094417236302377791, 12.57841503451396859638985580140, 13.38648061954026607274399290710, 13.81665555511748723162764296271, 14.447763733415123350623554450310, 15.29282833279120123985282063915, 15.77824840656992696379053586219, 16.4783667442920399678362054024, 16.82428287570725510956980091319, 17.86586540145970277531689790755

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