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Properties

Label 1-5200-5200.1387-r0-0-0
Degree $1$
Conductor $5200$
Sign $0.987 + 0.155i$
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  + (0.913 + 0.406i)3-s + (−0.866 − 0.5i)7-s + (0.669 + 0.743i)9-s + (−0.743 − 0.669i)11-s + (0.406 + 0.913i)17-s + (−0.406 − 0.913i)19-s + (−0.587 − 0.809i)21-s + (−0.743 − 0.669i)23-s + (0.309 + 0.951i)27-s + (−0.406 + 0.913i)29-s + (0.809 + 0.587i)31-s + (−0.406 − 0.913i)33-s + (0.978 − 0.207i)37-s + (0.978 − 0.207i)41-s + (0.5 − 0.866i)43-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)3-s + (−0.866 − 0.5i)7-s + (0.669 + 0.743i)9-s + (−0.743 − 0.669i)11-s + (0.406 + 0.913i)17-s + (−0.406 − 0.913i)19-s + (−0.587 − 0.809i)21-s + (−0.743 − 0.669i)23-s + (0.309 + 0.951i)27-s + (−0.406 + 0.913i)29-s + (0.809 + 0.587i)31-s + (−0.406 − 0.913i)33-s + (0.978 − 0.207i)37-s + (0.978 − 0.207i)41-s + (0.5 − 0.866i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $0.987 + 0.155i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (1387, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ 0.987 + 0.155i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.015720501 + 0.1574482659i\)
\(L(\frac12)\) \(\approx\) \(2.015720501 + 0.1574482659i\)
\(L(1)\) \(\approx\) \(1.264602061 + 0.09055757475i\)
\(L(1)\) \(\approx\) \(1.264602061 + 0.09055757475i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.913 + 0.406i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.743 - 0.669i)T \)
17 \( 1 + (0.406 + 0.913i)T \)
19 \( 1 + (-0.406 - 0.913i)T \)
23 \( 1 + (-0.743 - 0.669i)T \)
29 \( 1 + (-0.406 + 0.913i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.978 - 0.207i)T \)
41 \( 1 + (0.978 - 0.207i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.743 + 0.669i)T \)
61 \( 1 + (-0.207 + 0.978i)T \)
67 \( 1 + (0.104 - 0.994i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.669 - 0.743i)T \)
97 \( 1 + (0.994 - 0.104i)T \)
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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

−18.14488854186310206908167065832, −17.4575366663891261182791193990, −16.4876665972267428081420635327, −15.754234518729953079495195378020, −15.423485697223000790551103546728, −14.56961229534695643505554942501, −14.0252278263774708452208176850, −13.10525393745583301426463849143, −12.88996870770615358536561179521, −12.07083461953554224848830672247, −11.48970756237758587861011779388, −10.21304733051807591313364130758, −9.73433228473081365868070774015, −9.36010669037421387070764673143, −8.33600198736470578548912264918, −7.76320028588542728545073769763, −7.26300349490428016800015588433, −6.23730511870270021722346286114, −5.828881067306750178145988827726, −4.67911099698974213488446796857, −3.92651877296262126853306962084, −3.10313247681358153212057959830, −2.47621384534000210419562369856, −1.878634065043672245635399359388, −0.681234161421145667158572964553, 0.663483909229791906153954222822, 1.79661824969090614683005704956, 2.83791571052530199962959058415, 3.093882343930493765337955975111, 4.12414165751955273971118464540, 4.53502529117796542904990786675, 5.706059814980355916755735804665, 6.294244716044245490075334253646, 7.299217879581514491228799203213, 7.780812247226606921576634834662, 8.65640086912736741800037730326, 9.09534946960435928452891829144, 9.95137748973425583155084259138, 10.63728098394569875992368100309, 10.844404123946935598143974472475, 12.20987799527042624311442067073, 12.893771784360247365844167453040, 13.30317465643181087656764860375, 14.11079825926670905981323760925, 14.52726223376744210891650236881, 15.60529136367042707377189578814, 15.77653877949860647790712329236, 16.58606925351915194060259581961, 17.13014301266126743433103323687, 18.172594619848278626376907002172

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