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Properties

Label 1-5200-5200.3-r0-0-0
Degree $1$
Conductor $5200$
Sign $-0.941 + 0.337i$
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  + (−0.978 − 0.207i)3-s + (−0.866 + 0.5i)7-s + (0.913 + 0.406i)9-s + (0.406 + 0.913i)11-s + (0.207 + 0.978i)17-s + (−0.207 − 0.978i)19-s + (0.951 − 0.309i)21-s + (0.406 + 0.913i)23-s + (−0.809 − 0.587i)27-s + (−0.207 + 0.978i)29-s + (−0.309 + 0.951i)31-s + (−0.207 − 0.978i)33-s + (0.104 + 0.994i)37-s + (0.104 + 0.994i)41-s + (0.5 + 0.866i)43-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)3-s + (−0.866 + 0.5i)7-s + (0.913 + 0.406i)9-s + (0.406 + 0.913i)11-s + (0.207 + 0.978i)17-s + (−0.207 − 0.978i)19-s + (0.951 − 0.309i)21-s + (0.406 + 0.913i)23-s + (−0.809 − 0.587i)27-s + (−0.207 + 0.978i)29-s + (−0.309 + 0.951i)31-s + (−0.207 − 0.978i)33-s + (0.104 + 0.994i)37-s + (0.104 + 0.994i)41-s + (0.5 + 0.866i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $-0.941 + 0.337i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ -0.941 + 0.337i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1230036370 + 0.7072338086i\)
\(L(\frac12)\) \(\approx\) \(0.1230036370 + 0.7072338086i\)
\(L(1)\) \(\approx\) \(0.6685006803 + 0.1964847182i\)
\(L(1)\) \(\approx\) \(0.6685006803 + 0.1964847182i\)

Euler product

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

−17.58626322000206130541303080636, −16.746930704912881421287827475877, −16.5077375928984647459800818892, −16.010795043375152127453359486729, −15.13337254583712466292463652436, −14.358431914047960093787660814036, −13.55476161875564388961942129033, −13.0260915783541349336773471612, −12.17887829269401169447897160540, −11.742117161444780792251554567861, −10.85443432855533671457981911595, −10.4489071220239741269574290874, −9.619502015978974658457025301742, −9.14253053069721236519888848168, −8.103967806491613589121935117253, −7.21515190776379659819219478761, −6.6958162094446570550505431700, −5.8516046124795522142510529085, −5.5573841156277707070964612206, −4.357034049785505295963949883024, −3.897372616007463180727049835481, −3.10843864070652193437522494224, −2.04219429197216586849204105635, −0.82350764428029722155873970672, −0.29924279586488972302858493889, 1.17078260243043513731631297230, 1.78345965482755960805607722806, 2.87743954676267877541520885108, 3.65601616412726941562215270409, 4.65975173784016378980521582066, 5.12905468669440459055796622544, 6.09663113667526527370570699283, 6.533654268546732424838604446784, 7.16536801301440320553582604494, 7.944889403627916438953741092994, 9.03423219048928960323409282865, 9.52063905733210047377474212305, 10.272876546796164902552083976616, 10.929836315069333744729854124837, 11.66771487514156319827934597017, 12.30253811235955260320712891496, 12.9066877753023387305340396299, 13.23025324308891975969493853778, 14.384206394262195746567274017626, 15.183702064219735840393111943118, 15.61252360251290938917582211957, 16.40289159775880608186360110882, 16.97766278516408690261192389621, 17.604610391582618165813949457230, 18.156483403113266777182988520878

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