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Properties

Label 1-5200-5200.5117-r0-0-0
Degree $1$
Conductor $5200$
Sign $0.450 + 0.893i$
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  + (0.309 − 0.951i)3-s − 7-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (0.951 − 0.309i)17-s + (0.309 + 0.951i)19-s + (−0.309 + 0.951i)21-s + (0.587 + 0.809i)23-s + (−0.809 + 0.587i)27-s + (−0.951 − 0.309i)29-s + (−0.951 + 0.309i)31-s + (−0.309 − 0.951i)33-s + (−0.587 + 0.809i)37-s + (−0.587 + 0.809i)41-s + 43-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)3-s − 7-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (0.951 − 0.309i)17-s + (0.309 + 0.951i)19-s + (−0.309 + 0.951i)21-s + (0.587 + 0.809i)23-s + (−0.809 + 0.587i)27-s + (−0.951 − 0.309i)29-s + (−0.951 + 0.309i)31-s + (−0.309 − 0.951i)33-s + (−0.587 + 0.809i)37-s + (−0.587 + 0.809i)41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.450 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.450 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $0.450 + 0.893i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (5117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ 0.450 + 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6989597226 + 0.4304651894i\)
\(L(\frac12)\) \(\approx\) \(0.6989597226 + 0.4304651894i\)
\(L(1)\) \(\approx\) \(0.9228953243 - 0.2283993380i\)
\(L(1)\) \(\approx\) \(0.9228953243 - 0.2283993380i\)

Euler product

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

−17.644401488527127387672390662247, −17.04831522468623776004209943879, −16.46551177631824937994213314763, −15.92518713972858154528677125561, −15.200100517227527077751363509707, −14.61199348212753871516249183422, −14.09448565581125287380836213988, −13.16015940548311823830455016520, −12.58271445007156341777355207932, −11.83498303393576488590913237387, −10.97679851466022551233411544470, −10.4029437864684342185882851, −9.68595833391604700241748419353, −9.10177444635184325317141589450, −8.768546848316355968046969549534, −7.49543093796674158464201266266, −7.07480478087685941653835112011, −6.029063735511365224022513418340, −5.47870128444740985105876017521, −4.54929395857125079688354798286, −3.89708905914490275954599760038, −3.25431986844884059988739980565, −2.57360991425008822093564727065, −1.54786453492169709577391455468, −0.214631008367478072679651503077, 1.054848168763843588445994824466, 1.59063048600714154861553877442, 2.71923367120066498927428250117, 3.4481032527573394560519200820, 3.76216426469761411541352054506, 5.27308875088006217777512721300, 5.86721172259387748277100338151, 6.48108258763186277528887156973, 7.22487159246710764558916704406, 7.76026164207490124068483016210, 8.616624351357919759619066278535, 9.33718684730770028972237976616, 9.754937467012442052349323177377, 10.816114610594173247541393059911, 11.606646298691264286386430749129, 12.19965070286218049139642123131, 12.72535535573766583317118637204, 13.510389638222531149358375932692, 13.96993839472416138984505942120, 14.63405621941117218313766272690, 15.37154124369512053596685235851, 16.26884806141093202225531001033, 16.87383075334559706471380412268, 17.28392125047627096691367152214, 18.428118678564331244047295930727

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