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Properties

Label 1-5200-5200.3413-r0-0-0
Degree $1$
Conductor $5200$
Sign $-0.434 + 0.900i$
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 + (−0.5 + 0.866i)7-s + (0.669 + 0.743i)9-s + (0.669 − 0.743i)11-s + (0.406 + 0.913i)17-s + (−0.913 + 0.406i)19-s + (−0.809 + 0.587i)21-s + (0.743 + 0.669i)23-s + (0.309 + 0.951i)27-s + (−0.406 + 0.913i)29-s + (0.587 − 0.809i)31-s + (0.913 − 0.406i)33-s + (0.207 + 0.978i)37-s + (0.207 + 0.978i)41-s + (−0.5 + 0.866i)43-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)3-s + (−0.5 + 0.866i)7-s + (0.669 + 0.743i)9-s + (0.669 − 0.743i)11-s + (0.406 + 0.913i)17-s + (−0.913 + 0.406i)19-s + (−0.809 + 0.587i)21-s + (0.743 + 0.669i)23-s + (0.309 + 0.951i)27-s + (−0.406 + 0.913i)29-s + (0.587 − 0.809i)31-s + (0.913 − 0.406i)33-s + (0.207 + 0.978i)37-s + (0.207 + 0.978i)41-s + (−0.5 + 0.866i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $-0.434 + 0.900i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (3413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ -0.434 + 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.248187369 + 1.988919799i\)
\(L(\frac12)\) \(\approx\) \(1.248187369 + 1.988919799i\)
\(L(1)\) \(\approx\) \(1.322119618 + 0.5370831954i\)
\(L(1)\) \(\approx\) \(1.322119618 + 0.5370831954i\)

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 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.669 - 0.743i)T \)
17 \( 1 + (0.406 + 0.913i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (0.743 + 0.669i)T \)
29 \( 1 + (-0.406 + 0.913i)T \)
31 \( 1 + (0.587 - 0.809i)T \)
37 \( 1 + (0.207 + 0.978i)T \)
41 \( 1 + (0.207 + 0.978i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.669 - 0.743i)T \)
61 \( 1 + (-0.207 + 0.978i)T \)
67 \( 1 + (0.994 + 0.104i)T \)
71 \( 1 + (-0.994 + 0.104i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (0.587 - 0.809i)T \)
89 \( 1 + (-0.743 - 0.669i)T \)
97 \( 1 + (-0.104 - 0.994i)T \)
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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

−17.67875036329089638845302014720, −17.19758282080783819547269337565, −16.48977627964732348479602852258, −15.61561969291969887236493247840, −15.076520723198571373941070717697, −14.30419476204952658380337192841, −13.837768450856355325452922870094, −13.20900522394195755923094534665, −12.45340112110162390823912280209, −12.05405048704480463349464718278, −10.8963301487456991307717602182, −10.318126713947640820419827214511, −9.42004555695905670041983899675, −9.15380018928965156990285879999, −8.21862177095405388437326668483, −7.43733976832880669619748878304, −6.905279968852712735592153514178, −6.48139033192714259999560658599, −5.29202798687170878371730476196, −4.175431624126583556118073037449, −4.008355058479507344056356489460, −2.88870505841906068948148092801, −2.355757850129115850518340231220, −1.33804961624289638729705930033, −0.54132946609703731563604189373, 1.24235644215880906684065513260, 1.986855198961912134426600782654, 2.93568559063094567557349959711, 3.414698355735898656477041858110, 4.14214845643555803565617105707, 5.02043816667920146535392337433, 5.92350008318221943176486898597, 6.44142554226827949792294733292, 7.412838906665979326730882496969, 8.3382733937053257006006764664, 8.620053695811832387050790561891, 9.386933572064674526087987991119, 9.93803397365062885874986127311, 10.72598711230165792349738977786, 11.48283395602444473238424365988, 12.261691155056348126125861955640, 13.06688080641377029551724708523, 13.430081630532503123670935419665, 14.40099612907285550208280111870, 15.05806670252914807169066866405, 15.179029785737362345694147818533, 16.351848676345254023910805951, 16.577952647844172574021252276405, 17.43540080498805778781072256486, 18.5412946942362568930254574004

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