↑

Properties

Label 1-5200-5200.2397-r0-0-0
Degree $1$
Conductor $5200$
Sign $0.949 - 0.313i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $0.949 - 0.313i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (2397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ 0.949 - 0.313i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.418703822 - 0.2279375589i\)
\(L(\frac12)\) \(\approx\) \(1.418703822 - 0.2279375589i\)
\(L(1)\) \(\approx\) \(1.058233582 + 0.2293523168i\)
\(L(1)\) \(\approx\) \(1.058233582 + 0.2293523168i\)

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 \( 1 + (0.309 + 0.951i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.10703756157394962841898848431, −17.55492658010661751949941719679, −16.96038283315122467362607076704, −15.87026752727981974637345577195, −15.25978327201811757447026076593, −14.61846623347374078860754329907, −14.00502295135281798530367453146, −13.33614920250317415718039856489, −12.693588042365246141425351879645, −12.1629912221611565258265028918, −11.287456718757911713197360364812, −10.81566832657976904187862997814, −9.97466230222316642719167768512, −8.87893040333360785438452511979, −8.508546028742410020164496258306, −7.81858411856325212772720547143, −7.128240332228921153379301433897, −6.55255538627307501664498141653, −5.68152153353442585098611826800, −4.80244428354952291853041603318, −4.303581610082744749823662444164, −3.04488635925850558817240252905, −2.35584548953199691982052010583, −1.80900152273269454039910549955, −0.87070231409612750937292375824, 0.3956152068726342668657053751, 1.92286549238358805504979000415, 2.314618349658812257569285404295, 3.475116118186903814379266218244, 3.95517661003707297040551785558, 4.859052972875753742002725196741, 5.3752203724866254322307399346, 6.02699571350170370543996952799, 7.203970969747568524701391295192, 7.958204674195667951389386908483, 8.4958564836661035720863290368, 9.03532070526146192315274673102, 10.04998758196154864759230651667, 10.44627897452479576197413767486, 11.286701593231306845925116537188, 11.57090659035702574655940783093, 12.66866590202896276558072147731, 13.47410422118939565346899812383, 14.18554252117194885674791333363, 14.49681579645248826362364058129, 15.50404396527098643663942612632, 15.789856493171042389168782853973, 16.47491329198529829033844250292, 17.36368628092796213824595381212, 17.76710085181454193887013583639

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