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Properties

Label 1-5200-5200.1667-r0-0-0
Degree $1$
Conductor $5200$
Sign $0.0952 - 0.995i$
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  + (−0.669 − 0.743i)3-s + (0.866 − 0.5i)7-s + (−0.104 + 0.994i)9-s + (0.994 − 0.104i)11-s + (0.743 + 0.669i)17-s + (0.743 + 0.669i)19-s + (−0.951 − 0.309i)21-s + (−0.994 + 0.104i)23-s + (0.809 − 0.587i)27-s + (0.743 − 0.669i)29-s + (−0.309 − 0.951i)31-s + (−0.743 − 0.669i)33-s + (0.913 − 0.406i)37-s + (−0.913 + 0.406i)41-s + (−0.5 − 0.866i)43-s + ⋯
L(s)  = 1  + (−0.669 − 0.743i)3-s + (0.866 − 0.5i)7-s + (−0.104 + 0.994i)9-s + (0.994 − 0.104i)11-s + (0.743 + 0.669i)17-s + (0.743 + 0.669i)19-s + (−0.951 − 0.309i)21-s + (−0.994 + 0.104i)23-s + (0.809 − 0.587i)27-s + (0.743 − 0.669i)29-s + (−0.309 − 0.951i)31-s + (−0.743 − 0.669i)33-s + (0.913 − 0.406i)37-s + (−0.913 + 0.406i)41-s + (−0.5 − 0.866i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0952 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0952 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $0.0952 - 0.995i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (1667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ 0.0952 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.223998320 - 1.112438035i\)
\(L(\frac12)\) \(\approx\) \(1.223998320 - 1.112438035i\)
\(L(1)\) \(\approx\) \(0.9890590835 - 0.3221521096i\)
\(L(1)\) \(\approx\) \(0.9890590835 - 0.3221521096i\)

Euler product

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

−17.909242080776329175958067325385, −17.63802476758405478291191223983, −16.69057000112062793330177297875, −16.19661042085571936055390192729, −15.61497699004656859603847520865, −14.64923601740798403422568616488, −14.48849381402320698996438604654, −13.636957885860323618007305921504, −12.48358924848414357669928313889, −11.919966600766956638403566754267, −11.523866928242629684728659294051, −10.891845157333179454596822137235, −9.931913798740934195360805781086, −9.54188578622196438592326076384, −8.72504622766801791009784206913, −8.09041393273383411963834659448, −7.05068806936331699285572298097, −6.45450196443307341106622187820, −5.57842593571465048279688387824, −4.99256274932101165423319236328, −4.47456463251704855698190607301, −3.52809374030241373141459097960, −2.85664539971669544563178771451, −1.62412991450357606555072758056, −0.94686316279607456013544253197, 0.57166633236519253135959868764, 1.52340685636466207633038476893, 1.82120985233982599814177787370, 3.11523009380957742963850989760, 4.07097359311061061763950997605, 4.6176938355667396253256542724, 5.718250412385446824780461693493, 5.98925978922579890034252198665, 6.91334785954967855085412851956, 7.68431651692218926074080990963, 8.03325489540537092564637420940, 8.8882443454381756696762912496, 10.04910595212673470895595079127, 10.350445990287877342677301531011, 11.45641853527823288497347476511, 11.73145289462004083010683982323, 12.26311061107718284529927955702, 13.26106766689847939112342882860, 13.7654637771514603018193639711, 14.45352086004254794139146123030, 14.9580683087920727526470010659, 16.12611806317807476804644781984, 16.69736362719179370664728144631, 17.13942977293721752372791457070, 17.84496428677792538499080271475

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