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Properties

Label 1-5200-5200.2947-r0-0-0
Degree $1$
Conductor $5200$
Sign $0.636 - 0.771i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $0.636 - 0.771i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (2947, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ 0.636 - 0.771i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3026446655 - 0.1426311617i\)
\(L(\frac12)\) \(\approx\) \(0.3026446655 - 0.1426311617i\)
\(L(1)\) \(\approx\) \(0.5784484706 + 0.1210764315i\)
\(L(1)\) \(\approx\) \(0.5784484706 + 0.1210764315i\)

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

−18.24778692534066876646250929467, −17.392750544440584980926735199873, −16.81320671272544785198594277668, −16.16150080714098034541508146996, −15.46169969659506047437593799953, −14.962339565809891682496463055227, −13.77997021830780277058671732049, −13.16103138532915355578166087553, −12.88126536380428161220908242718, −12.18470097159255325944885847549, −11.30529217011306366745187410705, −10.88285589798952715561077725588, −10.10590459709918305686236167740, −9.17706781461926073997463664498, −8.67774441502939587972499254447, −7.613792425898095554993287807017, −7.14596135029559681317138276067, −6.41761389611949605275056864577, −5.783669026962548774460156963373, −5.07365230348620728024248421421, −4.413236491953670747642189795541, −3.12839364208934117514437699458, −2.52797655123791083547897887241, −1.84887637514701496015524778302, −0.5567835733512299188310376419, 0.16649329112385609943411333272, 1.341108500656034917058950981663, 2.52993303211283819983687092410, 3.352386866534312789502982654, 3.99163467021606309125742303169, 4.71914092158845102965923834417, 5.48351081317739039382148763152, 6.20939848384517412783814690351, 6.743906061745421094552891029651, 7.61549683784811634859293398230, 8.53079729474528412075442062476, 9.211715564540504160444458500121, 9.99526840769094019305310333988, 10.51799953592093982120944888055, 10.952966424214923412716535998511, 11.779425658265102075907998371455, 12.71695197915296388099372139348, 13.02705525992074741781401098981, 13.80513051239528050211944774159, 14.92373568842241969585556823157, 15.2395018478462482996298857157, 15.96208382830375592237951393268, 16.70066701107529760691880966454, 16.93081105037918313384040483854, 17.81620993061226582376310266942

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