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Properties

Label 1-5200-5200.5021-r0-0-0
Degree $1$
Conductor $5200$
Sign $0.337 + 0.941i$
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  + (0.207 − 0.978i)3-s + (0.5 + 0.866i)7-s + (−0.913 − 0.406i)9-s + (−0.406 − 0.913i)11-s + (−0.978 + 0.207i)17-s + (−0.207 − 0.978i)19-s + (0.951 − 0.309i)21-s + (−0.913 + 0.406i)23-s + (−0.587 + 0.809i)27-s + (0.207 − 0.978i)29-s + (0.309 − 0.951i)31-s + (−0.978 + 0.207i)33-s + (−0.994 + 0.104i)37-s + (0.104 + 0.994i)41-s + (−0.866 + 0.5i)43-s + ⋯
L(s)  = 1  + (0.207 − 0.978i)3-s + (0.5 + 0.866i)7-s + (−0.913 − 0.406i)9-s + (−0.406 − 0.913i)11-s + (−0.978 + 0.207i)17-s + (−0.207 − 0.978i)19-s + (0.951 − 0.309i)21-s + (−0.913 + 0.406i)23-s + (−0.587 + 0.809i)27-s + (0.207 − 0.978i)29-s + (0.309 − 0.951i)31-s + (−0.978 + 0.207i)33-s + (−0.994 + 0.104i)37-s + (0.104 + 0.994i)41-s + (−0.866 + 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $0.337 + 0.941i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (5021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ 0.337 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5072125779 + 0.3569206532i\)
\(L(\frac12)\) \(\approx\) \(0.5072125779 + 0.3569206532i\)
\(L(1)\) \(\approx\) \(0.8857492761 - 0.2316520805i\)
\(L(1)\) \(\approx\) \(0.8857492761 - 0.2316520805i\)

Euler product

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

−17.84559300257440440234885169415, −17.08657291437999598871757265960, −16.48501875464742082513413058282, −15.87924260711944406047469470217, −15.15152573433756481176929477969, −14.65178236289225858050597362159, −13.86163643630796162845690757809, −13.51366368338923017688776127386, −12.30925529731235496064337565837, −11.90183024495110727481418969540, −10.77012566490500314867019998057, −10.43321024869264603372263721715, −10.04934893583068173114159769359, −8.94026479026753967823673475287, −8.549416880122288230934705167647, −7.63770853979061939204340481447, −7.05005976610948711817933170618, −6.1243355871851851516054477407, −5.11066042965451107001809313140, −4.708484807785755487827881041341, −3.96102154871974676590321008483, −3.375111333054693289546232845504, −2.25815774665997849851706015494, −1.66597942678175498186474750004, −0.16499533503317946511739504319, 0.92517188793525110618395194633, 1.97172221476179543077752077812, 2.46245733133404853873156185028, 3.1788621095586077567485382056, 4.25493648860295460882425690804, 5.12654659361095103023148582500, 5.96020537735448697789290983956, 6.32786804444099680259622258550, 7.24490993727152011501669708272, 8.12359939285981271486372818914, 8.40422850068686110722008509721, 9.11206985237504306334937370742, 9.93242032399129773639084484162, 11.1772191153041983930171157929, 11.33441769124341916891348355212, 12.08971626776625782154402001863, 12.86658176331322475719308321219, 13.505773287199571090054633643936, 13.886967580707509289577968454842, 14.83526981787303675635359777601, 15.38158306306356371233850958485, 15.99346326204165234684349346779, 17.01790262839204233967225686113, 17.611205670235083629990871260049, 18.144011883708661738808297709771

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