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Properties

Label 1-5200-5200.627-r0-0-0
Degree $1$
Conductor $5200$
Sign $0.987 - 0.155i$
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 + (0.866 − 0.5i)7-s + (0.669 − 0.743i)9-s + (−0.743 + 0.669i)11-s + (−0.406 + 0.913i)17-s + (−0.406 + 0.913i)19-s + (−0.587 + 0.809i)21-s + (0.743 − 0.669i)23-s + (−0.309 + 0.951i)27-s + (−0.406 − 0.913i)29-s + (0.809 − 0.587i)31-s + (0.406 − 0.913i)33-s + (−0.978 − 0.207i)37-s + (0.978 + 0.207i)41-s + (−0.5 − 0.866i)43-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)3-s + (0.866 − 0.5i)7-s + (0.669 − 0.743i)9-s + (−0.743 + 0.669i)11-s + (−0.406 + 0.913i)17-s + (−0.406 + 0.913i)19-s + (−0.587 + 0.809i)21-s + (0.743 − 0.669i)23-s + (−0.309 + 0.951i)27-s + (−0.406 − 0.913i)29-s + (0.809 − 0.587i)31-s + (0.406 − 0.913i)33-s + (−0.978 − 0.207i)37-s + (0.978 + 0.207i)41-s + (−0.5 − 0.866i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $0.987 - 0.155i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (627, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ 0.987 - 0.155i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.154127925 - 0.09014912552i\)
\(L(\frac12)\) \(\approx\) \(1.154127925 - 0.09014912552i\)
\(L(1)\) \(\approx\) \(0.8321330887 + 0.06036659152i\)
\(L(1)\) \(\approx\) \(0.8321330887 + 0.06036659152i\)

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

−18.11952090472662231564227203845, −17.460263925344776322765650533595, −16.7370750169125488161629695532, −16.02022319092325849321119459650, −15.48737662943416740989430485177, −14.776294488192152128572129027957, −13.75202817587444536952392371638, −13.37981756309969603118231104571, −12.60958732744025320788788695346, −11.842286151373408211918886057123, −11.2921576556402358978175704663, −10.86913765793352453032695807449, −10.12837308778301020800724082617, −9.04407427260950933300020420643, −8.538800097689885785436527739451, −7.64452791115115692732801555615, −7.07659259267469071305682756084, −6.3408649330976110823068838572, −5.38095151599741411730178876595, −5.11350405315174593437914479148, −4.42511833959773809831980781934, −3.139732917875755056950068360004, −2.42234060171623771731532154451, −1.53810079917743908398712719145, −0.68733043514765460220130569190, 0.5101566730261329995112321675, 1.576487287485975428598517360693, 2.25116845651244461923055475636, 3.526943677274925655078519224237, 4.32430708294961356113636587077, 4.70296749719496140835821295407, 5.52732416936702046158894034167, 6.22590603707957721848627887407, 6.96339242919005948517699743515, 7.78432077400645442144405854524, 8.32871286844763063281060008555, 9.34464204947004752182204414274, 10.122178018311704256832593541744, 10.65775534491640419368187018424, 11.06282539327875153143343424356, 11.94046938926508492322502874797, 12.54438072557240201296725640574, 13.14992243009933808211500740348, 14.02874742896660291958801890239, 14.921701611355486461680074072051, 15.229704479376593325669333870056, 15.99540320445602771807978610390, 16.87797142552982795975769311333, 17.25111327311134867240582771965, 17.73002502019982802615022434094

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