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Properties

Label 1-5200-5200.2187-r0-0-0
Degree $1$
Conductor $5200$
Sign $0.769 + 0.638i$
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  + (−0.104 − 0.994i)3-s + (0.866 − 0.5i)7-s + (−0.978 + 0.207i)9-s + (−0.207 + 0.978i)11-s + (−0.994 − 0.104i)17-s + (0.994 + 0.104i)19-s + (−0.587 − 0.809i)21-s + (−0.207 + 0.978i)23-s + (0.309 + 0.951i)27-s + (0.994 − 0.104i)29-s + (0.809 + 0.587i)31-s + (0.994 + 0.104i)33-s + (−0.669 − 0.743i)37-s + (−0.669 − 0.743i)41-s + (0.5 + 0.866i)43-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)3-s + (0.866 − 0.5i)7-s + (−0.978 + 0.207i)9-s + (−0.207 + 0.978i)11-s + (−0.994 − 0.104i)17-s + (0.994 + 0.104i)19-s + (−0.587 − 0.809i)21-s + (−0.207 + 0.978i)23-s + (0.309 + 0.951i)27-s + (0.994 − 0.104i)29-s + (0.809 + 0.587i)31-s + (0.994 + 0.104i)33-s + (−0.669 − 0.743i)37-s + (−0.669 − 0.743i)41-s + (0.5 + 0.866i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $0.769 + 0.638i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (2187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ 0.769 + 0.638i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.203186315 + 0.4339729687i\)
\(L(\frac12)\) \(\approx\) \(1.203186315 + 0.4339729687i\)
\(L(1)\) \(\approx\) \(0.9912590566 - 0.1815747793i\)
\(L(1)\) \(\approx\) \(0.9912590566 - 0.1815747793i\)

Euler product

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

−17.915992090688999150047188479110, −17.14710033312801780326687810907, −16.55272436202781224141626580657, −15.793180322170484032676907752602, −15.37422189623658252432507397755, −14.73483209790530564152260520347, −13.81391422974024863026320247425, −13.66044314308781880399988384272, −12.30110367352431762171426738375, −11.746538403523284861492488457759, −11.176268689887885790734976188993, −10.544050327345165410069441595986, −9.93310683822594117887178989437, −8.9519293658228821175997166766, −8.55665686773128517410743452421, −8.01320756783920297259735302110, −6.864243030495152456764134069513, −6.02338864040898352266005668602, −5.44922474344352823094584498550, −4.66627132868781528789835483464, −4.229810434458045830924181742897, −3.07357645289927875156005823854, −2.67906421269352110035696054826, −1.536725852035758338653812165261, −0.36008100128728083894480480724, 1.02432385347414831133421664529, 1.62740091398500715871911543221, 2.380264744353513122811322296, 3.22396770617824360921585276157, 4.34371212271686999818025534478, 4.930050529929090378703636352487, 5.68217713345572173351033338033, 6.59244819934890312006740660294, 7.24995722672675681293019988677, 7.67869968525967487402551065703, 8.39355272540110426250982376233, 9.17217000862246314755860339372, 10.03479786970975174912994536323, 10.83301786885493480424526097236, 11.43923833544386879391332536160, 12.09051980351119585383707718662, 12.64605158847619062889494899282, 13.50608535023188199065376760297, 14.00983971388429816845136704558, 14.45051336057511395898532145636, 15.562039302212619577191305454282, 15.87213749542196635634714479325, 17.187822404523026188368030837771, 17.462910445194936060546972203556, 17.913635579896168151568603974953

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