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Properties

Label 1-200-200.27-r0-0-0
Degree $1$
Conductor $200$
Sign $0.684 - 0.728i$
Analytic cond. $0.928796$
Root an. cond. $0.928796$
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  + (−0.587 + 0.809i)3-s − i·7-s + (−0.309 − 0.951i)9-s + (0.309 − 0.951i)11-s + (−0.951 + 0.309i)13-s + (−0.587 − 0.809i)17-s + (0.809 − 0.587i)19-s + (0.809 + 0.587i)21-s + (0.951 + 0.309i)23-s + (0.951 + 0.309i)27-s + (−0.809 − 0.587i)29-s + (0.809 − 0.587i)31-s + (0.587 + 0.809i)33-s + (0.951 − 0.309i)37-s + (0.309 − 0.951i)39-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)3-s − i·7-s + (−0.309 − 0.951i)9-s + (0.309 − 0.951i)11-s + (−0.951 + 0.309i)13-s + (−0.587 − 0.809i)17-s + (0.809 − 0.587i)19-s + (0.809 + 0.587i)21-s + (0.951 + 0.309i)23-s + (0.951 + 0.309i)27-s + (−0.809 − 0.587i)29-s + (0.809 − 0.587i)31-s + (0.587 + 0.809i)33-s + (0.951 − 0.309i)37-s + (0.309 − 0.951i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.684 - 0.728i$
Analytic conductor: \(0.928796\)
Root analytic conductor: \(0.928796\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 200,\ (0:\ ),\ 0.684 - 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7494990598 - 0.3243372055i\)
\(L(\frac12)\) \(\approx\) \(0.7494990598 - 0.3243372055i\)
\(L(1)\) \(\approx\) \(0.8304687260 - 0.05743465635i\)
\(L(1)\) \(\approx\) \(0.8304687260 - 0.05743465635i\)

Euler product

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

−27.220753979245981813624124957183, −25.86947565462636305029648735202, −24.79356563396706024480838298355, −24.504484297599097546783600395652, −23.13245844161621521497110594598, −22.429793971995722475995796820308, −21.60318486494653518063936667084, −20.1589787287478950250124291358, −19.28077882439917848992380892609, −18.31335754843015045024538776094, −17.56144414896297970029363177573, −16.65338319274024167295953434091, −15.32148952885278189258944612009, −14.481152781961091831652415114204, −13.00115413750133161410627481738, −12.36101168603037293187498118467, −11.54539423657381225593249160949, −10.26278505597679500183634368067, −9.02271866689015038873717254629, −7.78357263155248175133889126777, −6.797536899328404393123585512978, −5.68839262501768797180832832204, −4.70496030233045628749839575609, −2.73228611678974902283581031659, −1.597147006499628074525741942721, 0.70751026559172448316068740264, 2.980425301631732533537491786115, 4.20866057971495749200689386749, 5.13025346000769720643410861394, 6.45119287315562127665108592309, 7.518688935377878261610904424650, 9.141419169909876477444601542895, 9.87610290403246612855779242831, 11.14511858991378970304063186473, 11.60342948753042657295983336959, 13.20535193655885946098793175189, 14.17197981948480795442401370237, 15.26611026332142210721941145644, 16.370522956943999830993579764628, 16.97127888670888521277495771374, 17.88179166315855743351899053760, 19.28454650537807388243891505326, 20.23702843946161169934120029223, 21.15825871007768280687653485886, 22.14212870494611552212872407264, 22.81759496673155289037827148339, 23.90729170720960452209067281919, 24.69637075171613885630720629988, 26.31863692532461111183515286323, 26.78966059438783086385612382274

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