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Properties

Label 1-200-200.131-r1-0-0
Degree $1$
Conductor $200$
Sign $0.535 + 0.844i$
Analytic cond. $21.4929$
Root an. cond. $21.4929$
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  + (0.309 − 0.951i)3-s − 7-s + (−0.809 − 0.587i)9-s + (−0.809 + 0.587i)11-s + (0.809 + 0.587i)13-s + (0.309 + 0.951i)17-s + (0.309 + 0.951i)19-s + (−0.309 + 0.951i)21-s + (0.809 − 0.587i)23-s + (−0.809 + 0.587i)27-s + (−0.309 + 0.951i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (0.809 + 0.587i)37-s + (0.809 − 0.587i)39-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)3-s − 7-s + (−0.809 − 0.587i)9-s + (−0.809 + 0.587i)11-s + (0.809 + 0.587i)13-s + (0.309 + 0.951i)17-s + (0.309 + 0.951i)19-s + (−0.309 + 0.951i)21-s + (0.809 − 0.587i)23-s + (−0.809 + 0.587i)27-s + (−0.309 + 0.951i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (0.809 + 0.587i)37-s + (0.809 − 0.587i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.535 + 0.844i$
Analytic conductor: \(21.4929\)
Root analytic conductor: \(21.4929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 200,\ (1:\ ),\ 0.535 + 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.010566811 + 0.5555638060i\)
\(L(\frac12)\) \(\approx\) \(1.010566811 + 0.5555638060i\)
\(L(1)\) \(\approx\) \(0.9515917494 - 0.07769500621i\)
\(L(1)\) \(\approx\) \(0.9515917494 - 0.07769500621i\)

Euler product

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

−26.50295120049098763061238647420, −25.704541652692474066951607783882, −24.94420070859002608224565909467, −23.39839570667207158886730019318, −22.70812172877624915035733063031, −21.698689279438185228223669648947, −20.87250489691061914199410043787, −19.95408306827023235315219395649, −19.027175516219352585713928890696, −17.88187819984667958119440144068, −16.499995086244376745911884912788, −15.90455017359143604236991707705, −15.14759017729638720362907359783, −13.70993447113202602312519077954, −13.120312059098905145862901485113, −11.485220816082407740772102921185, −10.57566120306837972709365227346, −9.58290952342709146130783144394, −8.73012245151757562446286088750, −7.46136092204688491461674808927, −5.94025910760982469417560421493, −4.97876513368106918503544536736, −3.45751433916472467897829498147, −2.809264169110000621506073988252, −0.39651838074974595363975969880, 1.31873628799363932254441788048, 2.67141491377310327627469545944, 3.84191785919302116402084238772, 5.69711709100122453254212130681, 6.609954297645564908690036971735, 7.64950312219785916594533919685, 8.7181810865794962227399427155, 9.84594415403781327451624492318, 11.09568683507663318474373197594, 12.53114242924453111591028007171, 12.909530391291307235244743095023, 14.02828818258979114721733412602, 15.09354619944443923294532384744, 16.273427828882776078061345301397, 17.255879423597134277035313675084, 18.687541435365082261363757431098, 18.786042830504125335884625263767, 20.14376488519479207999598952953, 20.86052869546559381336638490592, 22.29016449351910396480688020274, 23.27954347219578669297242341145, 23.82409909086624557374879054346, 25.11305988843245210508319845518, 25.80432435185185025933993665140, 26.40754020006331543770720784453

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