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Properties

Label 1-200-200.141-r0-0-0
Degree $1$
Conductor $200$
Sign $0.728 - 0.684i$
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  + (0.809 − 0.587i)3-s + 7-s + (0.309 − 0.951i)9-s + (−0.309 − 0.951i)11-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)17-s + (0.809 + 0.587i)19-s + (0.809 − 0.587i)21-s + (0.309 + 0.951i)23-s + (−0.309 − 0.951i)27-s + (0.809 − 0.587i)29-s + (−0.809 − 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.309 + 0.951i)37-s + (0.309 + 0.951i)39-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + 7-s + (0.309 − 0.951i)9-s + (−0.309 − 0.951i)11-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)17-s + (0.809 + 0.587i)19-s + (0.809 − 0.587i)21-s + (0.309 + 0.951i)23-s + (−0.309 − 0.951i)27-s + (0.809 − 0.587i)29-s + (−0.809 − 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.309 + 0.951i)37-s + (0.309 + 0.951i)39-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.728 - 0.684i$
Analytic conductor: \(0.928796\)
Root analytic conductor: \(0.928796\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 200,\ (0:\ ),\ 0.728 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.490816577 - 0.5902560389i\)
\(L(\frac12)\) \(\approx\) \(1.490816577 - 0.5902560389i\)
\(L(1)\) \(\approx\) \(1.369774500 - 0.3265760321i\)
\(L(1)\) \(\approx\) \(1.369774500 - 0.3265760321i\)

Euler product

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

−26.91529018176039279901232248169, −26.21965333410870982802391107646, −25.06889014570012005389322455780, −24.48618442031008753754389903808, −23.20433354727957471233866369594, −22.07335863031001894346153217807, −21.27211252663022249415715206514, −20.19080466390896484464107675709, −19.87228943905171852318815294266, −18.247000130030949977442700585149, −17.580264905347815410330989187863, −16.21571803115042795848954639245, −15.12893864165672256875303591956, −14.68155585453487881114538968719, −13.47777207015365745801312234261, −12.43832019466330139146363196755, −10.957751918970303458555268770578, −10.21526145341380552489751281848, −8.98173526340202755534012715555, −8.08122345886630399835600293358, −7.11043262294181123440268915855, −5.17864174132437569804140162229, −4.48861438930799776807832233174, −3.00386558897333523308527842705, −1.842407604649897411949522200615, 1.38271222462127771759948304491, 2.57810669334502157338657624515, 3.91818940429495813689360886828, 5.30987291212119296870583192782, 6.76475056973725224512337955098, 7.78580149682558607307478601417, 8.64751491573721352228703118210, 9.66091987131435491307531179239, 11.247605258747353312807497717198, 11.96639304529696212554821573165, 13.47066942697265335751054666881, 13.96353804680463150047940499760, 14.96329997372466547466010928168, 16.068350027940801340649887887108, 17.37903859889949517846922986924, 18.34275936112626843064955384569, 19.069568170787033990392994854225, 20.14332839007252056425068853387, 21.00967016623389789077483791990, 21.799841848676601729864415814173, 23.30636823567967247058632216810, 24.297324926117281025371647283851, 24.59795601944616418006339420959, 25.878792754440910040066733152693, 26.77663042553579115410405089496

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