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Properties

Label 1-200-200.19-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: $\chi_{200} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 200,\ (1:\ ),\ -0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06900779917 - 0.1255247207i\)
\(L(\frac12)\) \(\approx\) \(0.06900779917 - 0.1255247207i\)
\(L(1)\) \(\approx\) \(0.7352461878 + 0.1838218537i\)
\(L(1)\) \(\approx\) \(0.7352461878 + 0.1838218537i\)

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

−26.86950613696766972648943512230, −26.09796622360167829549279563901, −24.78264420746238995437606223820, −24.03362317523242635081831918138, −23.65567153033744244714738274373, −22.21324574256360405339228397315, −21.42807209664525277719975194814, −20.18973225832650442410260012293, −19.22876463082186894738706807508, −18.31063632991750884994703687284, −17.515750886158131175945189588390, −16.66110885670576340219921894323, −15.279183887291876960178699528088, −14.15200130745261054885133416513, −13.36167535529771347847642247832, −12.206851063120250252576687258455, −11.37123769732971486656193594036, −10.41067397124258548796073762637, −8.68326019787616095078026405439, −7.89248485960263109034224185654, −6.843119309395468767816325448033, −5.6243161477301546721592180293, −4.58495995774144954149087685502, −2.65543300571114044719815854094, −1.55288066555665008786927337303, 0.048414677537968595940096874611, 2.11702258031456383556470971050, 3.62072450005990040687054610353, 4.96383035626982521945602616877, 5.451103772024604760508762304183, 7.27393395512962824308617486392, 8.302052965737779044065315567801, 9.64259504907359065218786535582, 10.40884405614799331994531877679, 11.46844651354538108052527831619, 12.36318780645692650963079944413, 13.90081334299171231997282380102, 14.86778736872254892046368672027, 15.60847474247372413913654625, 16.689562169682925848982463156674, 17.682701612890196792575482516975, 18.37540556127670232261938342352, 20.16586821311228069182565325816, 20.5696970584774826844658160631, 21.61601172621841172905440669926, 22.4780418524751110829853776021, 23.39141925526879870496044768233, 24.41611440830396095508028253657, 25.5092992549052418234944569052, 26.56947227057019836312025383279

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