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Properties

Label 1-200-200.173-r1-0-0
Degree $1$
Conductor $200$
Sign $-0.684 + 0.728i$
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  + (−0.587 + 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

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

Invariants

Degree: \(1\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.684 + 0.728i$
Analytic conductor: \(21.4929\)
Root analytic conductor: \(21.4929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 200,\ (1:\ ),\ -0.684 + 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3537236518 + 0.8174071305i\)
\(L(\frac12)\) \(\approx\) \(0.3537236518 + 0.8174071305i\)
\(L(1)\) \(\approx\) \(0.7661667592 + 0.2594693205i\)
\(L(1)\) \(\approx\) \(0.7661667592 + 0.2594693205i\)

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

−26.115593338996324456193207219293, −25.23076376079936584581748376809, −24.37960979400080614427212866867, −23.57409630141340811863793610541, −22.6246105850330706892292795156, −21.67197202406629485923409595709, −20.753833203621667594491053600430, −19.149089021630181346528051512852, −18.73624161570987546901588420163, −17.87628059628069564497863374998, −16.646578058276971470598241468994, −15.89926502173037304883899853309, −14.55370174121718197427573844097, −13.38849990908789047535209749048, −12.63552361377119136235531043170, −11.458788920539094237258338357299, −10.89037749296768798471225969750, −9.115668009397072323483511235239, −8.26547204693846750136544925114, −6.95189086188283656146737169055, −5.92020944224691430501727281366, −5.10277986049647647872606274343, −3.1833423849556749457397750253, −1.87298648560772553803963759488, −0.34870447817693506492543101968, 1.34457463856853688814397155438, 3.45829696182408627925518751428, 4.31461674962863490015074193571, 5.51732673121122094743736173741, 6.671209378272809947109574846769, 7.93350536280597810352203974112, 9.30893985229766701013476787831, 10.4387412604863950747601322615, 10.86755977639817383581077721509, 12.28315118700524605515804401088, 13.26230374058279362583840464004, 14.64567278371843473069638557738, 15.4054057710720866825516474121, 16.57597628908465921347423762542, 17.202815878045207639368749681202, 18.17357328503071726323400580703, 19.52269635874196668640978649886, 20.752267619371609557953393477516, 21.039086488190464076328659999498, 22.48316426911124382737316669143, 23.21635589368983486359900039980, 23.74394179319848401761537339976, 25.46672706769159152919173482963, 26.07559638738150253358204454321, 27.15935093059406999511233808511

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