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Properties

Label 1-200-200.37-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: $\chi_{200} (37, \cdot )$
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

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

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