Properties

Label 1-99-99.31-r0-0-0
Degree $1$
Conductor $99$
Sign $0.815 - 0.578i$
Analytic cond. $0.459754$
Root an. cond. $0.459754$
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  + (0.913 − 0.406i)2-s + (0.669 − 0.743i)4-s + (0.913 + 0.406i)5-s + (−0.978 − 0.207i)7-s + (0.309 − 0.951i)8-s + 10-s + (−0.104 + 0.994i)13-s + (−0.978 + 0.207i)14-s + (−0.104 − 0.994i)16-s + (−0.809 + 0.587i)17-s + (0.309 − 0.951i)19-s + (0.913 − 0.406i)20-s + (−0.5 − 0.866i)23-s + (0.669 + 0.743i)25-s + (0.309 + 0.951i)26-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)2-s + (0.669 − 0.743i)4-s + (0.913 + 0.406i)5-s + (−0.978 − 0.207i)7-s + (0.309 − 0.951i)8-s + 10-s + (−0.104 + 0.994i)13-s + (−0.978 + 0.207i)14-s + (−0.104 − 0.994i)16-s + (−0.809 + 0.587i)17-s + (0.309 − 0.951i)19-s + (0.913 − 0.406i)20-s + (−0.5 − 0.866i)23-s + (0.669 + 0.743i)25-s + (0.309 + 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.815 - 0.578i$
Analytic conductor: \(0.459754\)
Root analytic conductor: \(0.459754\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 99,\ (0:\ ),\ 0.815 - 0.578i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.627489743 - 0.5190565833i\)
\(L(\frac12)\) \(\approx\) \(1.627489743 - 0.5190565833i\)
\(L(1)\) \(\approx\) \(1.629124589 - 0.3808043013i\)
\(L(1)\) \(\approx\) \(1.629124589 - 0.3808043013i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.913 - 0.406i)T \)
5 \( 1 + (0.913 + 0.406i)T \)
7 \( 1 + (-0.978 - 0.207i)T \)
13 \( 1 + (-0.104 + 0.994i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.978 - 0.207i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.669 + 0.743i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.669 - 0.743i)T \)
61 \( 1 + (-0.104 - 0.994i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.913 - 0.406i)T \)
83 \( 1 + (-0.104 - 0.994i)T \)
89 \( 1 + T \)
97 \( 1 + (0.913 - 0.406i)T \)
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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

−29.934198004446057772888775490348, −29.384852558157983739392890656822, −28.3440176009631079853866422626, −26.69296533289185179836403348947, −25.45964509471524241333817918218, −25.06520755740730892617391264797, −23.90106600461774771136623521087, −22.53386337604767643307513149243, −22.08086996195395840362393228558, −20.764135047728426300850100727506, −19.96553696025077080766180292176, −18.22605095414435135018409149264, −17.044118521917704128312148697651, −16.10803000124144934235877817883, −15.06189456347504179324407204895, −13.65671348890392918420690148454, −13.04106797715515482903805883946, −11.96438923852608832031603030737, −10.3085886793886658255769915078, −9.043565056380579017993583987609, −7.46855551358699223869423562583, −6.08561712693355282918891479977, −5.34728978107723808909694087939, −3.654924991932581476734535375631, −2.26720678821046352823756533570, 1.9521091757370593796784068178, 3.209631067049269452206597961233, 4.68394985081248898287004760791, 6.23227741673729814443600767401, 6.840276150876243358404262744193, 9.254931443619730196157566683191, 10.23157751944238948418235142110, 11.34550492197366526558428302941, 12.793882027668931996939249863787, 13.55974329655421568899838885421, 14.540471926149625330566361904632, 15.79361333761909694286806296570, 16.96445421571938369511279981848, 18.48930082368885436023090462665, 19.513755623762059308364095870336, 20.56613129513058825837795358118, 21.90887177163920934977004459140, 22.18575287949439551621796976396, 23.52757769087656591081971907659, 24.53083020811975046272244130466, 25.71129730680532279564982511617, 26.51803687722878725462153931712, 28.54372814628933050030762126370, 28.848040554080091401404420977456, 29.983273395100588527779758029942

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