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Properties

Label 1-99-99.40-r1-0-0
Degree $1$
Conductor $99$
Sign $-0.208 + 0.978i$
Analytic cond. $10.6390$
Root an. cond. $10.6390$
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  + (0.978 + 0.207i)2-s + (0.913 + 0.406i)4-s + (−0.978 + 0.207i)5-s + (0.104 + 0.994i)7-s + (0.809 + 0.587i)8-s − 10-s + (−0.669 + 0.743i)13-s + (−0.104 + 0.994i)14-s + (0.669 + 0.743i)16-s + (−0.309 + 0.951i)17-s + (0.809 + 0.587i)19-s + (−0.978 − 0.207i)20-s + (−0.5 − 0.866i)23-s + (0.913 − 0.406i)25-s + (−0.809 + 0.587i)26-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)2-s + (0.913 + 0.406i)4-s + (−0.978 + 0.207i)5-s + (0.104 + 0.994i)7-s + (0.809 + 0.587i)8-s − 10-s + (−0.669 + 0.743i)13-s + (−0.104 + 0.994i)14-s + (0.669 + 0.743i)16-s + (−0.309 + 0.951i)17-s + (0.809 + 0.587i)19-s + (−0.978 − 0.207i)20-s + (−0.5 − 0.866i)23-s + (0.913 − 0.406i)25-s + (−0.809 + 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.208 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.208 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.208 + 0.978i$
Analytic conductor: \(10.6390\)
Root analytic conductor: \(10.6390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 99,\ (1:\ ),\ -0.208 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.530419971 + 1.890451642i\)
\(L(\frac12)\) \(\approx\) \(1.530419971 + 1.890451642i\)
\(L(1)\) \(\approx\) \(1.485251565 + 0.7064357895i\)
\(L(1)\) \(\approx\) \(1.485251565 + 0.7064357895i\)

Euler product

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

−29.76464285580526419312841134952, −28.61052982814677727298064294483, −27.409434314666415989046273149142, −26.44753117175184966068590256688, −24.86322797461540541173924832239, −24.06931280233294703973005945, −23.07913076704178767097704759874, −22.42265962717647964788769525311, −20.94640816576623848632129469644, −20.00143316743260681821919638088, −19.45543991978897385869684589520, −17.62754194283285558334244982609, −16.21415743331869900342637465884, −15.46075364787998112125034085905, −14.19769653086180469162286691522, −13.21299043950997158106687683780, −11.985370943916836812652860972355, −11.13966135305088334847810043354, −9.86214145524213027298455066374, −7.79706118782988757832823075112, −7.00392984426408203719851264590, −5.18933927198405343557375970218, −4.16873926272240758094949901037, −2.94173538294176928693213379062, −0.79717358796301082575908138126, 2.2297798365410413439434505296, 3.653708916653056946406869867146, 4.86714724406179267190520228852, 6.2393897201326261443429484243, 7.48509828114418043375574791671, 8.66808625269216187938655447057, 10.628190582912525289899576667187, 11.96808111124175300413407835009, 12.33424964893438318394494510098, 14.03756948380809655468316086950, 15.009147077473239752295083979890, 15.7825596303434852697258862496, 16.89518844307747147261253961477, 18.583937965114269107201041899987, 19.59882111371905839465330845559, 20.744524837957469785687504477695, 21.97759371039192211279509592556, 22.56969368550946003975984738243, 23.951671798897805031452622389401, 24.4138213773360898023393682733, 25.76330189118238865306186344535, 26.777900780645909184311699706435, 28.13065356572815161824718219779, 29.0989928397575955167718298411, 30.43815215317487353427194141753

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