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Properties

Label 1-99-99.7-r1-0-0
Degree $1$
Conductor $99$
Sign $-0.788 + 0.615i$
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  + (−0.669 + 0.743i)2-s + (−0.104 − 0.994i)4-s + (0.669 + 0.743i)5-s + (−0.913 − 0.406i)7-s + (0.809 + 0.587i)8-s − 10-s + (0.978 + 0.207i)13-s + (0.913 − 0.406i)14-s + (−0.978 + 0.207i)16-s + (−0.309 + 0.951i)17-s + (0.809 + 0.587i)19-s + (0.669 − 0.743i)20-s + (−0.5 + 0.866i)23-s + (−0.104 + 0.994i)25-s + (−0.809 + 0.587i)26-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (−0.104 − 0.994i)4-s + (0.669 + 0.743i)5-s + (−0.913 − 0.406i)7-s + (0.809 + 0.587i)8-s − 10-s + (0.978 + 0.207i)13-s + (0.913 − 0.406i)14-s + (−0.978 + 0.207i)16-s + (−0.309 + 0.951i)17-s + (0.809 + 0.587i)19-s + (0.669 − 0.743i)20-s + (−0.5 + 0.866i)23-s + (−0.104 + 0.994i)25-s + (−0.809 + 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.788 + 0.615i$
Analytic conductor: \(10.6390\)
Root analytic conductor: \(10.6390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 99,\ (1:\ ),\ -0.788 + 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3001865811 + 0.8722012604i\)
\(L(\frac12)\) \(\approx\) \(0.3001865811 + 0.8722012604i\)
\(L(1)\) \(\approx\) \(0.6458144472 + 0.4027560033i\)
\(L(1)\) \(\approx\) \(0.6458144472 + 0.4027560033i\)

Euler product

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

−29.011882552390980724156865994113, −28.526417757846904720958584275470, −27.54265693024382437714575075880, −26.15135193364656524230995235028, −25.47757225691715558260421691262, −24.422596073213303641314649626908, −22.66516967393205806577333074930, −21.859517284393921613482825031357, −20.636497995388402998504690709663, −20.02743083194564966013791255271, −18.620286221031763070459974080638, −17.892199684472815219466832060958, −16.54532578433025045903790043606, −15.881807419791988960768987188297, −13.69729218957009064390802662782, −12.91096182928154362002767080586, −11.85391236223301677670221636597, −10.45428075600824413906715855432, −9.31817355986088969281097925748, −8.67595489956514538401695387138, −6.9474085177468950611458456268, −5.36754681835059490472243716532, −3.59231606769337777899477208808, −2.154779542821513101348835821746, −0.51559900842817742622569221997, 1.61778106026359062133507829442, 3.610854371745213057488989642334, 5.74823641304121540663430676083, 6.50623284356052233853508477017, 7.69769640792034510121853206651, 9.23657572132016315605515510693, 10.10999947075686121870385198502, 11.12717344530329304446105755790, 13.21757872655044870959381995975, 14.06919564015423899138911120319, 15.29871881840874988054877473552, 16.34023445752799180026175015548, 17.37948201919605242347689234419, 18.414834870954978610322738384452, 19.2375677051744941925602571778, 20.47546184688558023975180402936, 22.05303285227343688412607279289, 22.96410489240109853112003479484, 23.99577284499026091961687854367, 25.36617953220345706992072131047, 25.96937392509166786074928695335, 26.68280972792627096109949593669, 28.04847487035792942712202071647, 29.00472709966800094872934560855, 29.85972447870807997620485747163

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