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Properties

Label 1-99-99.95-r0-0-0
Degree $1$
Conductor $99$
Sign $0.978 + 0.208i$
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  + (−0.978 − 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

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

Invariants

Degree: \(1\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.978 + 0.208i$
Analytic conductor: \(0.459754\)
Root analytic conductor: \(0.459754\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 99,\ (0:\ ),\ 0.978 + 0.208i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7767891057 + 0.08175363204i\)
\(L(\frac12)\) \(\approx\) \(0.7767891057 + 0.08175363204i\)
\(L(1)\) \(\approx\) \(0.8126138515 + 0.02219017734i\)
\(L(1)\) \(\approx\) \(0.8126138515 + 0.02219017734i\)

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

−29.71041261563042176124263297528, −28.95418084784090413698841476331, −27.83119458961534896695957619563, −26.65906933550754783094772269156, −26.05896500730550569151751006678, −24.92691999070739111834935147211, −24.08320890427569428279165831657, −22.69410910556401923159665589289, −21.32071053525420191012084314036, −20.314860177437468705997414969749, −19.368653812119919113738872324, −17.98699891780364367518112990513, −17.35375844305643092991197569511, −16.44091666037244208262904626629, −14.966634445544093853333205672383, −13.97101896543653419920733861562, −12.527497356782352977300116891770, −10.75553104819948697445400322208, −10.22874261621192480779628705016, −8.99944907982651050396914502955, −7.59000413591222670969594826885, −6.58154983387654052265277498555, −5.20660669131850584407001154655, −2.93528511734713996802268332867, −1.31042940411552294824237566523, 1.68327122912165713607046036543, 2.85469458608846763392206617844, 5.238457199307969193809266757521, 6.49441511576069705263045454095, 7.930921067521144102344136111135, 9.34939313172647231768326425448, 9.74441776290344713156860688704, 11.446363594552818243710512651815, 12.30443822184308095765117743176, 13.81885144675170186407881474194, 15.22647004941731004768192683476, 16.43026270358316249240822816293, 17.40134613192786891993336667090, 18.37548958577008692559045911371, 19.20090102300820629923401330602, 20.67395537350581958319452952761, 21.33913775496172796726807552510, 22.393059415621435034959893815578, 24.38607175297457128441123949334, 24.95701013891727919641467804182, 25.890707313839717483019436630071, 26.99710344178810341470424358070, 28.061089009164977232923373159978, 28.97925125836406838703429702260, 29.54900170717397640518903058038

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