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Properties

Label 1-99-99.92-r1-0-0
Degree $1$
Conductor $99$
Sign $-0.773 - 0.634i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.773 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.773 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.773 - 0.634i$
Analytic conductor: \(10.6390\)
Root analytic conductor: \(10.6390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 99,\ (1:\ ),\ -0.773 - 0.634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04711430083 - 0.1317222320i\)
\(L(\frac12)\) \(\approx\) \(0.04711430083 - 0.1317222320i\)
\(L(1)\) \(\approx\) \(0.5510021773 + 0.07685811219i\)
\(L(1)\) \(\approx\) \(0.5510021773 + 0.07685811219i\)

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

−29.89550785070827045513682664116, −29.368473348443343206581291501672, −27.76320239688543929459062624134, −27.18261715968796291699165636311, −26.44557368607166753604098935728, −25.188005302266789008814408601061, −23.81709538955687321363487600018, −22.63914529042576048891285520374, −21.61703855775187259101041952557, −20.5080117114924448408331206684, −19.54728397954760232184225485968, −18.59182384661578896787903611943, −17.618890331414595191691104305447, −16.558525606142656799466219109034, −15.082922889886664481748745109612, −13.94561810466427535156900317620, −12.37367334253100128114802729017, −11.34171335323379526908856430146, −10.61198367343460983375157589250, −9.233582670552874641004349546167, −7.811235233892162808622232640164, −7.10944997515951172792939317753, −4.70166371630792577703803144829, −3.39046271360856303476770079830, −1.896017359712416387429134469129, 0.07501840823422884555696042342, 1.85366239720955667533020456140, 4.45374590582240573851287697009, 5.42791151125268325675744956113, 7.08576192991859917669632203255, 8.251897145308415396006153026664, 8.9350990363317532457354361662, 10.5056585986559652955100002863, 11.7121650280163838353546239690, 13.07892661484616064010549372649, 14.825336613445225247794994349529, 15.23830937410555415014893509622, 16.727358918599415248945335820690, 17.35524024966911884412875451928, 18.67355370683225434990510960885, 19.64927011197909473143015278760, 20.65080357956359041805653873827, 22.11928455420683596527303231540, 23.608182348525984743429907080645, 24.221741737249553294897428138401, 25.014967172047719119072881422777, 26.352759892891980814643026164088, 27.35396115591132124147658160572, 27.985214642019626072670250655731, 28.921781473251425086795469808317

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