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Properties

Label 1-99-99.41-r0-0-0
Degree $1$
Conductor $99$
Sign $-0.615 + 0.788i$
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  + (0.669 + 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

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

Invariants

Degree: \(1\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.615 + 0.788i$
Analytic conductor: \(0.459754\)
Root analytic conductor: \(0.459754\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 99,\ (0:\ ),\ -0.615 + 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4898288350 + 1.003941666i\)
\(L(\frac12)\) \(\approx\) \(0.4898288350 + 1.003941666i\)
\(L(1)\) \(\approx\) \(0.8951499942 + 0.7549105015i\)
\(L(1)\) \(\approx\) \(0.8951499942 + 0.7549105015i\)

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

−29.507576219833981587368296118823, −28.800820948343724387488698442999, −27.819566864781486567045906290842, −26.81258764382110023162621429293, −25.27325702845854947765461517099, −24.13467367359769079853858973622, −23.12167758219989719435518925511, −22.54806040655582100364642676365, −20.9687110386731942077013806558, −20.31430464206063422593000520296, −19.334991396428240323612758470466, −18.35120959172219137049002070522, −16.41963736464034380487843499523, −15.792036704174685518555744717329, −14.23792341137872847889778174076, −13.175815177105829513390518290844, −12.29252509045439489083162132873, −11.223036395091245509029632247371, −9.91955637089759072668679204305, −8.76774218743570895513635285564, −6.97800799290622928017844163486, −5.50356260432876560961568056291, −4.165199817724815838314681026023, −3.1338716809577441614531339084, −1.01405192946047354218048425028, 2.98754885143829296256813423913, 3.85862498465423362260759663542, 5.65399850340929376851119127377, 6.67848907760289786183782860652, 7.79285245258613315449835803567, 9.10751999010381992592008787730, 10.860948794258228686291924251320, 12.09603925692418101566009637166, 13.15666231771682981015358147191, 14.32827400716910882689352829755, 15.56624898055811833090337882188, 15.9577918670037813891985896367, 17.49733562098809440088558264046, 18.66108255115476196342711924539, 19.77271042478908477122871852882, 21.3044914565016576771934451706, 22.29348180432868045671015220685, 23.077357781854729484085935016291, 23.904054042672889622860068313690, 25.345391384917036875509573252918, 25.96842590645936856406193556554, 26.949768784762140239192744791276, 28.23671461686415338675797351727, 29.649186087661453024516862939485, 30.67603908979738174644078400068

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