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Properties

Label 1-99-99.32-r0-0-0
Degree $1$
Conductor $99$
Sign $0.342 + 0.939i$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + 8-s − 10-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 17-s − 19-s + (0.5 − 0.866i)20-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 26-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + 8-s − 10-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 17-s − 19-s + (0.5 − 0.866i)20-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(0.459754\)
Root analytic conductor: \(0.459754\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 99,\ (0:\ ),\ 0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7114354947 + 0.4981524963i\)
\(L(\frac12)\) \(\approx\) \(0.7114354947 + 0.4981524963i\)
\(L(1)\) \(\approx\) \(0.8148834573 + 0.3814039884i\)
\(L(1)\) \(\approx\) \(0.8148834573 + 0.3814039884i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−29.73311026503316888645237521567, −28.511004547828461118238724135768, −28.01032921274703323478547818296, −27.075255588305057313792802637180, −25.50520659460191880615914495354, −25.00060652514239518004593500373, −23.45096898482929298350645799793, −22.09924838819708722341315960882, −21.09041401137618106342581400656, −20.56577672808417917119819381912, −19.203996613214099341653821731530, −18.17851690481460124758534684615, −17.27248211087698777830560527031, −16.198978451997647766003481949269, −14.60809393450394363624237537361, −13.06437268624552902308022436243, −12.4267763582876905253253656489, −11.16098236568774588777796745337, −9.91486224309373574533049786584, −8.77847635966066878569064458651, −8.0148494082748976407028715759, −5.79390385693861197079072681815, −4.515034346670144754860877074409, −2.739500164748515075765218909709, −1.33431798093472058628095084409, 1.66111610522805616379320785197, 3.93261266859087398981281365932, 5.55127006670665499231414022906, 6.78128709607474239997558543134, 7.65792373074251517264811384482, 9.131013638035364790050205070407, 10.32101910268802337168169348222, 11.18824795574279542473790244499, 13.372944431725085989610424991478, 14.259151237161110063542761727134, 15.05452437546675296996801965152, 16.57764293398832160408901157153, 17.31338615090604738905826128760, 18.43922061588955605721425008512, 19.22351385890523347352222963149, 20.741775419317828356909066069162, 21.980887244633145961593449592265, 23.324103241242882074824831931637, 23.80858452583650324793712186753, 25.375164895319315797756007839848, 25.94019375437309691523494929062, 26.951541774435975137458661586228, 27.791300876198571010103663694826, 29.16827450804960318898788345848, 30.08423753099817190459794218465

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