Properties

Label 1-99-99.68-r0-0-0
Degree $1$
Conductor $99$
Sign $0.540 - 0.841i$
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  + (0.913 − 0.406i)2-s + (0.669 − 0.743i)4-s + (−0.913 − 0.406i)5-s + (0.978 + 0.207i)7-s + (0.309 − 0.951i)8-s − 10-s + (0.104 − 0.994i)13-s + (0.978 − 0.207i)14-s + (−0.104 − 0.994i)16-s + (−0.809 + 0.587i)17-s + (−0.309 + 0.951i)19-s + (−0.913 + 0.406i)20-s + (0.5 + 0.866i)23-s + (0.669 + 0.743i)25-s + (−0.309 − 0.951i)26-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)2-s + (0.669 − 0.743i)4-s + (−0.913 − 0.406i)5-s + (0.978 + 0.207i)7-s + (0.309 − 0.951i)8-s − 10-s + (0.104 − 0.994i)13-s + (0.978 − 0.207i)14-s + (−0.104 − 0.994i)16-s + (−0.809 + 0.587i)17-s + (−0.309 + 0.951i)19-s + (−0.913 + 0.406i)20-s + (0.5 + 0.866i)23-s + (0.669 + 0.743i)25-s + (−0.309 − 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.540 - 0.841i$
Analytic conductor: \(0.459754\)
Root analytic conductor: \(0.459754\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 99,\ (0:\ ),\ 0.540 - 0.841i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.372457454 - 0.7492157435i\)
\(L(\frac12)\) \(\approx\) \(1.372457454 - 0.7492157435i\)
\(L(1)\) \(\approx\) \(1.457986860 - 0.5216298586i\)
\(L(1)\) \(\approx\) \(1.457986860 - 0.5216298586i\)

Euler product

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

−30.62282474847942097137326176118, −29.464382126926348221192196262731, −28.05179031306223201531852114497, −26.77350928563547646984235799819, −26.11572373007363037647809127593, −24.5037747544452321890860069264, −23.96391704254669218061351230372, −22.96663612869947363872397739311, −21.96860606802351079176118665047, −20.854412401062811959847914634272, −19.88845302414345893070070608711, −18.47991044468459046599317412734, −17.14402193017599610913068535731, −16.01368626582483396812993727216, −14.97785534704634143942205007448, −14.20540724993512080584081555526, −12.9294980088268836953411690513, −11.45117071066658450419997897316, −11.128839744711962472617894656939, −8.753801995082465880514583906634, −7.52787934280713278660858022514, −6.640263354927876887378783781291, −4.85157422590167755604411061695, −4.02234633910117394629498728522, −2.37787210554348468543368013898, 1.59490774131691463076170592320, 3.408227302811940080530499094653, 4.59079751163166935581379553378, 5.67684582116985077009282715948, 7.406444391583994110570532678488, 8.59512809477493975985816476229, 10.47954463339311538498939215197, 11.44497718329914859056512889856, 12.38109977541732088964199034489, 13.454833373716247840764480379907, 14.95132392372732685411106376744, 15.39802690997146770071238257172, 16.868944900954785785154637348208, 18.3956993166240360298629517191, 19.635655395829960302985862689341, 20.45157172258619997024616844876, 21.378805395672378433981060430533, 22.59856114656484891556614983775, 23.55763942263706060518242233487, 24.33888334724459461470619774543, 25.28669370243764867200265106219, 27.16116746966147579971667598617, 27.80138593269626121478928343882, 28.877954516797474408002814322575, 30.146129907645598081256244202713

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