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Properties

Label 1-99-99.25-r0-0-0
Degree $1$
Conductor $99$
Sign $0.0111 + 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0111 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0111 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.0111 + 0.999i$
Analytic conductor: \(0.459754\)
Root analytic conductor: \(0.459754\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 99,\ (0:\ ),\ 0.0111 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3383705635 + 0.3421586541i\)
\(L(\frac12)\) \(\approx\) \(0.3383705635 + 0.3421586541i\)
\(L(1)\) \(\approx\) \(0.5421413238 + 0.1874494048i\)
\(L(1)\) \(\approx\) \(0.5421413238 + 0.1874494048i\)

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

−30.005991147218985657097849748608, −28.434324323235136212625691760524, −27.604849893152228057223173421174, −26.72202950649130454240329422036, −25.96886827903973733576249689550, −24.686610434562015578167198684133, −23.516751078636817604284070025993, −22.55090204885744943176462996482, −20.83430975105193978182755319410, −20.12905594946311009562811615358, −19.17693739015679248385140705507, −18.17849182074494829636526195479, −16.96771577871973789143905913171, −16.02345814450995913637024774718, −15.033024671198363450739676015740, −13.34064413743798640333913989184, −11.93937269674484535110744264679, −10.94486271932952727915277098562, −10.03761980960323485016231052130, −8.460603608739471774266639478985, −7.58259039150894318913026881646, −6.486081704814932325732725336051, −4.19691046428554717619163619369, −2.88485814136192363869274916891, −0.688710905620312400505734838760, 1.78486359535053587161088633436, 3.62750843694828395923812951632, 5.59404224865048564322902139041, 6.88477439082133108542802630835, 8.299930895302125940279658015478, 8.89354041276842569541903398381, 10.44316026883232743802042251149, 11.630812375018585336041313917113, 12.47266570616066931687290804278, 14.53549063206877265234749878694, 15.586508423190078970651880780, 16.27312205905846535713366677969, 17.546752403462994876137257216856, 18.89992133137870086317936045716, 19.2437565025776471638702621232, 20.598036851793345528027205111001, 21.66986882104673682478083891439, 23.35929150603491656967713992391, 24.0796518870157794282012195258, 25.30705719896401581079038747039, 26.10369838730935656304341381871, 27.36287522162681849081905236205, 28.02570518670590097666227074242, 28.80938411406162238134476176803, 30.18135384181414628450367861978

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