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Properties

Label 1-55-55.9-r0-0-0
Degree $1$
Conductor $55$
Sign $-0.0457 + 0.998i$
Analytic cond. $0.255418$
Root an. cond. $0.255418$
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  + (0.809 + 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)6-s + (−0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s − 12-s + (0.809 + 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.809 + 0.587i)16-s + (0.809 − 0.587i)17-s + (−0.309 − 0.951i)18-s + (0.309 − 0.951i)19-s + 21-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)6-s + (−0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s − 12-s + (0.809 + 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.809 + 0.587i)16-s + (0.809 − 0.587i)17-s + (−0.309 − 0.951i)18-s + (0.309 − 0.951i)19-s + 21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $-0.0457 + 0.998i$
Analytic conductor: \(0.255418\)
Root analytic conductor: \(0.255418\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 55,\ (0:\ ),\ -0.0457 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8055629363 + 0.8432950798i\)
\(L(\frac12)\) \(\approx\) \(0.8055629363 + 0.8432950798i\)
\(L(1)\) \(\approx\) \(1.085091677 + 0.7171116045i\)
\(L(1)\) \(\approx\) \(1.085091677 + 0.7171116045i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 - T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−32.58042077598538697649499351693, −31.48344553484452551866746932007, −30.56100015802860386285787097841, −29.64371813512130743061158656417, −28.55745413538793429848730555789, −27.8324968248891632168755589845, −25.52262261774298955052847480706, −24.71757741972086719119397120725, −23.486324002974834964626632314289, −22.65311537764720482519201746181, −21.49615637791904040020666912816, −20.10562826252193757446034382676, −18.92084681700847649836044439887, −18.15222765751261926359729480020, −16.24667799512585669444531774535, −14.81997977545213891209309468625, −13.51270995892512904815236874150, −12.46346382275905890707247036511, −11.65667010478030453288367465531, −10.13750316534113046012229199940, −8.211460933569137105657218071898, −6.30764028851213591744645568769, −5.4935460026456768975921614539, −3.29308429217669922499465760383, −1.70537787624743339898265490860, 3.36572190717465049567477552300, 4.447962708189683302867701363724, 5.86155700227286617348716766344, 7.27187171566337502694350402233, 9.05238759570647553182154853137, 10.663550204464247709206950157131, 11.87987807581467680401765487733, 13.531391025858777994425594549758, 14.513408319263702293994370914346, 15.9858724938748600036336777701, 16.5191217523917225258332050118, 17.825320437179512348498429266710, 20.0577806245872561613131407525, 21.00937600685276332154743819335, 22.115627765471367028490525457418, 23.11849167186352608485148078398, 23.94510967187141155281548577038, 25.739535041181610967874262773955, 26.33072154598392247809897530799, 27.602148298900419123070881452, 29.0626586388278156239006204535, 30.20808645298552645469016913385, 31.51681263970658493507655898097, 32.55790224901415653091285221708, 33.2352960413878782894242346531

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