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Properties

Label 1-55-55.49-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: $\chi_{55} (49, \cdot )$
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

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

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