Properties

Degree 1
Conductor $ 5 \cdot 11 $
Sign $-0.0457 + 0.998i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(55\)    =    \(5 \cdot 11\)
\( \varepsilon \)  =  $-0.0457 + 0.998i$
motivic weight  =  \(0\)
character  :  $\chi_{55} (9, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 55,\ (0:\ ),\ -0.0457 + 0.998i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8055629363 + 0.8432950798i$
$L(\frac12,\chi)$  $\approx$  $0.8055629363 + 0.8432950798i$
$L(\chi,1)$  $\approx$  1.085091677 + 0.7171116045i
$L(1,\chi)$  $\approx$  1.085091677 + 0.7171116045i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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