Properties

Degree 1
Conductor 101
Sign $-0.878 - 0.478i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.187 − 0.982i)2-s + (0.425 − 0.904i)3-s + (−0.929 − 0.368i)4-s + (−0.187 − 0.982i)5-s + (−0.809 − 0.587i)6-s + (0.992 + 0.125i)7-s + (−0.535 + 0.844i)8-s + (−0.637 − 0.770i)9-s − 10-s + (0.637 + 0.770i)11-s + (−0.728 + 0.684i)12-s + (−0.992 + 0.125i)13-s + (0.309 − 0.951i)14-s + (−0.968 − 0.248i)15-s + (0.728 + 0.684i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s,χ)  = 1  + (0.187 − 0.982i)2-s + (0.425 − 0.904i)3-s + (−0.929 − 0.368i)4-s + (−0.187 − 0.982i)5-s + (−0.809 − 0.587i)6-s + (0.992 + 0.125i)7-s + (−0.535 + 0.844i)8-s + (−0.637 − 0.770i)9-s − 10-s + (0.637 + 0.770i)11-s + (−0.728 + 0.684i)12-s + (−0.992 + 0.125i)13-s + (0.309 − 0.951i)14-s + (−0.968 − 0.248i)15-s + (0.728 + 0.684i)16-s + (−0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.878 - 0.478i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.878 - 0.478i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(101\)
\( \varepsilon \)  =  $-0.878 - 0.478i$
motivic weight  =  \(0\)
character  :  $\chi_{101} (21, \cdot )$
Sato-Tate  :  $\mu(50)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 101,\ (0:\ ),\ -0.878 - 0.478i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2849386991 - 1.118071127i$
$L(\frac12,\chi)$  $\approx$  $0.2849386991 - 1.118071127i$
$L(\chi,1)$  $\approx$  0.7184798533 - 0.9274126163i
$L(1,\chi)$  $\approx$  0.7184798533 - 0.9274126163i

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

−30.96069001377246754753753838959, −29.460735894477357568042327180625, −27.51427345499947627433844769251, −27.00768937374226259650113480934, −26.47934634386566131916590694023, −25.109679753346131328695541065465, −24.36937001128742110317768739691, −22.8900433536140550074803232514, −22.12320005176369344414841210919, −21.30870219320560426699217821225, −19.82832951785892858642956125978, −18.53351761506886292610078127982, −17.39169226648007776988599885724, −16.33417192157023619886995374450, −15.17066296522452360391472173891, −14.454177309581895999228970131108, −13.82000950093026218482954581578, −11.73427211333315343832790235950, −10.52751777258695654146223954971, −9.19998989044767482860115322832, −8.07390653577983595451433992569, −6.96422890880826976955640075491, −5.37203434669652996033592785146, −4.240501493374071455986226746659, −2.95350357458542581999238843450, 1.25246122002676641808098371389, 2.32170489666040503982011461160, 4.19852644415880012463418989771, 5.30299286344406646219851926319, 7.32832777296406674262613888002, 8.634396208866087940727781980532, 9.413014215590783942194157584996, 11.31455195231088910188914803036, 12.17158990868276905999736618130, 12.9659586459725329063304492139, 14.14063888039878118524157797373, 15.09641718304878648261121064356, 17.37249325172139578496658874569, 17.76004675580060346482503890592, 19.34922202636197910782456805046, 19.94295465539455510995057048172, 20.79574066027766589950813390113, 21.96578610288045637388874727545, 23.38868858933810130202168692853, 24.214149819011592400201628758142, 24.967731066880605310830152318299, 26.63872973167742997267322511696, 27.68052560605568025667754421538, 28.61971685242908710870111099567, 29.47529700609699970018229052856

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