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Properties

Label 1-101-101.21-r0-0-0
Degree $1$
Conductor $101$
Sign $-0.878 - 0.478i$
Analytic cond. $0.469042$
Root an. cond. $0.469042$
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  + (0.187 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $-0.878 - 0.478i$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ -0.878 - 0.478i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2849386991 - 1.118071127i\)
\(L(\frac12)\) \(\approx\) \(0.2849386991 - 1.118071127i\)
\(L(1)\) \(\approx\) \(0.7184798533 - 0.9274126163i\)
\(L(1)\) \(\approx\) \(0.7184798533 - 0.9274126163i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (0.187 - 0.982i)T \)
3 \( 1 + (0.425 - 0.904i)T \)
5 \( 1 + (-0.187 - 0.982i)T \)
7 \( 1 + (0.992 + 0.125i)T \)
11 \( 1 + (0.637 + 0.770i)T \)
13 \( 1 + (-0.992 + 0.125i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.728 - 0.684i)T \)
23 \( 1 + (0.876 + 0.481i)T \)
29 \( 1 + (0.992 - 0.125i)T \)
31 \( 1 + (-0.992 - 0.125i)T \)
37 \( 1 + (-0.425 - 0.904i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + (0.0627 - 0.998i)T \)
47 \( 1 + (0.0627 + 0.998i)T \)
53 \( 1 + (0.929 - 0.368i)T \)
59 \( 1 + (-0.728 - 0.684i)T \)
61 \( 1 + (0.929 + 0.368i)T \)
67 \( 1 + (0.425 + 0.904i)T \)
71 \( 1 + (-0.425 + 0.904i)T \)
73 \( 1 + (-0.876 - 0.481i)T \)
79 \( 1 + (0.876 - 0.481i)T \)
83 \( 1 + (-0.876 + 0.481i)T \)
89 \( 1 + (-0.728 + 0.684i)T \)
97 \( 1 + (-0.929 - 0.368i)T \)
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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

−30.96069001377246754753753838959, −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