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Properties

Label 1-101-101.55-r1-0-0
Degree $1$
Conductor $101$
Sign $-0.144 - 0.989i$
Analytic cond. $10.8539$
Root an. cond. $10.8539$
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  + (−0.684 + 0.728i)2-s + (−0.982 − 0.187i)3-s + (−0.0627 − 0.998i)4-s + (0.728 − 0.684i)5-s + (0.809 − 0.587i)6-s + (−0.481 + 0.876i)7-s + (0.770 + 0.637i)8-s + (0.929 + 0.368i)9-s + i·10-s + (0.368 − 0.929i)11-s + (−0.125 + 0.992i)12-s + (−0.876 + 0.481i)13-s + (−0.309 − 0.951i)14-s + (−0.844 + 0.535i)15-s + (−0.992 + 0.125i)16-s + (0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (−0.684 + 0.728i)2-s + (−0.982 − 0.187i)3-s + (−0.0627 − 0.998i)4-s + (0.728 − 0.684i)5-s + (0.809 − 0.587i)6-s + (−0.481 + 0.876i)7-s + (0.770 + 0.637i)8-s + (0.929 + 0.368i)9-s + i·10-s + (0.368 − 0.929i)11-s + (−0.125 + 0.992i)12-s + (−0.876 + 0.481i)13-s + (−0.309 − 0.951i)14-s + (−0.844 + 0.535i)15-s + (−0.992 + 0.125i)16-s + (0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $-0.144 - 0.989i$
Analytic conductor: \(10.8539\)
Root analytic conductor: \(10.8539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (1:\ ),\ -0.144 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2933216444 - 0.3392613582i\)
\(L(\frac12)\) \(\approx\) \(0.2933216444 - 0.3392613582i\)
\(L(1)\) \(\approx\) \(0.5367229983 + 0.009332826616i\)
\(L(1)\) \(\approx\) \(0.5367229983 + 0.009332826616i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (-0.684 + 0.728i)T \)
3 \( 1 + (-0.982 - 0.187i)T \)
5 \( 1 + (0.728 - 0.684i)T \)
7 \( 1 + (-0.481 + 0.876i)T \)
11 \( 1 + (0.368 - 0.929i)T \)
13 \( 1 + (-0.876 + 0.481i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.992 - 0.125i)T \)
23 \( 1 + (0.425 - 0.904i)T \)
29 \( 1 + (-0.481 - 0.876i)T \)
31 \( 1 + (0.876 + 0.481i)T \)
37 \( 1 + (-0.187 - 0.982i)T \)
41 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 + (-0.968 - 0.248i)T \)
47 \( 1 + (-0.968 + 0.248i)T \)
53 \( 1 + (-0.998 - 0.0627i)T \)
59 \( 1 + (-0.125 - 0.992i)T \)
61 \( 1 + (-0.998 + 0.0627i)T \)
67 \( 1 + (0.982 - 0.187i)T \)
71 \( 1 + (-0.187 + 0.982i)T \)
73 \( 1 + (-0.904 - 0.425i)T \)
79 \( 1 + (-0.425 - 0.904i)T \)
83 \( 1 + (0.904 - 0.425i)T \)
89 \( 1 + (0.125 - 0.992i)T \)
97 \( 1 + (0.0627 + 0.998i)T \)
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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

−29.6903983219890998389384482096, −29.11098861820541132495052054639, −27.83700676157697374983125184362, −27.11291815808115722385461611335, −26.02035577898280299704231653999, −25.10949152539969270539004855018, −23.21223174547285543134734147639, −22.50812151507627575348758278543, −21.60023222754076450254919558283, −20.514696599308097054161436573489, −19.28079617023294481482492874035, −18.157397173101959396796761808144, −17.22903911558155359378166526333, −16.737806761230565190765161073214, −15.037469405954892014983302356874, −13.373449308661500058834072436376, −12.356120681414343133636212022450, −11.158451410459155688121006881499, −9.99739663719464033228947373860, −9.774180438248192097632519138524, −7.44410094769177758001522386856, −6.60448870140409579811681045775, −4.782386090504252647889569743248, −3.23155069056728306519921102490, −1.45839308289844756025382746341, 0.289441071069884612418251757350, 1.883495230396561137965902503268, 4.83622536075765917303664658403, 5.8901102931906749329725454384, 6.57959431337333662515240855283, 8.34123682243730080900426259639, 9.41973283995005777803007476515, 10.46242433411946351324067884931, 11.941161972070491717641428782948, 13.05242255100915102468805969315, 14.49626800907473027071825053504, 15.89532525819481526207725080635, 16.83293156934066879330920405636, 17.31408256590446320897136266718, 18.71923000193972350133664991452, 19.294400472513066220153748823361, 21.23943387955103487219603585179, 22.11826949595522886293865889727, 23.38659423505680430677915113156, 24.512975782584697497037899219165, 24.9042938073343231688647304133, 26.23652097225548953820439833515, 27.48801061758446232341122433795, 28.362019684337241182275572035066, 28.96623747278663312138768369039

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