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Properties

Label 1-101-101.87-r0-0-0
Degree $1$
Conductor $101$
Sign $-0.692 - 0.721i$
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  + (−0.809 − 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + 6-s + (−0.809 + 0.587i)7-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + 10-s + (0.309 − 0.951i)11-s + (−0.809 − 0.587i)12-s + (−0.809 − 0.587i)13-s + 14-s + (0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + 6-s + (−0.809 + 0.587i)7-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + 10-s + (0.309 − 0.951i)11-s + (−0.809 − 0.587i)12-s + (−0.809 − 0.587i)13-s + 14-s + (0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $-0.692 - 0.721i$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ -0.692 - 0.721i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05040887592 - 0.1182633987i\)
\(L(\frac12)\) \(\approx\) \(0.05040887592 - 0.1182633987i\)
\(L(1)\) \(\approx\) \(0.3547168239 - 0.03569410856i\)
\(L(1)\) \(\approx\) \(0.3547168239 - 0.03569410856i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−29.82883041733769901156799629570, −29.13468379858348241020805488623, −27.93153802412086257723814919880, −27.546661402047149046005728915034, −26.093086603494840857910291006595, −25.11602268502924069567699239857, −23.9279620955327366335917268587, −23.41254536906394319696084027759, −22.42889644457000327182497308103, −20.391515266187037799179961409629, −19.42306492197800988724948919277, −18.74588136622116708452861538041, −17.20866105273777733885890693516, −16.75407240269901440945629273788, −15.78196533243579733222337357659, −14.375109094948295733232432753479, −12.72283880739339785487479512858, −11.85230762704557731635609986831, −10.455986285730620794663503379799, −9.363122742118366131216907750018, −7.67714497162519422833318230582, −7.123108071861432228892592065095, −5.76324505942967824235400469568, −4.330576878019266281492041006753, −1.59557221465535422019684939460, 0.18608994542959256937362878185, 2.893850583690359733023055297060, 3.945722560887398358855523646128, 5.92923631849639493879949112089, 7.22310990046518599360589274553, 8.70409773990748725260202094922, 9.9425587782463243454384879687, 10.84605166012881855607607709420, 11.870620763368195621942799031512, 12.65112303744348601041334841854, 14.86471977273249913546692886682, 16.00095430543449810820518017607, 16.686623803697333202346387287746, 17.99590935151144486106748987801, 19.00574568603659565715585868062, 19.7617318356762587384464583923, 21.29261063180050014991170021580, 22.13011211591155870619758424866, 22.851296893076458422860685753009, 24.36289910449504782574654800645, 25.869651841701704918340478938, 26.69463977170285867536843187921, 27.61856439451557463513832700591, 28.217289387696547844424406385954, 29.531956695322105981774341029562

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