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Properties

Label 1-101-101.99-r1-0-0
Degree $1$
Conductor $101$
Sign $-0.433 - 0.901i$
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  + (−0.998 − 0.0627i)2-s + (0.368 + 0.929i)3-s + (0.992 + 0.125i)4-s + (0.0627 + 0.998i)5-s + (−0.309 − 0.951i)6-s + (−0.844 − 0.535i)7-s + (−0.982 − 0.187i)8-s + (−0.728 + 0.684i)9-s − i·10-s + (−0.684 − 0.728i)11-s + (0.248 + 0.968i)12-s + (−0.535 − 0.844i)13-s + (0.809 + 0.587i)14-s + (−0.904 + 0.425i)15-s + (0.968 + 0.248i)16-s + (−0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0627i)2-s + (0.368 + 0.929i)3-s + (0.992 + 0.125i)4-s + (0.0627 + 0.998i)5-s + (−0.309 − 0.951i)6-s + (−0.844 − 0.535i)7-s + (−0.982 − 0.187i)8-s + (−0.728 + 0.684i)9-s − i·10-s + (−0.684 − 0.728i)11-s + (0.248 + 0.968i)12-s + (−0.535 − 0.844i)13-s + (0.809 + 0.587i)14-s + (−0.904 + 0.425i)15-s + (0.968 + 0.248i)16-s + (−0.309 + 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $-0.433 - 0.901i$
Analytic conductor: \(10.8539\)
Root analytic conductor: \(10.8539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (1:\ ),\ -0.433 - 0.901i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02976226707 - 0.04735648999i\)
\(L(\frac12)\) \(\approx\) \(0.02976226707 - 0.04735648999i\)
\(L(1)\) \(\approx\) \(0.5128599726 + 0.1660481759i\)
\(L(1)\) \(\approx\) \(0.5128599726 + 0.1660481759i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (-0.998 - 0.0627i)T \)
3 \( 1 + (0.368 + 0.929i)T \)
5 \( 1 + (0.0627 + 0.998i)T \)
7 \( 1 + (-0.844 - 0.535i)T \)
11 \( 1 + (-0.684 - 0.728i)T \)
13 \( 1 + (-0.535 - 0.844i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (0.968 - 0.248i)T \)
23 \( 1 + (0.637 - 0.770i)T \)
29 \( 1 + (-0.844 + 0.535i)T \)
31 \( 1 + (0.535 - 0.844i)T \)
37 \( 1 + (-0.929 - 0.368i)T \)
41 \( 1 + (0.951 - 0.309i)T \)
43 \( 1 + (-0.876 + 0.481i)T \)
47 \( 1 + (-0.876 - 0.481i)T \)
53 \( 1 + (-0.125 - 0.992i)T \)
59 \( 1 + (0.248 - 0.968i)T \)
61 \( 1 + (-0.125 + 0.992i)T \)
67 \( 1 + (-0.368 + 0.929i)T \)
71 \( 1 + (-0.929 + 0.368i)T \)
73 \( 1 + (0.770 + 0.637i)T \)
79 \( 1 + (-0.637 - 0.770i)T \)
83 \( 1 + (-0.770 + 0.637i)T \)
89 \( 1 + (-0.248 - 0.968i)T \)
97 \( 1 + (-0.992 - 0.125i)T \)
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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

−29.457809789425846748403915209303, −28.90483613991479274792995387891, −28.22710228012920808456001680314, −26.720143127763634300348929363795, −25.7234182543164126211714578775, −24.90709958184134206819520465973, −24.238665841984363922248826133310, −23.00109471039966883472080491703, −21.13582953706959912366795779620, −20.19091494666472701209354684001, −19.3574230917691859981726352029, −18.41533485690553380004014154544, −17.42817364132377091705090303443, −16.29257254283663914372828184483, −15.30699199393182251753190448997, −13.62537518853867080612508350712, −12.43741450146941167989799748698, −11.726829199350065676934069048424, −9.634246799494592962341442184889, −9.10416442568810692663824361058, −7.77304607905530778163518232307, −6.81875051344386640308698507835, −5.37321319259998892246808427316, −2.85324215287292407784607896527, −1.5865483682690610500451468593, 0.0305947437328368798455097751, 2.685596476844001355464978042495, 3.47596020896337376495649795620, 5.74497373307429545647453403900, 7.14895667407082208902287439340, 8.32024532545415568719376775698, 9.72228641425863277964452099099, 10.42346758309968448515678282146, 11.19434711354431428250688074023, 13.127633988561531622766716389458, 14.645103790818209398876161401102, 15.57320022218732177101491852761, 16.48615349094113524114347898296, 17.6104955671196960597967109000, 18.92484013810567150844367883235, 19.69195072496111125656971828272, 20.72749602385034378266848735690, 21.89483502613920291536186508832, 22.773821378510621483346945448471, 24.46479073269083105698672065090, 25.77800539459833076459192971639, 26.42217310133175683468430935450, 26.86286753148259686500212031767, 28.13557485909380438227563504106, 29.20405074652954347446513242334

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