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Properties

Label 1-101-101.11-r1-0-0
Degree $1$
Conductor $101$
Sign $0.0231 + 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0231 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0231 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.0231 + 0.999i$
Analytic conductor: \(10.8539\)
Root analytic conductor: \(10.8539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (1:\ ),\ 0.0231 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.595847826 + 2.656539447i\)
\(L(\frac12)\) \(\approx\) \(2.595847826 + 2.656539447i\)
\(L(1)\) \(\approx\) \(1.930752002 + 1.172881233i\)
\(L(1)\) \(\approx\) \(1.930752002 + 1.172881233i\)

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

−29.675136792885359730750973793061, −28.47790233352293974017262346819, −27.487647877633470448972387560889, −26.27403785764577803309713245153, −25.081320403079178745804377832852, −24.162785738660468257596710811354, −23.16677892053866878219528379189, −21.52943670390586148250512960496, −21.0566064582163796289325273308, −20.14259380848336794810177358646, −19.34768354569872825465067681917, −17.860727987964113894204811655528, −16.5178776312855580790667464477, −14.84927860870755978527330844043, −14.28575083511644329474933841971, −13.135157145457579874408147590137, −12.39285253988412038178932519789, −10.46062277558241888454081733923, −9.856976108994147593762530113668, −8.528045822113789119369490355716, −6.91171549576865320513750540521, −4.95467741643626435494272597690, −4.23738689206570628654876763669, −2.49908513116299501612067762217, −1.40443369629440173097956439298, 2.36136883452317664341357454104, 3.21435993853615919656290152215, 5.09121025314040824566794674774, 6.26037208079144166202271486837, 7.61485287138395127779353294069, 8.55565722246750667085041240457, 9.88788777974385049117726452588, 11.70871923467144434615318217922, 13.03820514448774289342036294672, 13.944800823467273609130603469651, 14.81319872171125746333489443216, 15.52788254889340390897039192308, 17.14359580363980431494902561169, 18.26449950774082761809635256800, 19.19407680109665456366891098671, 21.12280581859219020047706284560, 21.337973709037219693198309561851, 22.55729670340289936490657596147, 23.967882232828948395692558131001, 24.94259342962584286618314555887, 25.42370758152954131774191276534, 26.51444196933094533506191613013, 27.374856840477256597925115289244, 29.50383120655057520678888953281, 29.994406443755589162042728976510

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