Properties

Degree 1
Conductor 101
Sign $0.0231 - 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(101\)
\( \varepsilon \)  =  $0.0231 - 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{101} (46, \cdot )$
Sato-Tate  :  $\mu(100)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 101,\ (1:\ ),\ 0.0231 - 0.999i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.595847826 - 2.656539447i$
$L(\frac12,\chi)$  $\approx$  $2.595847826 - 2.656539447i$
$L(\chi,1)$  $\approx$  1.930752002 - 1.172881233i
$L(1,\chi)$  $\approx$  1.930752002 - 1.172881233i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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