Properties

Degree 1
Conductor 53
Sign $-0.995 + 0.0974i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.663 + 0.748i)2-s + (−0.239 + 0.970i)3-s + (−0.120 + 0.992i)4-s + (−0.464 + 0.885i)5-s + (−0.885 + 0.464i)6-s + (0.748 − 0.663i)7-s + (−0.822 + 0.568i)8-s + (−0.885 − 0.464i)9-s + (−0.970 + 0.239i)10-s + (0.354 − 0.935i)11-s + (−0.935 − 0.354i)12-s + (0.120 + 0.992i)13-s + (0.992 + 0.120i)14-s + (−0.748 − 0.663i)15-s + (−0.970 − 0.239i)16-s + (−0.568 + 0.822i)17-s + ⋯
L(s,χ)  = 1  + (0.663 + 0.748i)2-s + (−0.239 + 0.970i)3-s + (−0.120 + 0.992i)4-s + (−0.464 + 0.885i)5-s + (−0.885 + 0.464i)6-s + (0.748 − 0.663i)7-s + (−0.822 + 0.568i)8-s + (−0.885 − 0.464i)9-s + (−0.970 + 0.239i)10-s + (0.354 − 0.935i)11-s + (−0.935 − 0.354i)12-s + (0.120 + 0.992i)13-s + (0.992 + 0.120i)14-s + (−0.748 − 0.663i)15-s + (−0.970 − 0.239i)16-s + (−0.568 + 0.822i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.995 + 0.0974i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.995 + 0.0974i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(53\)
\( \varepsilon \)  =  $-0.995 + 0.0974i$
motivic weight  =  \(0\)
character  :  $\chi_{53} (22, \cdot )$
Sato-Tate  :  $\mu(52)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 53,\ (1:\ ),\ -0.995 + 0.0974i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.08398917263 + 1.719204211i$
$L(\frac12,\chi)$  $\approx$  $0.08398917263 + 1.719204211i$
$L(\chi,1)$  $\approx$  0.7636303680 + 1.056330321i
$L(1,\chi)$  $\approx$  0.7636303680 + 1.056330321i

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

−32.20157594113143458818048563920, −30.975463085918037768877784444784, −30.61729233978230793635301284780, −29.13009379400584448444971455534, −28.258689914638458338301721714797, −27.45164865526024728068947249748, −24.86544161525855734010834848042, −24.53583600587501213041166262272, −23.20939821448646245158118130118, −22.34408707274663145520569208319, −20.57415730380003167081245176208, −20.014140288672390067810234299341, −18.55467250291734211393852095592, −17.59757835975270394714567681303, −15.66555050985921680554894817643, −14.31519703678846779225643697951, −12.89632711129503542316847951576, −12.15933857647147128966674205349, −11.21075126475371884841185099858, −9.18530796347847624574783539607, −7.66762920784399821664277060344, −5.74698939540895657198993274005, −4.61053333125827324229436883048, −2.38760518192168568018038571410, −0.864610723824046613682768647261, 3.40635317811632050353762480088, 4.38053539485920063793822484232, 5.98254754197649779039224544427, 7.40151676514073096139655319941, 8.91179307287261369257329822394, 10.90862663976469880714560503282, 11.64742895634198169823647467495, 13.89171184895624092707195909783, 14.5966197405253935896467027831, 15.79501175256772159134134108439, 16.7904996125700885580298859263, 18.00020672654085707194557265746, 19.90927513668940826388676654425, 21.45223148653821591938137100452, 22.02129775348037174107505974606, 23.33497747957582465271525221518, 24.08234047955186716642472486932, 25.91527964570394885713632437166, 26.72493440534073150926923536884, 27.373711523720840562324410339007, 29.29244299298191526091003186809, 30.63108818806059525058006305419, 31.41695072182452242727861219832, 32.82217337885696460462020983877, 33.491245273368252014802179011856

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