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Properties

Label 1-53-53.22-r1-0-0
Degree $1$
Conductor $53$
Sign $-0.995 + 0.0974i$
Analytic cond. $5.69564$
Root an. cond. $5.69564$
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  + (0.663 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $-0.995 + 0.0974i$
Analytic conductor: \(5.69564\)
Root analytic conductor: \(5.69564\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (1:\ ),\ -0.995 + 0.0974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08398917263 + 1.719204211i\)
\(L(\frac12)\) \(\approx\) \(0.08398917263 + 1.719204211i\)
\(L(1)\) \(\approx\) \(0.7636303680 + 1.056330321i\)
\(L(1)\) \(\approx\) \(0.7636303680 + 1.056330321i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad53 \( 1 \)
good2 \( 1 + (0.663 + 0.748i)T \)
3 \( 1 + (-0.239 + 0.970i)T \)
5 \( 1 + (-0.464 + 0.885i)T \)
7 \( 1 + (0.748 - 0.663i)T \)
11 \( 1 + (0.354 - 0.935i)T \)
13 \( 1 + (0.120 + 0.992i)T \)
17 \( 1 + (-0.568 + 0.822i)T \)
19 \( 1 + (0.992 - 0.120i)T \)
23 \( 1 + iT \)
29 \( 1 + (0.354 + 0.935i)T \)
31 \( 1 + (-0.935 + 0.354i)T \)
37 \( 1 + (0.970 + 0.239i)T \)
41 \( 1 + (0.935 + 0.354i)T \)
43 \( 1 + (0.970 - 0.239i)T \)
47 \( 1 + (0.885 - 0.464i)T \)
59 \( 1 + (-0.885 + 0.464i)T \)
61 \( 1 + (-0.822 + 0.568i)T \)
67 \( 1 + (0.992 + 0.120i)T \)
71 \( 1 + (-0.239 - 0.970i)T \)
73 \( 1 + (-0.822 - 0.568i)T \)
79 \( 1 + (-0.663 + 0.748i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.568 - 0.822i)T \)
97 \( 1 + (0.885 + 0.464i)T \)
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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

−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