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Properties

Label 1-23-23.20-r1-0-0
Degree $1$
Conductor $23$
Sign $0.0137 + 0.999i$
Analytic cond. $2.47169$
Root an. cond. $2.47169$
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  + (−0.959 + 0.281i)2-s + (−0.654 − 0.755i)3-s + (0.841 − 0.540i)4-s + (0.142 + 0.989i)5-s + (0.841 + 0.540i)6-s + (−0.415 + 0.909i)7-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (−0.415 − 0.909i)10-s + (0.959 + 0.281i)11-s + (−0.959 − 0.281i)12-s + (0.415 + 0.909i)13-s + (0.142 − 0.989i)14-s + (0.654 − 0.755i)15-s + (0.415 − 0.909i)16-s + (−0.841 − 0.540i)17-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)2-s + (−0.654 − 0.755i)3-s + (0.841 − 0.540i)4-s + (0.142 + 0.989i)5-s + (0.841 + 0.540i)6-s + (−0.415 + 0.909i)7-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (−0.415 − 0.909i)10-s + (0.959 + 0.281i)11-s + (−0.959 − 0.281i)12-s + (0.415 + 0.909i)13-s + (0.142 − 0.989i)14-s + (0.654 − 0.755i)15-s + (0.415 − 0.909i)16-s + (−0.841 − 0.540i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0137 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0137 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(23\)
Sign: $0.0137 + 0.999i$
Analytic conductor: \(2.47169\)
Root analytic conductor: \(2.47169\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 23,\ (1:\ ),\ 0.0137 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4090902206 + 0.4147358290i\)
\(L(\frac12)\) \(\approx\) \(0.4090902206 + 0.4147358290i\)
\(L(1)\) \(\approx\) \(0.5441219459 + 0.1794606337i\)
\(L(1)\) \(\approx\) \(0.5441219459 + 0.1794606337i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.959 + 0.281i)T \)
3 \( 1 + (-0.654 - 0.755i)T \)
5 \( 1 + (0.142 + 0.989i)T \)
7 \( 1 + (-0.415 + 0.909i)T \)
11 \( 1 + (0.959 + 0.281i)T \)
13 \( 1 + (0.415 + 0.909i)T \)
17 \( 1 + (-0.841 - 0.540i)T \)
19 \( 1 + (-0.841 + 0.540i)T \)
29 \( 1 + (0.841 + 0.540i)T \)
31 \( 1 + (-0.654 + 0.755i)T \)
37 \( 1 + (0.142 - 0.989i)T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.415 + 0.909i)T \)
59 \( 1 + (0.415 + 0.909i)T \)
61 \( 1 + (0.654 - 0.755i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (0.841 - 0.540i)T \)
79 \( 1 + (-0.415 - 0.909i)T \)
83 \( 1 + (0.142 - 0.989i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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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

−38.40384992819929295669797533152, −37.13619250516231703951003325528, −35.76911920333364067430423616265, −34.868996286180610125642262501875, −33.19968294615686015783989700963, −32.42523067921103371798964536200, −30.07233559476544474075228030171, −28.92600875706850260462480424141, −27.878447462380804413873326033220, −26.94110073083399489155801008077, −25.49833306469668824435328141866, −23.8657460661749322730612970427, −22.0704727617995745792937880225, −20.654144817223323611838144843912, −19.712265520366555550562020374770, −17.47548545794902528393727030749, −16.83459567762552641116418487961, −15.58873603349660210206814123447, −12.88108573964420989398239487829, −11.255781240002591564957671265331, −9.95914322759170426351235099824, −8.63475817596672732671615466727, −6.35513516932204405990366033252, −4.018004610345029742647779218478, −0.68927167805141068339308489506, 2.1034098104138519049761141064, 6.13379978100420175032299046450, 6.98738309119494467691872104202, 9.014338616699156319356322119226, 10.8429973908163320662908692284, 12.05890842432505160889361686841, 14.355339081803838916594847975054, 16.031078844188307854525406654365, 17.554314776131283231811918154921, 18.58587426121303333333273805663, 19.45068481377651720179245271194, 21.85331309735800151087057901110, 23.24163342298545838720588965730, 24.825141253384908541362326749065, 25.71346992981429228080468869533, 27.339411431460373633006559649210, 28.62776828532853142100634256183, 29.58484743325249939823549778357, 30.90328098998247631688314238088, 33.23864795570691626396469131912, 34.243681912018048976041859119818, 35.19632877423891364690366041777, 36.14060008901563562887897633342, 37.84362650788025549533657533426, 38.53633776886653723208406945272

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