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Properties

Label 1-4020-4020.23-r1-0-0
Degree $1$
Conductor $4020$
Sign $0.0184 + 0.999i$
Analytic cond. $432.008$
Root an. cond. $432.008$
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  + (0.998 + 0.0475i)7-s + (0.981 + 0.189i)11-s + (−0.371 + 0.928i)13-s + (−0.814 + 0.580i)17-s + (0.0475 + 0.998i)19-s + (0.690 + 0.723i)23-s + (−0.5 − 0.866i)29-s + (−0.928 + 0.371i)31-s + (0.866 + 0.5i)37-s + (0.995 − 0.0950i)41-s + (−0.909 − 0.415i)43-s + (0.971 − 0.235i)47-s + (0.995 + 0.0950i)49-s + (0.909 − 0.415i)53-s + (0.142 − 0.989i)59-s + ⋯
L(s)  = 1  + (0.998 + 0.0475i)7-s + (0.981 + 0.189i)11-s + (−0.371 + 0.928i)13-s + (−0.814 + 0.580i)17-s + (0.0475 + 0.998i)19-s + (0.690 + 0.723i)23-s + (−0.5 − 0.866i)29-s + (−0.928 + 0.371i)31-s + (0.866 + 0.5i)37-s + (0.995 − 0.0950i)41-s + (−0.909 − 0.415i)43-s + (0.971 − 0.235i)47-s + (0.995 + 0.0950i)49-s + (0.909 − 0.415i)53-s + (0.142 − 0.989i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0184 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0184 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.0184 + 0.999i$
Analytic conductor: \(432.008\)
Root analytic conductor: \(432.008\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4020,\ (1:\ ),\ 0.0184 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.950797780 + 1.987141285i\)
\(L(\frac12)\) \(\approx\) \(1.950797780 + 1.987141285i\)
\(L(1)\) \(\approx\) \(1.248926045 + 0.2635593596i\)
\(L(1)\) \(\approx\) \(1.248926045 + 0.2635593596i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
67 \( 1 \)
good7 \( 1 + (0.998 + 0.0475i)T \)
11 \( 1 + (0.981 + 0.189i)T \)
13 \( 1 + (-0.371 + 0.928i)T \)
17 \( 1 + (-0.814 + 0.580i)T \)
19 \( 1 + (0.0475 + 0.998i)T \)
23 \( 1 + (0.690 + 0.723i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.928 + 0.371i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (0.995 - 0.0950i)T \)
43 \( 1 + (-0.909 - 0.415i)T \)
47 \( 1 + (0.971 - 0.235i)T \)
53 \( 1 + (0.909 - 0.415i)T \)
59 \( 1 + (0.142 - 0.989i)T \)
61 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (0.580 - 0.814i)T \)
73 \( 1 + (0.189 + 0.981i)T \)
79 \( 1 + (-0.786 + 0.618i)T \)
83 \( 1 + (0.945 - 0.327i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−18.016995807241755808556615775107, −17.554412042216895408689513497006, −16.83030551770988553437796683350, −16.1968659124780937745875524060, −15.146953089874340301416195456623, −14.84174445111252467748823714100, −14.15048486667278980524709529466, −13.323831999720663524458890784252, −12.713974202891762079055325861918, −11.85905136541047117336342991013, −11.08704658145741154388393029841, −10.875892680185723114726957469683, −9.74292771975511895219443545995, −8.94333341861281720639373346963, −8.58134599056342656416959814591, −7.4249113985009660273536969808, −7.14238952687259225147922092998, −6.09215842715540944874358998360, −5.29068641897113093871431585532, −4.62573371650157756384607637352, −3.94011351156042387041357042269, −2.85754420830248112559803569110, −2.20390969074098994714912181369, −1.12002070439264114694700852967, −0.46572572449332198685821162537, 0.940448722804371756679747273051, 1.795141670231289412985115943315, 2.25508399383815313892036416775, 3.70499252502081030294493468110, 4.10878535642907713420064865298, 4.95712616804665261853100660402, 5.7396588066178090947128329736, 6.591919341084438524919192697739, 7.26060203168256650579607730775, 8.02300546203335383505385290307, 8.80867765915000347222915979119, 9.36700021072834053319481254615, 10.15845880614408635786198929358, 11.16290466054680474986657323018, 11.49077150895035549277690308865, 12.195237902757090196235183455217, 13.009707252979860144745031771195, 13.83842700711171860520766222020, 14.496909571294107416654027290127, 14.90489597463067864046072747057, 15.66128448699659523859207532744, 16.73957407192787437160843017226, 17.02476219376890030659478236182, 17.71869513852549596248864285761, 18.49242837804140746697937736219

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