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Properties

Label 1-4020-4020.119-r1-0-0
Degree $1$
Conductor $4020$
Sign $0.973 - 0.227i$
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  + (−0.415 + 0.909i)7-s + (0.142 − 0.989i)11-s + (−0.959 − 0.281i)13-s + (−0.654 + 0.755i)17-s + (−0.415 − 0.909i)19-s + (−0.841 + 0.540i)23-s − 29-s + (−0.959 + 0.281i)31-s − 37-s + (−0.654 + 0.755i)41-s + (0.654 − 0.755i)43-s + (−0.841 + 0.540i)47-s + (−0.654 − 0.755i)49-s + (0.654 + 0.755i)53-s + (−0.959 + 0.281i)59-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)7-s + (0.142 − 0.989i)11-s + (−0.959 − 0.281i)13-s + (−0.654 + 0.755i)17-s + (−0.415 − 0.909i)19-s + (−0.841 + 0.540i)23-s − 29-s + (−0.959 + 0.281i)31-s − 37-s + (−0.654 + 0.755i)41-s + (0.654 − 0.755i)43-s + (−0.841 + 0.540i)47-s + (−0.654 − 0.755i)49-s + (0.654 + 0.755i)53-s + (−0.959 + 0.281i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.973 - 0.227i$
Analytic conductor: \(432.008\)
Root analytic conductor: \(432.008\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4020,\ (1:\ ),\ 0.973 - 0.227i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4249491123 - 0.04891200064i\)
\(L(\frac12)\) \(\approx\) \(0.4249491123 - 0.04891200064i\)
\(L(1)\) \(\approx\) \(0.7026232980 + 0.05415663131i\)
\(L(1)\) \(\approx\) \(0.7026232980 + 0.05415663131i\)

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 \( 1 + (-0.415 + 0.909i)T \)
11 \( 1 + (0.142 - 0.989i)T \)
13 \( 1 + (-0.959 - 0.281i)T \)
17 \( 1 + (-0.654 + 0.755i)T \)
19 \( 1 + (-0.415 - 0.909i)T \)
23 \( 1 + (-0.841 + 0.540i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.959 + 0.281i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (-0.841 + 0.540i)T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (-0.959 + 0.281i)T \)
61 \( 1 + (0.142 + 0.989i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.142 + 0.989i)T \)
79 \( 1 + (-0.959 - 0.281i)T \)
83 \( 1 + (0.142 - 0.989i)T \)
89 \( 1 + (-0.841 - 0.540i)T \)
97 \( 1 + T \)
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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

−18.25612354676193179446702487014, −17.62406569116440265024414699671, −16.866448340161569081149047443124, −16.48246311325781825715561891865, −15.63273779712394425952823505453, −14.82408664735425140888650429684, −14.2835150718248486600314488209, −13.60808083028553308238405001672, −12.72109490476382104794694378471, −12.33748221989359886073085882024, −11.44852214662846095303594080127, −10.67443106380062823622703037813, −9.87759453830133822844803786937, −9.59494390795829383359611306232, −8.6049518744417785066505712973, −7.645244155636246255876634582379, −7.12224044263748481967545359039, −6.57767482989997222738831425065, −5.56173888749793871981608912711, −4.677980898756992065016984882328, −4.106773709964650314257400987, −3.35085547628100521910715777349, −2.18590552500413295436679094766, −1.70659979382934450907149897653, −0.27558418497651870557978844922, 0.176510654718848853917987679903, 1.57875528974291110822776373493, 2.352249921643856544789551343618, 3.124874582237845508689402785808, 3.88417532724976327575461400286, 4.87134351251161557875805598044, 5.66013374579726263388924410354, 6.161616571403504278699298774017, 7.04061272651883789127555073510, 7.81163545337560921015067857997, 8.829021551857487934033262537391, 8.98831419749430153951718327653, 9.98596708238509492731631696816, 10.71071944020386872271600112497, 11.47575774517551147152012696607, 12.07966959254737918262135413282, 12.88828665111746200938692825107, 13.33752463440945559394085174411, 14.27412347200974046142409889082, 14.97859942439850449971070456548, 15.52676544027141673097364950723, 16.17676577976451701270395926203, 16.98697367819386181024491421479, 17.542881546094324113803605728151, 18.359254421090873905056281820668

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