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Properties

Label 1-1584-1584.139-r0-0-0
Degree $1$
Conductor $1584$
Sign $-0.608 + 0.793i$
Analytic cond. $7.35606$
Root an. cond. $7.35606$
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  + (−0.207 − 0.978i)5-s + (0.104 + 0.994i)7-s + (0.743 + 0.669i)13-s + (−0.309 + 0.951i)17-s + (−0.587 + 0.809i)19-s + (−0.5 − 0.866i)23-s + (−0.913 + 0.406i)25-s + (0.994 − 0.104i)29-s + (−0.669 + 0.743i)31-s + (0.951 − 0.309i)35-s + (−0.587 − 0.809i)37-s + (−0.104 + 0.994i)41-s + (−0.866 − 0.5i)43-s + (−0.913 + 0.406i)47-s + (−0.978 + 0.207i)49-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)5-s + (0.104 + 0.994i)7-s + (0.743 + 0.669i)13-s + (−0.309 + 0.951i)17-s + (−0.587 + 0.809i)19-s + (−0.5 − 0.866i)23-s + (−0.913 + 0.406i)25-s + (0.994 − 0.104i)29-s + (−0.669 + 0.743i)31-s + (0.951 − 0.309i)35-s + (−0.587 − 0.809i)37-s + (−0.104 + 0.994i)41-s + (−0.866 − 0.5i)43-s + (−0.913 + 0.406i)47-s + (−0.978 + 0.207i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1584\)    =    \(2^{4} \cdot 3^{2} \cdot 11\)
Sign: $-0.608 + 0.793i$
Analytic conductor: \(7.35606\)
Root analytic conductor: \(7.35606\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1584} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1584,\ (0:\ ),\ -0.608 + 0.793i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3073657180 + 0.6234957629i\)
\(L(\frac12)\) \(\approx\) \(0.3073657180 + 0.6234957629i\)
\(L(1)\) \(\approx\) \(0.8635154020 + 0.1070732107i\)
\(L(1)\) \(\approx\) \(0.8635154020 + 0.1070732107i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + (-0.207 - 0.978i)T \)
7 \( 1 + (0.104 + 0.994i)T \)
13 \( 1 + (0.743 + 0.669i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.587 + 0.809i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.994 - 0.104i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (-0.913 + 0.406i)T \)
53 \( 1 + (-0.951 + 0.309i)T \)
59 \( 1 + (0.406 - 0.913i)T \)
61 \( 1 + (-0.743 + 0.669i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (0.743 - 0.669i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.978 - 0.207i)T \)
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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

−20.14980705230884807235464337834, −19.56970892524482734955315823856, −18.71943420055131856456044398965, −17.89074727048234683233518282853, −17.50764148809669810904691852989, −16.44064770645462328155096948946, −15.63631454160388402106889350360, −15.09191118568838449790543523853, −14.04071605294731440587530588562, −13.65094465716318316219312064592, −12.83477932419673847076181652779, −11.55342353279731354201025569910, −11.17713575010899192906872134554, −10.34494041223076078861345037110, −9.75180409427002177643588821957, −8.5588496653060209391488565661, −7.769570781088883892250555446496, −6.99017308804020124332294679081, −6.44188635282755541932580303920, −5.322562492501213058865534407415, −4.31381190534439670119482209617, −3.49432199721292387284337457458, −2.76712725275380522903611081398, −1.56664130079813268421916014536, −0.24728363060802983184094590064, 1.45626039541439859138814797503, 2.0395394219247863669294472356, 3.3905347756359599001289866555, 4.274198533036436295250686367860, 5.018365413396173374949126178, 6.005737381929754480065656661943, 6.53197314107739439821763210514, 7.98750922731029896760646643910, 8.52086797172537085545279599216, 9.01004117489169659384330988129, 10.024858523718672974669038129426, 10.95873612119771992464760024009, 11.82790886727632405565325852724, 12.48807819027488362836806907197, 12.9739945238068602016558097449, 14.06742172302273872095937574280, 14.772518711818749704380900232728, 15.70875372328471018186908991077, 16.20055641597862854511991822089, 16.93831624918715907253099741879, 17.82318441203971720399179201160, 18.55982558849925543274410416262, 19.299079487053223160501290407077, 19.97951127194887895495936338348, 20.91306827683238719577298572785

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