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Properties

Label 1-4032-4032.3181-r1-0-0
Degree $1$
Conductor $4032$
Sign $-0.448 + 0.893i$
Analytic cond. $433.298$
Root an. cond. $433.298$
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  + (0.923 − 0.382i)5-s + (0.382 + 0.923i)11-s + (−0.130 − 0.991i)13-s + (−0.866 + 0.5i)17-s + (0.793 − 0.608i)19-s + (0.707 + 0.707i)23-s + (0.707 − 0.707i)25-s + (0.608 + 0.793i)29-s + (−0.5 + 0.866i)31-s + (−0.130 + 0.991i)37-s + (−0.258 + 0.965i)41-s + (0.991 + 0.130i)43-s + (−0.866 + 0.5i)47-s + (0.608 − 0.793i)53-s + (0.707 + 0.707i)55-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)5-s + (0.382 + 0.923i)11-s + (−0.130 − 0.991i)13-s + (−0.866 + 0.5i)17-s + (0.793 − 0.608i)19-s + (0.707 + 0.707i)23-s + (0.707 − 0.707i)25-s + (0.608 + 0.793i)29-s + (−0.5 + 0.866i)31-s + (−0.130 + 0.991i)37-s + (−0.258 + 0.965i)41-s + (0.991 + 0.130i)43-s + (−0.866 + 0.5i)47-s + (0.608 − 0.793i)53-s + (0.707 + 0.707i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-0.448 + 0.893i$
Analytic conductor: \(433.298\)
Root analytic conductor: \(433.298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (3181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4032,\ (1:\ ),\ -0.448 + 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9160425251 + 1.484278621i\)
\(L(\frac12)\) \(\approx\) \(0.9160425251 + 1.484278621i\)
\(L(1)\) \(\approx\) \(1.198785426 + 0.09560027446i\)
\(L(1)\) \(\approx\) \(1.198785426 + 0.09560027446i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.923 - 0.382i)T \)
11 \( 1 + (0.382 + 0.923i)T \)
13 \( 1 + (-0.130 - 0.991i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.793 - 0.608i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (0.608 + 0.793i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.130 + 0.991i)T \)
41 \( 1 + (-0.258 + 0.965i)T \)
43 \( 1 + (0.991 + 0.130i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.608 - 0.793i)T \)
59 \( 1 + (-0.130 + 0.991i)T \)
61 \( 1 + (-0.991 + 0.130i)T \)
67 \( 1 + (-0.608 - 0.793i)T \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 + (0.258 - 0.965i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (0.793 - 0.608i)T \)
89 \( 1 + (-0.258 - 0.965i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−18.07436710057844083839218479951, −17.42656854096767730406295607491, −16.67600861653944444160348905244, −16.248062444999767103080243534450, −15.31417157023810114507271328667, −14.493975965757907616092918766134, −13.89805350947295977129283346870, −13.58767183960133696230622250903, −12.64989888443970447372216532326, −11.80205281008155506121218493231, −11.147120128709479508063763670989, −10.56908400352347068830008691457, −9.62309163319268804598335526302, −9.15976997709565343420968253499, −8.51759106010039912549773969115, −7.39988921740824179466266343171, −6.78842218128106519585661703783, −6.07834538139729648154728871711, −5.492445332909049864400477823891, −4.53052067855439698608296807761, −3.74084683399194774591930904218, −2.77821429558924972819054332265, −2.15048196789743612963426159565, −1.24304128588051717724866150489, −0.2413299241901907786777855122, 1.11213796270863425852503854101, 1.58662561678935767395355519305, 2.66066057016850474938050563473, 3.28022761728518044866398758353, 4.55574136761331392655945173085, 4.9811445903502232938398881219, 5.759555150249714070370641040825, 6.61543624440662428609969740197, 7.195282538622036625649773402261, 8.12646398966669674032910279877, 8.987011973505088262275516075756, 9.42956731976660737804146489677, 10.23791694993621455181375591943, 10.77684535128149439513930478188, 11.74398872587988130114878353903, 12.48964771213977669215129395505, 13.11772760448882090578988767114, 13.56463217417211778888793920322, 14.51058202704836695623798796758, 15.07188352457012553915569407010, 15.77874141438591264985790105915, 16.56853648595311612159375700781, 17.35624131158679395192188261075, 17.84783537334914054476041973982, 18.105923147865478418922479173192

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