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Properties

Label 1-4032-4032.187-r0-0-0
Degree $1$
Conductor $4032$
Sign $0.893 - 0.448i$
Analytic cond. $18.7245$
Root an. cond. $18.7245$
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

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

Invariants

Degree: \(1\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.893 - 0.448i$
Analytic conductor: \(18.7245\)
Root analytic conductor: \(18.7245\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4032,\ (0:\ ),\ 0.893 - 0.448i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.264620871 - 0.2993779511i\)
\(L(\frac12)\) \(\approx\) \(1.264620871 - 0.2993779511i\)
\(L(1)\) \(\approx\) \(0.9131654977 - 0.1190636698i\)
\(L(1)\) \(\approx\) \(0.9131654977 - 0.1190636698i\)

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

−18.46927535286017004961911241575, −17.96888726009873889051425663677, −17.12571549543706484555792557416, −16.45617985520493071784174098416, −15.666250350586950051816513393226, −15.21369593726102975715105231727, −14.51898972100905960365006628456, −13.829373627400580895495496880002, −13.02277368591708655058012365684, −12.1672113798881716150162149477, −11.6989977224316653993619867749, −11.06092633018564443579533910428, −10.32238955723604597893185011295, −9.376925049745045267581973575622, −8.92561944020852099663490927798, −7.8515385609274231807321438198, −7.42259474748524404634684253476, −6.585996872460311126755806802278, −6.06416222351003080021516684676, −4.70374787149804208412607102758, −4.29099840485509869877247718151, −3.658154155485631485127591100024, −2.51148042593009703766465473550, −1.92820526619202645148163195686, −0.62822558498079692660801498070, 0.633719592903355927014967101345, 1.38932427644254640254410063135, 2.67711639219001958867400586053, 3.5052394387821154039285748806, 3.93294220055477978936829049901, 5.06363560041577012135213225454, 5.5323276280432975203587650128, 6.5169847403480778316498531684, 7.32499894506802448458759403103, 8.02551937341878216032776995679, 8.584651958160240226809068782198, 9.29022384631207711979155883787, 10.17132597775606888534272181667, 11.05117058406659463190064777775, 11.477816459274275988675983664220, 12.25785473339983688716738452085, 12.83740377108520852300655231262, 13.73760259705068752331640670807, 14.219573975861076606964525635408, 15.211908548083078320956659701295, 15.86143914083665309223517101216, 16.122768848999112666514169367181, 17.051704880982045810957298390208, 17.73128170909316794193842835896, 18.46907055668024949037908730209

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