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Properties

Label 2-1584-396.131-c0-0-4
Degree $2$
Conductor $1584$
Sign $0.984 - 0.173i$
Analytic cond. $0.790518$
Root an. cond. $0.889111$
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  + 3-s + 9-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s + (−1.5 − 0.866i)31-s + (0.5 + 0.866i)33-s + 37-s + (0.5 + 0.866i)47-s + (0.5 − 0.866i)49-s − 1.73i·53-s + (−0.5 + 0.866i)59-s + (−0.5 + 0.866i)69-s − 71-s + ⋯
L(s)  = 1  + 3-s + 9-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s + (−1.5 − 0.866i)31-s + (0.5 + 0.866i)33-s + 37-s + (0.5 + 0.866i)47-s + (0.5 − 0.866i)49-s − 1.73i·53-s + (−0.5 + 0.866i)59-s + (−0.5 + 0.866i)69-s − 71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1584\)    =    \(2^{4} \cdot 3^{2} \cdot 11\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1584} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1584,\ (\ :0),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.605104168\)
\(L(\frac12)\) \(\approx\) \(1.605104168\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + 1.73iT - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - 1.73iT - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.594911039207034380735889395916, −8.928278308650131740756195877906, −7.974605110637929069470931560570, −7.42883980202713283721052709631, −6.58663632256802057483562325581, −5.52893864536373114378324646795, −4.32012713862446569056126639139, −3.78139968569781875441758450907, −2.53872046684487447374174716494, −1.64718827139036342634569102306, 1.44406315573415977471035187523, 2.65237737974868495483185104364, 3.57993385667201976004482263457, 4.32189125892702422368976306557, 5.53252426697632345920761677076, 6.44950415523688628297209988765, 7.36977966218864435314461518079, 8.034836568769722430490701989476, 8.966519709655018988873380451885, 9.249165799761302718392500285879

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