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Properties

Label 2-1584-396.131-c0-0-5
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  + (0.5 − 0.866i)3-s + (−1.5 − 0.866i)5-s + (−0.499 − 0.866i)9-s + (−0.5 − 0.866i)11-s + (−1.5 + 0.866i)15-s + (−1 + 1.73i)23-s + (1 + 1.73i)25-s − 0.999·27-s + (−1.5 − 0.866i)31-s − 0.999·33-s + 37-s + 1.73i·45-s + (−0.5 − 0.866i)47-s + (0.5 − 0.866i)49-s − 1.73i·53-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−1.5 − 0.866i)5-s + (−0.499 − 0.866i)9-s + (−0.5 − 0.866i)11-s + (−1.5 + 0.866i)15-s + (−1 + 1.73i)23-s + (1 + 1.73i)25-s − 0.999·27-s + (−1.5 − 0.866i)31-s − 0.999·33-s + 37-s + 1.73i·45-s + (−0.5 − 0.866i)47-s + (0.5 − 0.866i)49-s − 1.73i·53-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

\(L(\frac{1}{2})\) \(\approx\) \(0.6402232368\)
\(L(\frac12)\) \(\approx\) \(0.6402232368\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 - T^{2} \)
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

−8.956818278373550980187930164221, −8.240687078333895823286765134799, −7.79628643733384447214566888347, −7.18060995829898612462842968557, −5.96839232154977873238055153423, −5.15315531974899285030900604524, −3.83244940666484969633267890851, −3.41169035121625682594763965288, −1.90238716150179542920800972922, −0.45851820237222443246986195535, 2.40455163356452094904377937206, 3.18162323136225797344425236502, 4.21531777166481769441752980449, 4.56054349148080190180254586369, 5.90620267666003285770051464916, 7.08091482825613951525010269124, 7.66470624005081495052362281293, 8.320151031910143279395228532018, 9.150850837170978975787406030078, 10.16996673499990272747726006692

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