Properties

Label 2-1584-44.43-c1-0-3
Degree $2$
Conductor $1584$
Sign $-i$
Analytic cond. $12.6483$
Root an. cond. $3.55644$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 3.31i·11-s + 3.31i·23-s + 4·25-s + 9.94i·31-s + 7·37-s + 6.63i·47-s − 7·49-s − 6·53-s + 9.94i·55-s + 3.31i·59-s + 9.94i·67-s + 16.5i·71-s + 9·89-s − 17·97-s + ⋯
L(s)  = 1  − 1.34·5-s − 1.00i·11-s + 0.691i·23-s + 0.800·25-s + 1.78i·31-s + 1.15·37-s + 0.967i·47-s − 49-s − 0.824·53-s + 1.34i·55-s + 0.431i·59-s + 1.21i·67-s + 1.96i·71-s + 0.953·89-s − 1.72·97-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1584\)    =    \(2^{4} \cdot 3^{2} \cdot 11\)
Sign: $-i$
Analytic conductor: \(12.6483\)
Root analytic conductor: \(3.55644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1584} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1584,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7612397610\)
\(L(\frac12)\) \(\approx\) \(0.7612397610\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + 3.31iT \)
good5 \( 1 + 3T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 3.31iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 9.94iT - 31T^{2} \)
37 \( 1 - 7T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 6.63iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 3.31iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 9.94iT - 67T^{2} \)
71 \( 1 - 16.5iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + 17T + 97T^{2} \)
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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

−9.527298031399968695009072617038, −8.636712062768920253185706800042, −8.065230724546502233192443461282, −7.35385081331080536077665418146, −6.47058745551317505987186758689, −5.48711054168476884362949155134, −4.50280814318955923080287379462, −3.63200033997258241259008051521, −2.89569895439985644400896794507, −1.12974241868404701799265172265, 0.34302960194193152541892960370, 2.05847254630549191612834740248, 3.28772959447658657385014592230, 4.24757181404877892307860964303, 4.77413391941721413371555392182, 6.06369749379060009767879803291, 6.95320088838568664856591127885, 7.76427541762664832793694490961, 8.158487958521229707799719767839, 9.268837417625597732973522039058

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