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Properties

Label 2-1584-396.131-c0-0-1
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 − 1.73i·15-s + (−0.5 + 0.866i)23-s + (1 + 1.73i)25-s + 0.999·27-s + (0.499 − 0.866i)33-s − 2·37-s + (−1.49 + 0.866i)45-s + (0.5 + 0.866i)47-s + (0.5 − 0.866i)49-s − 1.73i·53-s + 1.73i·55-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 − 1.73i·15-s + (−0.5 + 0.866i)23-s + (1 + 1.73i)25-s + 0.999·27-s + (0.499 − 0.866i)33-s − 2·37-s + (−1.49 + 0.866i)45-s + (0.5 + 0.866i)47-s + (0.5 − 0.866i)49-s − 1.73i·53-s + 1.73i·55-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

\(L(\frac{1}{2})\) \(\approx\) \(1.214017480\)
\(L(\frac12)\) \(\approx\) \(1.214017480\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + 2T + T^{2} \)
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 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - 2T + T^{2} \)
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 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.794482013199296151920306141129, −8.949524808936430771990717424295, −7.82896546326616602006560301314, −6.89598811674799316809379674695, −6.59259474497869233013772107584, −5.66690582311765000748908396685, −5.03939840352492441292943197013, −3.47514294068236071154019634833, −2.20262741527899467462116539434, −1.67679823383568835411134555908, 1.14117576903481033194684540153, 2.53490149227862443791957095002, 3.82334763384938534448562630607, 4.73697386428357090464533218322, 5.60614851564553322351396492174, 5.98334232833906613663986046442, 6.89520366631960081883580826146, 8.497205804823390330509431790775, 8.897691833414538556266242044033, 9.563507425096312070858750098366

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