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Properties

Label 2-10584-1.1-c1-0-22
Degree $2$
Conductor $10584$
Sign $-1$
Analytic cond. $84.5136$
Root an. cond. $9.19313$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 2·11-s − 5·13-s + 6·17-s + 4·19-s − 6·23-s + 11·25-s − 6·29-s + 7·31-s + 7·37-s + 2·41-s − 7·43-s − 2·47-s − 6·53-s + 8·55-s + 6·59-s + 9·61-s + 20·65-s − 7·67-s + 8·71-s − 10·73-s + 79-s + 14·83-s − 24·85-s + 12·89-s − 16·95-s + 15·97-s + ⋯
L(s)  = 1  − 1.78·5-s − 0.603·11-s − 1.38·13-s + 1.45·17-s + 0.917·19-s − 1.25·23-s + 11/5·25-s − 1.11·29-s + 1.25·31-s + 1.15·37-s + 0.312·41-s − 1.06·43-s − 0.291·47-s − 0.824·53-s + 1.07·55-s + 0.781·59-s + 1.15·61-s + 2.48·65-s − 0.855·67-s + 0.949·71-s − 1.17·73-s + 0.112·79-s + 1.53·83-s − 2.60·85-s + 1.27·89-s − 1.64·95-s + 1.52·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10584\)    =    \(2^{3} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(84.5136\)
Root analytic conductor: \(9.19313\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 10584,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 9 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
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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

−16.47566884415560, −16.38729692910717, −15.71499506608631, −15.15006626286149, −14.62361676460693, −14.24747718280245, −13.28070523716202, −12.68006993060018, −11.97925179335690, −11.83678075704314, −11.27143495002924, −10.30168288149096, −9.946305560970474, −9.243188279327161, −8.205646762000441, −7.782227566894105, −7.610037928334655, −6.823414381076248, −5.840078808382175, −5.054197248173195, −4.555670855565835, −3.675577425971648, −3.183341589523019, −2.305032191563194, −0.9148176393277150, 0, 0.9148176393277150, 2.305032191563194, 3.183341589523019, 3.675577425971648, 4.555670855565835, 5.054197248173195, 5.840078808382175, 6.823414381076248, 7.610037928334655, 7.782227566894105, 8.205646762000441, 9.243188279327161, 9.946305560970474, 10.30168288149096, 11.27143495002924, 11.83678075704314, 11.97925179335690, 12.68006993060018, 13.28070523716202, 14.24747718280245, 14.62361676460693, 15.15006626286149, 15.71499506608631, 16.38729692910717, 16.47566884415560

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