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Properties

Label 2-10584-1.1-c1-0-20
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 $0$

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: \(0\)
Selberg data: \((2,\ 10584,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.247735476\)
\(L(\frac12)\) \(\approx\) \(3.247735476\)
\(L(\frac{3}{2})\) 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

−16.81346105554114, −16.07433982354070, −15.25464361373484, −14.81159070400054, −14.11873042969323, −13.78609633280113, −13.10527396006816, −12.79599006902204, −11.79939754363859, −11.49119889445820, −10.45729861686688, −10.07737598212602, −9.569162248651861, −8.972888215558623, −8.522064350216556, −7.341338608862158, −6.854946579892224, −6.289756766469493, −5.632231469940657, −4.785910581282790, −4.543474781622124, −3.009102093730685, −2.592851982810520, −1.765912211849104, −0.8669081358047007, 0.8669081358047007, 1.765912211849104, 2.592851982810520, 3.009102093730685, 4.543474781622124, 4.785910581282790, 5.632231469940657, 6.289756766469493, 6.854946579892224, 7.341338608862158, 8.522064350216556, 8.972888215558623, 9.569162248651861, 10.07737598212602, 10.45729861686688, 11.49119889445820, 11.79939754363859, 12.79599006902204, 13.10527396006816, 13.78609633280113, 14.11873042969323, 14.81159070400054, 15.25464361373484, 16.07433982354070, 16.81346105554114

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