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Properties

Label 2-10584-1.1-c1-0-23
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  + 2·5-s + 3·11-s + 6·13-s + 4·17-s + 4·19-s − 4·23-s − 25-s + 5·29-s + 7·31-s − 2·41-s + 8·43-s + 2·47-s − 10·53-s + 6·55-s + 9·59-s − 8·61-s + 12·65-s − 6·67-s + 12·71-s − 11·73-s + 79-s + 15·83-s + 8·85-s + 10·89-s + 8·95-s − 5·97-s + 101-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.904·11-s + 1.66·13-s + 0.970·17-s + 0.917·19-s − 0.834·23-s − 1/5·25-s + 0.928·29-s + 1.25·31-s − 0.312·41-s + 1.21·43-s + 0.291·47-s − 1.37·53-s + 0.809·55-s + 1.17·59-s − 1.02·61-s + 1.48·65-s − 0.733·67-s + 1.42·71-s − 1.28·73-s + 0.112·79-s + 1.64·83-s + 0.867·85-s + 1.05·89-s + 0.820·95-s − 0.507·97-s + 0.0995·101-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

\(L(1)\) \(\approx\) \(3.516509547\)
\(L(\frac12)\) \(\approx\) \(3.516509547\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−16.46221593611716, −16.02918148167997, −15.58339070053371, −14.70163094840339, −14.05244096423692, −13.79732342293024, −13.36450421013287, −12.42017383494086, −11.96253147077412, −11.40557554278118, −10.62094717558324, −10.10427598645560, −9.485643343934955, −9.012542900073065, −8.211408425815438, −7.741970867542710, −6.705448954778738, −6.162043515909218, −5.813939302342201, −4.956032695206977, −4.029284573667454, −3.448624296026330, −2.574211926678426, −1.480995241125940, −1.016715689954122, 1.016715689954122, 1.480995241125940, 2.574211926678426, 3.448624296026330, 4.029284573667454, 4.956032695206977, 5.813939302342201, 6.162043515909218, 6.705448954778738, 7.741970867542710, 8.211408425815438, 9.012542900073065, 9.485643343934955, 10.10427598645560, 10.62094717558324, 11.40557554278118, 11.96253147077412, 12.42017383494086, 13.36450421013287, 13.79732342293024, 14.05244096423692, 14.70163094840339, 15.58339070053371, 16.02918148167997, 16.46221593611716

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