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Properties

Label 2-10584-1.1-c1-0-8
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 + 4·11-s − 13-s + 4·17-s + 19-s + 4·23-s + 11·25-s + 4·31-s − 9·37-s − 8·43-s + 12·47-s − 8·53-s − 16·55-s − 4·59-s + 5·61-s + 4·65-s + 11·67-s + 8·71-s − 73-s − 5·79-s − 8·83-s − 16·85-s − 12·89-s − 4·95-s − 5·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 1.78·5-s + 1.20·11-s − 0.277·13-s + 0.970·17-s + 0.229·19-s + 0.834·23-s + 11/5·25-s + 0.718·31-s − 1.47·37-s − 1.21·43-s + 1.75·47-s − 1.09·53-s − 2.15·55-s − 0.520·59-s + 0.640·61-s + 0.496·65-s + 1.34·67-s + 0.949·71-s − 0.117·73-s − 0.562·79-s − 0.878·83-s − 1.73·85-s − 1.27·89-s − 0.410·95-s − 0.507·97-s + 0.0995·101-s + 0.0985·103-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

\(L(1)\) \(\approx\) \(1.443108272\)
\(L(\frac12)\) \(\approx\) \(1.443108272\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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

−16.59935244762531, −15.86314465135690, −15.42438984721031, −14.99898020630311, −14.24638991246039, −13.97813153057237, −12.89306814733701, −12.25947200687567, −12.04585800686552, −11.39061777440111, −10.96618235615714, −10.12622294672294, −9.474733223736423, −8.658377860826964, −8.320760198362604, −7.520911599218233, −7.057627764026628, −6.509367516881759, −5.434593502228473, −4.766104293402060, −4.019844287987757, −3.515263827979389, −2.869723179070413, −1.459418924673641, −0.6060128709505486, 0.6060128709505486, 1.459418924673641, 2.869723179070413, 3.515263827979389, 4.019844287987757, 4.766104293402060, 5.434593502228473, 6.509367516881759, 7.057627764026628, 7.520911599218233, 8.320760198362604, 8.658377860826964, 9.474733223736423, 10.12622294672294, 10.96618235615714, 11.39061777440111, 12.04585800686552, 12.25947200687567, 12.89306814733701, 13.97813153057237, 14.24638991246039, 14.99898020630311, 15.42438984721031, 15.86314465135690, 16.59935244762531

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