Properties

Label 2-72e2-1.1-c1-0-3
Degree $2$
Conductor $5184$
Sign $1$
Analytic cond. $41.3944$
Root an. cond. $6.43385$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 3.44·7-s − 1.44·11-s − 5.89·13-s − 4.89·17-s + 4·19-s − 5.44·23-s − 4·25-s + 0.101·29-s − 2.55·31-s + 3.44·35-s + 0.898·37-s + 11.8·41-s − 2.34·43-s − 6.34·47-s + 4.89·49-s + 8.89·53-s + 1.44·55-s − 14.3·59-s + 7.89·61-s + 5.89·65-s + 12.3·67-s + 7.79·71-s − 4.89·73-s + 5·77-s − 13.4·79-s + 0.550·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.30·7-s − 0.437·11-s − 1.63·13-s − 1.18·17-s + 0.917·19-s − 1.13·23-s − 0.800·25-s + 0.0187·29-s − 0.458·31-s + 0.583·35-s + 0.147·37-s + 1.85·41-s − 0.358·43-s − 0.926·47-s + 0.699·49-s + 1.22·53-s + 0.195·55-s − 1.86·59-s + 1.01·61-s + 0.731·65-s + 1.50·67-s + 0.925·71-s − 0.573·73-s + 0.569·77-s − 1.51·79-s + 0.0604·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(41.3944\)
Root analytic conductor: \(6.43385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5184,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5131296979\)
\(L(\frac12)\) \(\approx\) \(0.5131296979\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + T + 5T^{2} \)
7 \( 1 + 3.44T + 7T^{2} \)
11 \( 1 + 1.44T + 11T^{2} \)
13 \( 1 + 5.89T + 13T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 5.44T + 23T^{2} \)
29 \( 1 - 0.101T + 29T^{2} \)
31 \( 1 + 2.55T + 31T^{2} \)
37 \( 1 - 0.898T + 37T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 + 2.34T + 43T^{2} \)
47 \( 1 + 6.34T + 47T^{2} \)
53 \( 1 - 8.89T + 53T^{2} \)
59 \( 1 + 14.3T + 59T^{2} \)
61 \( 1 - 7.89T + 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 - 7.79T + 71T^{2} \)
73 \( 1 + 4.89T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 - 0.550T + 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + 3.89T + 97T^{2} \)
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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

−8.030200922843012301921071779697, −7.48692935031461603583495669770, −6.85824021793040372397054799092, −6.10017260962653808080682863647, −5.32765191197100769243608382505, −4.44861806040112754251153922618, −3.72049838356548262665664224823, −2.80479861760478796836196446731, −2.13554095068885957293860379543, −0.35876913074410584393685452131, 0.35876913074410584393685452131, 2.13554095068885957293860379543, 2.80479861760478796836196446731, 3.72049838356548262665664224823, 4.44861806040112754251153922618, 5.32765191197100769243608382505, 6.10017260962653808080682863647, 6.85824021793040372397054799092, 7.48692935031461603583495669770, 8.030200922843012301921071779697

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