Properties

Label 2-72e2-1.1-c1-0-85
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 $1$

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: $\chi_{5184} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5184,\ (\ :1/2),\ -1)\)

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 \( 1 \)
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

−8.011610246749906490895334891958, −7.31144338011062910065244253599, −6.35125090859543457081741332730, −5.55751681320044101093018990181, −4.92482838914868723279164283972, −4.36823002282681997660380240995, −3.19084127629040496665354144147, −2.17698492914301764935021916367, −1.61737496797709842929059814747, 0, 1.61737496797709842929059814747, 2.17698492914301764935021916367, 3.19084127629040496665354144147, 4.36823002282681997660380240995, 4.92482838914868723279164283972, 5.55751681320044101093018990181, 6.35125090859543457081741332730, 7.31144338011062910065244253599, 8.011610246749906490895334891958

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