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Properties

Label 2-72e2-1.1-c1-0-68
Degree $2$
Conductor $5184$
Sign $-1$
Analytic cond. $41.3944$
Root an. cond. $6.43385$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 2·7-s − 2·11-s − 13-s + 3·17-s − 2·19-s + 6·23-s − 4·25-s − 29-s − 8·31-s − 2·35-s − 37-s − 2·41-s − 10·43-s + 4·47-s − 3·49-s + 10·53-s + 2·55-s − 4·59-s − 9·61-s + 65-s + 14·67-s + 10·71-s − 9·73-s − 4·77-s + 10·79-s − 12·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.755·7-s − 0.603·11-s − 0.277·13-s + 0.727·17-s − 0.458·19-s + 1.25·23-s − 4/5·25-s − 0.185·29-s − 1.43·31-s − 0.338·35-s − 0.164·37-s − 0.312·41-s − 1.52·43-s + 0.583·47-s − 3/7·49-s + 1.37·53-s + 0.269·55-s − 0.520·59-s − 1.15·61-s + 0.124·65-s + 1.71·67-s + 1.18·71-s − 1.05·73-s − 0.455·77-s + 1.12·79-s − 1.31·83-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: yes
Arithmetic: yes
Character: Trivial
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 \( 1 + T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 9 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.83183883422768713644232091354, −7.30754259073006344252649542526, −6.51301414706917366998863156629, −5.36986537425335647613270246539, −5.12656573806582728748000894922, −4.09193195998284307936737221915, −3.36102040400281792770735695951, −2.34932291764629378386923963817, −1.38515225110294082706245571114, 0, 1.38515225110294082706245571114, 2.34932291764629378386923963817, 3.36102040400281792770735695951, 4.09193195998284307936737221915, 5.12656573806582728748000894922, 5.36986537425335647613270246539, 6.51301414706917366998863156629, 7.30754259073006344252649542526, 7.83183883422768713644232091354

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