Properties

Label 2-5400-1.1-c1-0-58
Degree $2$
Conductor $5400$
Sign $-1$
Analytic cond. $43.1192$
Root an. cond. $6.56652$
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  − 2·11-s + 3·17-s − 19-s + 3·23-s − 4·29-s − 5·31-s − 10·37-s − 6·41-s + 6·43-s + 8·47-s − 7·49-s + 3·53-s + 5·61-s + 2·67-s − 2·71-s − 6·73-s − 11·79-s + 9·83-s − 10·89-s − 8·97-s + 12·101-s + 12·103-s − 4·107-s + 9·109-s − 6·113-s + ⋯
L(s)  = 1  − 0.603·11-s + 0.727·17-s − 0.229·19-s + 0.625·23-s − 0.742·29-s − 0.898·31-s − 1.64·37-s − 0.937·41-s + 0.914·43-s + 1.16·47-s − 49-s + 0.412·53-s + 0.640·61-s + 0.244·67-s − 0.237·71-s − 0.702·73-s − 1.23·79-s + 0.987·83-s − 1.05·89-s − 0.812·97-s + 1.19·101-s + 1.18·103-s − 0.386·107-s + 0.862·109-s − 0.564·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5400\)    =    \(2^{3} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(43.1192\)
Root analytic conductor: \(6.56652\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−7.66799247541723016901781805637, −7.26178345444521246986748164023, −6.39999755757971246516471452479, −5.48358742314242556721498817529, −5.10726254791347361757990363869, −4.00791804556625027557393879766, −3.30759479173743955199544638022, −2.37800474256130148896006176014, −1.37350549586058851111756287117, 0, 1.37350549586058851111756287117, 2.37800474256130148896006176014, 3.30759479173743955199544638022, 4.00791804556625027557393879766, 5.10726254791347361757990363869, 5.48358742314242556721498817529, 6.39999755757971246516471452479, 7.26178345444521246986748164023, 7.66799247541723016901781805637

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