Properties

Label 2-5400-1.1-c1-0-47
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  − 2·7-s − 4·11-s + 2·13-s + 5·17-s − 5·19-s + 23-s + 2·29-s + 7·31-s + 6·37-s − 4·43-s + 4·47-s − 3·49-s + 9·53-s − 14·59-s − 11·61-s − 14·67-s + 12·73-s + 8·77-s − 3·79-s − 83-s − 4·91-s − 16·97-s − 12·101-s − 4·103-s − 12·107-s − 19·109-s + 6·113-s + ⋯
L(s)  = 1  − 0.755·7-s − 1.20·11-s + 0.554·13-s + 1.21·17-s − 1.14·19-s + 0.208·23-s + 0.371·29-s + 1.25·31-s + 0.986·37-s − 0.609·43-s + 0.583·47-s − 3/7·49-s + 1.23·53-s − 1.82·59-s − 1.40·61-s − 1.71·67-s + 1.40·73-s + 0.911·77-s − 0.337·79-s − 0.109·83-s − 0.419·91-s − 1.62·97-s − 1.19·101-s − 0.394·103-s − 1.16·107-s − 1.81·109-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: Trivial
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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−7.981382171554326217645556243477, −7.09600632258773302546659919319, −6.27600889627054566556417922941, −5.79895872444129673803553393069, −4.89252147239304591192913415964, −4.10725787094981855919921282343, −3.10912288243369646692729134226, −2.59932060746352882152800764632, −1.26958903158419492463048187751, 0, 1.26958903158419492463048187751, 2.59932060746352882152800764632, 3.10912288243369646692729134226, 4.10725787094981855919921282343, 4.89252147239304591192913415964, 5.79895872444129673803553393069, 6.27600889627054566556417922941, 7.09600632258773302546659919319, 7.981382171554326217645556243477

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