Properties

Label 2-43560-1.1-c1-0-1
Degree $2$
Conductor $43560$
Sign $1$
Analytic cond. $347.828$
Root an. cond. $18.6501$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 2·7-s − 4·13-s − 2·17-s − 4·19-s − 8·23-s + 25-s − 10·29-s + 4·31-s − 2·35-s + 8·43-s − 8·47-s − 3·49-s − 6·53-s + 14·59-s + 14·61-s − 4·65-s − 4·67-s − 12·71-s − 6·73-s + 12·79-s + 4·83-s − 2·85-s + 12·89-s + 8·91-s − 4·95-s − 14·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 1.10·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s − 1.85·29-s + 0.718·31-s − 0.338·35-s + 1.21·43-s − 1.16·47-s − 3/7·49-s − 0.824·53-s + 1.82·59-s + 1.79·61-s − 0.496·65-s − 0.488·67-s − 1.42·71-s − 0.702·73-s + 1.35·79-s + 0.439·83-s − 0.216·85-s + 1.27·89-s + 0.838·91-s − 0.410·95-s − 1.42·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43560\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(347.828\)
Root analytic conductor: \(18.6501\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 43560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4993120978\)
\(L(\frac12)\) \(\approx\) \(0.4993120978\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.64144303811192, −14.31404368211007, −13.51058057977360, −13.07820097175093, −12.78811005585312, −12.10239581687416, −11.65299334330258, −11.02951252177860, −10.28309745615702, −10.06326884136871, −9.331535017591498, −9.176817883619970, −8.172266827163266, −7.881871044547400, −7.056027886702727, −6.597079438043118, −6.071440710963655, −5.492563192990125, −4.856132507443906, −4.070184732653736, −3.695762129261614, −2.629207621715733, −2.302569094189798, −1.538492227839355, −0.2386648175201205, 0.2386648175201205, 1.538492227839355, 2.302569094189798, 2.629207621715733, 3.695762129261614, 4.070184732653736, 4.856132507443906, 5.492563192990125, 6.071440710963655, 6.597079438043118, 7.056027886702727, 7.881871044547400, 8.172266827163266, 9.176817883619970, 9.331535017591498, 10.06326884136871, 10.28309745615702, 11.02951252177860, 11.65299334330258, 12.10239581687416, 12.78811005585312, 13.07820097175093, 13.51058057977360, 14.31404368211007, 14.64144303811192

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