Properties

Degree $2$
Conductor $259920$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 4·7-s + 6·11-s − 4·13-s − 6·17-s + 6·23-s + 25-s − 2·29-s + 4·35-s − 8·37-s − 10·41-s + 4·43-s − 2·47-s + 9·49-s − 10·53-s + 6·55-s − 12·59-s + 2·61-s − 4·65-s + 8·67-s − 6·73-s + 24·77-s − 4·79-s + 2·83-s − 6·85-s + 14·89-s − 16·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s + 1.80·11-s − 1.10·13-s − 1.45·17-s + 1.25·23-s + 1/5·25-s − 0.371·29-s + 0.676·35-s − 1.31·37-s − 1.56·41-s + 0.609·43-s − 0.291·47-s + 9/7·49-s − 1.37·53-s + 0.809·55-s − 1.56·59-s + 0.256·61-s − 0.496·65-s + 0.977·67-s − 0.702·73-s + 2.73·77-s − 0.450·79-s + 0.219·83-s − 0.650·85-s + 1.48·89-s − 1.67·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(259920\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 19^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{259920} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 259920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.451237340\)
\(L(\frac12)\) \(\approx\) \(3.451237340\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
19 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−12.79294437928752, −12.28801705518973, −11.77383113663903, −11.44266973378123, −11.13274765809583, −10.51618907549901, −10.15210018711814, −9.277331067661983, −9.151153011816048, −8.796413461360836, −8.186841295058787, −7.620196025064141, −7.115840963090403, −6.623306785020152, −6.391615946139337, −5.514256164149810, −5.015815146088849, −4.685767466127527, −4.278694763085302, −3.558981967253657, −2.972384200626922, −2.135292594239468, −1.758170236812473, −1.377149553315182, −0.4885073943012223, 0.4885073943012223, 1.377149553315182, 1.758170236812473, 2.135292594239468, 2.972384200626922, 3.558981967253657, 4.278694763085302, 4.685767466127527, 5.015815146088849, 5.514256164149810, 6.391615946139337, 6.623306785020152, 7.115840963090403, 7.620196025064141, 8.186841295058787, 8.796413461360836, 9.151153011816048, 9.277331067661983, 10.15210018711814, 10.51618907549901, 11.13274765809583, 11.44266973378123, 11.77383113663903, 12.28801705518973, 12.79294437928752

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