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 − 2·7-s + 2·13-s + 6·17-s + 6·23-s + 25-s + 4·29-s − 2·35-s + 10·37-s + 8·41-s − 2·43-s − 2·47-s − 3·49-s + 2·53-s + 14·61-s + 2·65-s − 4·67-s + 12·71-s + 6·73-s + 8·79-s + 2·83-s + 6·85-s + 8·89-s − 4·91-s − 6·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s + 0.554·13-s + 1.45·17-s + 1.25·23-s + 1/5·25-s + 0.742·29-s − 0.338·35-s + 1.64·37-s + 1.24·41-s − 0.304·43-s − 0.291·47-s − 3/7·49-s + 0.274·53-s + 1.79·61-s + 0.248·65-s − 0.488·67-s + 1.42·71-s + 0.702·73-s + 0.900·79-s + 0.219·83-s + 0.650·85-s + 0.847·89-s − 0.419·91-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-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

\(L(1)\) \(\approx\) \(4.265170927\)
\(L(\frac12)\) \(\approx\) \(4.265170927\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−12.84971559107469, −12.52567142912777, −11.89848315470130, −11.43213342077009, −10.91952771426192, −10.53115848630109, −9.911350676042824, −9.591843589114514, −9.289063565389056, −8.628973361454171, −8.115326146800221, −7.735144714146553, −7.050881288185529, −6.661711903905233, −6.100403296663957, −5.803610906359500, −5.117867693437806, −4.779877726264080, −3.907879398396265, −3.565431611169295, −2.891984208297409, −2.582357566222643, −1.745027252302753, −0.8968048695071398, −0.7423835940260190, 0.7423835940260190, 0.8968048695071398, 1.745027252302753, 2.582357566222643, 2.891984208297409, 3.565431611169295, 3.907879398396265, 4.779877726264080, 5.117867693437806, 5.803610906359500, 6.100403296663957, 6.661711903905233, 7.050881288185529, 7.735144714146553, 8.115326146800221, 8.628973361454171, 9.289063565389056, 9.591843589114514, 9.911350676042824, 10.53115848630109, 10.91952771426192, 11.43213342077009, 11.89848315470130, 12.52567142912777, 12.84971559107469

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