Properties

Degree $2$
Conductor $215600$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 9-s + 11-s − 4·13-s − 4·19-s − 6·23-s − 4·27-s − 6·29-s + 8·31-s + 2·33-s − 2·37-s − 8·39-s − 6·41-s + 8·43-s − 6·47-s + 6·53-s − 8·57-s − 12·59-s − 2·61-s − 10·67-s − 12·69-s + 12·71-s − 16·73-s − 8·79-s − 11·81-s − 12·87-s − 6·89-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 0.917·19-s − 1.25·23-s − 0.769·27-s − 1.11·29-s + 1.43·31-s + 0.348·33-s − 0.328·37-s − 1.28·39-s − 0.937·41-s + 1.21·43-s − 0.875·47-s + 0.824·53-s − 1.05·57-s − 1.56·59-s − 0.256·61-s − 1.22·67-s − 1.44·69-s + 1.42·71-s − 1.87·73-s − 0.900·79-s − 1.22·81-s − 1.28·87-s − 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 215600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(215600\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{2} \cdot 11\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{215600} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 215600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.054938396\)
\(L(\frac12)\) \(\approx\) \(1.054938396\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good3 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.11587502459434, −12.53121746317711, −12.09460847949919, −11.69570493787837, −11.18940511651164, −10.42296570456521, −10.15579608665242, −9.664202367231142, −9.111388416377491, −8.832017623047454, −8.236284808109355, −7.808762091323212, −7.483315937565003, −6.850952158785471, −6.274029991535724, −5.831000985677297, −5.168887332280262, −4.504814678089314, −4.116581153516764, −3.585497102938512, −2.879204326501277, −2.535277906755531, −1.914860889960638, −1.440467668532548, −0.2420023211056799, 0.2420023211056799, 1.440467668532548, 1.914860889960638, 2.535277906755531, 2.879204326501277, 3.585497102938512, 4.116581153516764, 4.504814678089314, 5.168887332280262, 5.831000985677297, 6.274029991535724, 6.850952158785471, 7.483315937565003, 7.808762091323212, 8.236284808109355, 8.832017623047454, 9.111388416377491, 9.664202367231142, 10.15579608665242, 10.42296570456521, 11.18940511651164, 11.69570493787837, 12.09460847949919, 12.53121746317711, 13.11587502459434

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