Properties

Degree $2$
Conductor $43120$
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 − 5-s + 9-s + 11-s + 4·13-s + 2·15-s − 4·19-s + 6·23-s + 25-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 − 45-s + 6·47-s − 6·53-s − 55-s + 8·57-s − 12·59-s − 2·61-s − 4·65-s + 10·67-s − 12·69-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.516·15-s − 0.917·19-s + 1.25·23-s + 1/5·25-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 − 0.149·45-s + 0.875·47-s − 0.824·53-s − 0.134·55-s + 1.05·57-s − 1.56·59-s − 0.256·61-s − 0.496·65-s + 1.22·67-s − 1.44·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43120\)    =    \(2^{4} \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{43120} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 43120,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.179456983\)
\(L(\frac12)\) \(\approx\) \(1.179456983\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
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

−14.93043586430255, −13.98562859718106, −13.76473452421277, −12.88902830708611, −12.65309028744101, −12.02771024648962, −11.42775560722985, −11.10268357998618, −10.81669653425646, −10.10594118189696, −9.501304782506279, −8.719948466743006, −8.447058555565232, −7.762923299189426, −6.972108061422238, −6.480408070202109, −6.168606381052839, −5.423942361180331, −4.866974317088540, −4.345555512538527, −3.574622705222735, −3.049384527032988, −2.011144814773939, −1.154590564901268, −0.4827364686881440, 0.4827364686881440, 1.154590564901268, 2.011144814773939, 3.049384527032988, 3.574622705222735, 4.345555512538527, 4.866974317088540, 5.423942361180331, 6.168606381052839, 6.480408070202109, 6.972108061422238, 7.762923299189426, 8.447058555565232, 8.719948466743006, 9.501304782506279, 10.10594118189696, 10.81669653425646, 11.10268357998618, 11.42775560722985, 12.02771024648962, 12.65309028744101, 12.88902830708611, 13.76473452421277, 13.98562859718106, 14.93043586430255

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