Properties

Degree $2$
Conductor $242760$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 7-s + 9-s + 4·11-s − 2·13-s − 15-s − 4·19-s + 21-s + 25-s + 27-s + 10·29-s + 4·33-s − 35-s − 6·37-s − 2·39-s + 6·41-s − 4·43-s − 45-s − 8·47-s + 49-s + 6·53-s − 4·55-s − 4·57-s − 4·59-s + 10·61-s + 63-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s − 0.917·19-s + 0.218·21-s + 1/5·25-s + 0.192·27-s + 1.85·29-s + 0.696·33-s − 0.169·35-s − 0.986·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.824·53-s − 0.539·55-s − 0.529·57-s − 0.520·59-s + 1.28·61-s + 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 242760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(242760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 17^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{242760} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 242760,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.662114949\)
\(L(\frac12)\) \(\approx\) \(3.662114949\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
17 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−12.74537968761906, −12.41790477551095, −11.98391638839073, −11.59768008118569, −10.98905846229921, −10.64021879875030, −9.957378178988006, −9.661324710153609, −9.083050511626016, −8.569811809051343, −8.207027882632054, −7.934685570330809, −7.004346254568110, −6.844834517263042, −6.435399709787033, −5.661685261829489, −5.021936827465157, −4.538921610220708, −4.127136941129015, −3.574361531254277, −3.066769286865792, −2.326800096322473, −1.911761348904222, −1.133155097068498, −0.5452285488883666, 0.5452285488883666, 1.133155097068498, 1.911761348904222, 2.326800096322473, 3.066769286865792, 3.574361531254277, 4.127136941129015, 4.538921610220708, 5.021936827465157, 5.661685261829489, 6.435399709787033, 6.844834517263042, 7.004346254568110, 7.934685570330809, 8.207027882632054, 8.569811809051343, 9.083050511626016, 9.661324710153609, 9.957378178988006, 10.64021879875030, 10.98905846229921, 11.59768008118569, 11.98391638839073, 12.41790477551095, 12.74537968761906

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