Properties

Degree $2$
Conductor $12480$
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 + 9-s + 4·11-s − 13-s + 15-s − 6·17-s + 4·19-s − 8·23-s + 25-s − 27-s − 6·29-s + 8·31-s − 4·33-s + 10·37-s + 39-s − 6·41-s + 4·43-s − 45-s − 7·49-s + 6·51-s + 10·53-s − 4·55-s − 4·57-s + 4·59-s + 2·61-s + 65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 0.258·15-s − 1.45·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.696·33-s + 1.64·37-s + 0.160·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s − 49-s + 0.840·51-s + 1.37·53-s − 0.539·55-s − 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.124·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12480\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 13\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{12480} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 12480,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.252337852\)
\(L(\frac12)\) \(\approx\) \(1.252337852\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−16.22992780081376, −15.96664838716216, −15.07451027906615, −14.85653496975215, −13.97134405172880, −13.51030131724721, −12.90988036254917, −12.05479124604983, −11.72608741330487, −11.41775908372907, −10.62720585503268, −9.924591430132350, −9.428296971250434, −8.771551711242449, −8.047515709529025, −7.407423328989697, −6.744982850394037, −6.193484619658994, −5.607601555217530, −4.537028299549830, −4.278629670801758, −3.485781008330903, −2.446515071722422, −1.565722509155510, −0.5341218031293773, 0.5341218031293773, 1.565722509155510, 2.446515071722422, 3.485781008330903, 4.278629670801758, 4.537028299549830, 5.607601555217530, 6.193484619658994, 6.744982850394037, 7.407423328989697, 8.047515709529025, 8.771551711242449, 9.428296971250434, 9.924591430132350, 10.62720585503268, 11.41775908372907, 11.72608741330487, 12.05479124604983, 12.90988036254917, 13.51030131724721, 13.97134405172880, 14.85653496975215, 15.07451027906615, 15.96664838716216, 16.22992780081376

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