Properties

Degree $2$
Conductor $66240$
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 + 4·11-s + 2·13-s + 6·17-s − 4·19-s + 23-s + 25-s − 2·29-s + 2·37-s − 10·41-s + 4·43-s − 7·49-s + 6·53-s + 4·55-s − 4·59-s + 10·61-s + 2·65-s + 12·67-s + 8·71-s + 10·73-s − 8·79-s − 4·83-s + 6·85-s − 18·89-s − 4·95-s + 2·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.20·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.371·29-s + 0.328·37-s − 1.56·41-s + 0.609·43-s − 49-s + 0.824·53-s + 0.539·55-s − 0.520·59-s + 1.28·61-s + 0.248·65-s + 1.46·67-s + 0.949·71-s + 1.17·73-s − 0.900·79-s − 0.439·83-s + 0.650·85-s − 1.90·89-s − 0.410·95-s + 0.203·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66240\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{66240} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 66240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.421618764\)
\(L(\frac12)\) \(\approx\) \(3.421618764\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
23 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.07409662314174, −13.95195102064758, −13.06741364908826, −12.79031625843100, −12.19281851159892, −11.70230373798662, −11.16619227685546, −10.73149036047558, −9.922172993062462, −9.767126257906991, −9.144982716760192, −8.400698479332227, −8.299663908072936, −7.390219182491916, −6.803104363374411, −6.425726569788953, −5.779218127007744, −5.337675442833463, −4.625735881974783, −3.843555884365168, −3.566271101559334, −2.754983911049880, −1.924977185902809, −1.366143028772428, −0.6571973606871682, 0.6571973606871682, 1.366143028772428, 1.924977185902809, 2.754983911049880, 3.566271101559334, 3.843555884365168, 4.625735881974783, 5.337675442833463, 5.779218127007744, 6.425726569788953, 6.803104363374411, 7.390219182491916, 8.299663908072936, 8.400698479332227, 9.144982716760192, 9.767126257906991, 9.922172993062462, 10.73149036047558, 11.16619227685546, 11.70230373798662, 12.19281851159892, 12.79031625843100, 13.06741364908826, 13.95195102064758, 14.07409662314174

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