Properties

Label 2-2160-1.1-c3-0-33
Degree $2$
Conductor $2160$
Sign $1$
Analytic cond. $127.444$
Root an. cond. $11.2891$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s + 22·7-s + 12·11-s + 38·13-s − 105·17-s + 157·19-s + 117·23-s + 25·25-s + 66·29-s + 25·31-s − 110·35-s + 314·37-s − 504·41-s − 380·43-s + 252·47-s + 141·49-s + 3·53-s − 60·55-s + 318·59-s + 293·61-s − 190·65-s + 322·67-s + 120·71-s + 44·73-s + 264·77-s − 917·79-s − 309·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.18·7-s + 0.328·11-s + 0.810·13-s − 1.49·17-s + 1.89·19-s + 1.06·23-s + 1/5·25-s + 0.422·29-s + 0.144·31-s − 0.531·35-s + 1.39·37-s − 1.91·41-s − 1.34·43-s + 0.782·47-s + 0.411·49-s + 0.00777·53-s − 0.147·55-s + 0.701·59-s + 0.614·61-s − 0.362·65-s + 0.587·67-s + 0.200·71-s + 0.0705·73-s + 0.390·77-s − 1.30·79-s − 0.408·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $1$
Analytic conductor: \(127.444\)
Root analytic conductor: \(11.2891\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.746351556\)
\(L(\frac12)\) \(\approx\) \(2.746351556\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
good7 \( 1 - 22 T + p^{3} T^{2} \)
11 \( 1 - 12 T + p^{3} T^{2} \)
13 \( 1 - 38 T + p^{3} T^{2} \)
17 \( 1 + 105 T + p^{3} T^{2} \)
19 \( 1 - 157 T + p^{3} T^{2} \)
23 \( 1 - 117 T + p^{3} T^{2} \)
29 \( 1 - 66 T + p^{3} T^{2} \)
31 \( 1 - 25 T + p^{3} T^{2} \)
37 \( 1 - 314 T + p^{3} T^{2} \)
41 \( 1 + 504 T + p^{3} T^{2} \)
43 \( 1 + 380 T + p^{3} T^{2} \)
47 \( 1 - 252 T + p^{3} T^{2} \)
53 \( 1 - 3 T + p^{3} T^{2} \)
59 \( 1 - 318 T + p^{3} T^{2} \)
61 \( 1 - 293 T + p^{3} T^{2} \)
67 \( 1 - 322 T + p^{3} T^{2} \)
71 \( 1 - 120 T + p^{3} T^{2} \)
73 \( 1 - 44 T + p^{3} T^{2} \)
79 \( 1 + 917 T + p^{3} T^{2} \)
83 \( 1 + 309 T + p^{3} T^{2} \)
89 \( 1 - 1272 T + p^{3} T^{2} \)
97 \( 1 - 1328 T + p^{3} T^{2} \)
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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

−8.601591150513406660143105609660, −8.073984795241922239785229248594, −7.16796282077513354834222171152, −6.53246131072541104879264250647, −5.34185839777789793165124437247, −4.76298043839384812564481024238, −3.87223186896078592539082835511, −2.90333938493657302144748009078, −1.65853882783165728899243122584, −0.805261026022207019403575232192, 0.805261026022207019403575232192, 1.65853882783165728899243122584, 2.90333938493657302144748009078, 3.87223186896078592539082835511, 4.76298043839384812564481024238, 5.34185839777789793165124437247, 6.53246131072541104879264250647, 7.16796282077513354834222171152, 8.073984795241922239785229248594, 8.601591150513406660143105609660

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