Properties

Label 2-2160-1.1-c3-0-39
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 + 34·7-s + 48·11-s − 70·13-s − 27·17-s − 119·19-s + 51·23-s + 25·25-s − 30·29-s + 133·31-s + 170·35-s + 218·37-s + 156·41-s + 88·43-s + 516·47-s + 813·49-s − 639·53-s + 240·55-s + 654·59-s + 461·61-s − 350·65-s − 182·67-s − 900·71-s + 704·73-s + 1.63e3·77-s + 1.37e3·79-s − 915·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.83·7-s + 1.31·11-s − 1.49·13-s − 0.385·17-s − 1.43·19-s + 0.462·23-s + 1/5·25-s − 0.192·29-s + 0.770·31-s + 0.821·35-s + 0.968·37-s + 0.594·41-s + 0.312·43-s + 1.60·47-s + 2.37·49-s − 1.65·53-s + 0.588·55-s + 1.44·59-s + 0.967·61-s − 0.667·65-s − 0.331·67-s − 1.50·71-s + 1.12·73-s + 2.41·77-s + 1.95·79-s − 1.21·83-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

\(L(2)\) \(\approx\) \(3.290423378\)
\(L(\frac12)\) \(\approx\) \(3.290423378\)
\(L(\frac{5}{2})\) 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 \( 1 - 34 T + p^{3} T^{2} \)
11 \( 1 - 48 T + p^{3} T^{2} \)
13 \( 1 + 70 T + p^{3} T^{2} \)
17 \( 1 + 27 T + p^{3} T^{2} \)
19 \( 1 + 119 T + p^{3} T^{2} \)
23 \( 1 - 51 T + p^{3} T^{2} \)
29 \( 1 + 30 T + p^{3} T^{2} \)
31 \( 1 - 133 T + p^{3} T^{2} \)
37 \( 1 - 218 T + p^{3} T^{2} \)
41 \( 1 - 156 T + p^{3} T^{2} \)
43 \( 1 - 88 T + p^{3} T^{2} \)
47 \( 1 - 516 T + p^{3} T^{2} \)
53 \( 1 + 639 T + p^{3} T^{2} \)
59 \( 1 - 654 T + p^{3} T^{2} \)
61 \( 1 - 461 T + p^{3} T^{2} \)
67 \( 1 + 182 T + p^{3} T^{2} \)
71 \( 1 + 900 T + p^{3} T^{2} \)
73 \( 1 - 704 T + p^{3} T^{2} \)
79 \( 1 - 1375 T + p^{3} T^{2} \)
83 \( 1 + 915 T + p^{3} T^{2} \)
89 \( 1 + 1116 T + p^{3} T^{2} \)
97 \( 1 + 16 T + p^{3} T^{2} \)
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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

−8.742447437581169516040276495903, −7.982998979768060653871640791432, −7.20538982798364093098368932434, −6.41531091827097797623877816479, −5.45504609680205565212839012156, −4.57117138062820557170739171584, −4.19614472394475229459152879871, −2.51878556701793052305078865888, −1.88145669138915693141323995422, −0.853516504137245965233691057524, 0.853516504137245965233691057524, 1.88145669138915693141323995422, 2.51878556701793052305078865888, 4.19614472394475229459152879871, 4.57117138062820557170739171584, 5.45504609680205565212839012156, 6.41531091827097797623877816479, 7.20538982798364093098368932434, 7.982998979768060653871640791432, 8.742447437581169516040276495903

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