Properties

Label 2-2160-1.1-c3-0-78
Degree $2$
Conductor $2160$
Sign $-1$
Analytic cond. $127.444$
Root an. cond. $11.2891$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s + 30.4·7-s − 37.9·11-s − 43.3·13-s + 29.8·17-s + 28.6·19-s + 51.2·23-s + 25·25-s − 178.·29-s + 237.·31-s − 152.·35-s − 12.7·37-s − 222.·41-s − 464.·43-s + 400.·47-s + 586.·49-s − 249.·53-s + 189.·55-s − 779.·59-s + 609.·61-s + 216.·65-s + 172.·67-s − 1.04e3·71-s − 38.6·73-s − 1.15e3·77-s − 1.19e3·79-s + 150.·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.64·7-s − 1.04·11-s − 0.925·13-s + 0.425·17-s + 0.345·19-s + 0.464·23-s + 0.200·25-s − 1.14·29-s + 1.37·31-s − 0.736·35-s − 0.0564·37-s − 0.848·41-s − 1.64·43-s + 1.24·47-s + 1.71·49-s − 0.646·53-s + 0.465·55-s − 1.71·59-s + 1.28·61-s + 0.414·65-s + 0.315·67-s − 1.74·71-s − 0.0619·73-s − 1.71·77-s − 1.70·79-s + 0.199·83-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: no
Arithmetic: yes
Character: $\chi_{2160} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2160,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 5T \)
good7 \( 1 - 30.4T + 343T^{2} \)
11 \( 1 + 37.9T + 1.33e3T^{2} \)
13 \( 1 + 43.3T + 2.19e3T^{2} \)
17 \( 1 - 29.8T + 4.91e3T^{2} \)
19 \( 1 - 28.6T + 6.85e3T^{2} \)
23 \( 1 - 51.2T + 1.21e4T^{2} \)
29 \( 1 + 178.T + 2.43e4T^{2} \)
31 \( 1 - 237.T + 2.97e4T^{2} \)
37 \( 1 + 12.7T + 5.06e4T^{2} \)
41 \( 1 + 222.T + 6.89e4T^{2} \)
43 \( 1 + 464.T + 7.95e4T^{2} \)
47 \( 1 - 400.T + 1.03e5T^{2} \)
53 \( 1 + 249.T + 1.48e5T^{2} \)
59 \( 1 + 779.T + 2.05e5T^{2} \)
61 \( 1 - 609.T + 2.26e5T^{2} \)
67 \( 1 - 172.T + 3.00e5T^{2} \)
71 \( 1 + 1.04e3T + 3.57e5T^{2} \)
73 \( 1 + 38.6T + 3.89e5T^{2} \)
79 \( 1 + 1.19e3T + 4.93e5T^{2} \)
83 \( 1 - 150.T + 5.71e5T^{2} \)
89 \( 1 + 119.T + 7.04e5T^{2} \)
97 \( 1 - 1.26e3T + 9.12e5T^{2} \)
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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

−8.164469161423018861752560007304, −7.67047225044445807845785818649, −7.08050854175346862095510645867, −5.73537117788599248122141209537, −4.96926929817038081508935351691, −4.56304193048878833325781966818, −3.28231741178723094787648609750, −2.29964731430367246191271985723, −1.29226625441494257434469020694, 0, 1.29226625441494257434469020694, 2.29964731430367246191271985723, 3.28231741178723094787648609750, 4.56304193048878833325781966818, 4.96926929817038081508935351691, 5.73537117788599248122141209537, 7.08050854175346862095510645867, 7.67047225044445807845785818649, 8.164469161423018861752560007304

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