Properties

Label 2-2160-1.1-c3-0-69
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 − 29.9·7-s + 15.9·11-s + 41.9·13-s + 36.9·17-s + 10.9·19-s − 114.·23-s + 25·25-s − 145.·29-s + 46.4·31-s − 149.·35-s + 74·37-s − 335.·41-s − 84.4·43-s + 255.·47-s + 552.·49-s + 399.·53-s + 79.5·55-s + 2.56·59-s + 568.·61-s + 209.·65-s − 769.·67-s + 441.·71-s + 966.·73-s − 476.·77-s − 503.·79-s − 1.27e3·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.61·7-s + 0.436·11-s + 0.894·13-s + 0.526·17-s + 0.131·19-s − 1.03·23-s + 0.200·25-s − 0.932·29-s + 0.269·31-s − 0.722·35-s + 0.328·37-s − 1.27·41-s − 0.299·43-s + 0.792·47-s + 1.60·49-s + 1.03·53-s + 0.195·55-s + 0.00565·59-s + 1.19·61-s + 0.399·65-s − 1.40·67-s + 0.738·71-s + 1.54·73-s − 0.704·77-s − 0.716·79-s − 1.68·83-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: Trivial
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 \( 1 + 29.9T + 343T^{2} \)
11 \( 1 - 15.9T + 1.33e3T^{2} \)
13 \( 1 - 41.9T + 2.19e3T^{2} \)
17 \( 1 - 36.9T + 4.91e3T^{2} \)
19 \( 1 - 10.9T + 6.85e3T^{2} \)
23 \( 1 + 114.T + 1.21e4T^{2} \)
29 \( 1 + 145.T + 2.43e4T^{2} \)
31 \( 1 - 46.4T + 2.97e4T^{2} \)
37 \( 1 - 74T + 5.06e4T^{2} \)
41 \( 1 + 335.T + 6.89e4T^{2} \)
43 \( 1 + 84.4T + 7.95e4T^{2} \)
47 \( 1 - 255.T + 1.03e5T^{2} \)
53 \( 1 - 399.T + 1.48e5T^{2} \)
59 \( 1 - 2.56T + 2.05e5T^{2} \)
61 \( 1 - 568.T + 2.26e5T^{2} \)
67 \( 1 + 769.T + 3.00e5T^{2} \)
71 \( 1 - 441.T + 3.57e5T^{2} \)
73 \( 1 - 966.T + 3.89e5T^{2} \)
79 \( 1 + 503.T + 4.93e5T^{2} \)
83 \( 1 + 1.27e3T + 5.71e5T^{2} \)
89 \( 1 - 0.240T + 7.04e5T^{2} \)
97 \( 1 + 6.71T + 9.12e5T^{2} \)
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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

−8.505850086077686099154693266962, −7.44598604184156958853828082463, −6.59783665235281406461533407623, −6.08088216311097203108533086639, −5.37565806150177925520242694434, −3.96130688351542645903293794539, −3.45948866939964856929130822098, −2.41222090958923483756220530516, −1.20153187780864716376027545647, 0, 1.20153187780864716376027545647, 2.41222090958923483756220530516, 3.45948866939964856929130822098, 3.96130688351542645903293794539, 5.37565806150177925520242694434, 6.08088216311097203108533086639, 6.59783665235281406461533407623, 7.44598604184156958853828082463, 8.505850086077686099154693266962

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