Properties

Label 2-2160-1.1-c3-0-16
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s − 35.5·7-s + 44.7·11-s + 77.8·13-s − 120.·17-s − 121.·19-s − 152.·23-s + 25·25-s + 187.·29-s + 161.·31-s − 177.·35-s + 13.3·37-s + 188.·41-s + 81.5·43-s − 48.6·47-s + 923.·49-s − 707.·53-s + 223.·55-s − 16.4·59-s − 743.·61-s + 389.·65-s + 59.2·67-s − 144.·71-s + 657.·73-s − 1.59e3·77-s + 454.·79-s + 165.·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.92·7-s + 1.22·11-s + 1.66·13-s − 1.71·17-s − 1.46·19-s − 1.38·23-s + 0.200·25-s + 1.20·29-s + 0.934·31-s − 0.859·35-s + 0.0592·37-s + 0.716·41-s + 0.289·43-s − 0.150·47-s + 2.69·49-s − 1.83·53-s + 0.548·55-s − 0.0363·59-s − 1.56·61-s + 0.742·65-s + 0.108·67-s − 0.241·71-s + 1.05·73-s − 2.35·77-s + 0.646·79-s + 0.218·83-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: \(0\)
Selberg data: \((2,\ 2160,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.594192109\)
\(L(\frac12)\) \(\approx\) \(1.594192109\)
\(L(\frac{5}{2})\) 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 \( 1 + 35.5T + 343T^{2} \)
11 \( 1 - 44.7T + 1.33e3T^{2} \)
13 \( 1 - 77.8T + 2.19e3T^{2} \)
17 \( 1 + 120.T + 4.91e3T^{2} \)
19 \( 1 + 121.T + 6.85e3T^{2} \)
23 \( 1 + 152.T + 1.21e4T^{2} \)
29 \( 1 - 187.T + 2.43e4T^{2} \)
31 \( 1 - 161.T + 2.97e4T^{2} \)
37 \( 1 - 13.3T + 5.06e4T^{2} \)
41 \( 1 - 188.T + 6.89e4T^{2} \)
43 \( 1 - 81.5T + 7.95e4T^{2} \)
47 \( 1 + 48.6T + 1.03e5T^{2} \)
53 \( 1 + 707.T + 1.48e5T^{2} \)
59 \( 1 + 16.4T + 2.05e5T^{2} \)
61 \( 1 + 743.T + 2.26e5T^{2} \)
67 \( 1 - 59.2T + 3.00e5T^{2} \)
71 \( 1 + 144.T + 3.57e5T^{2} \)
73 \( 1 - 657.T + 3.89e5T^{2} \)
79 \( 1 - 454.T + 4.93e5T^{2} \)
83 \( 1 - 165.T + 5.71e5T^{2} \)
89 \( 1 - 535.T + 7.04e5T^{2} \)
97 \( 1 + 436.T + 9.12e5T^{2} \)
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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

−8.995443231769473771126914036239, −8.141806120930525218885359744977, −6.62464073526742028746026769065, −6.32110421439244311670709599002, −6.17470739182867370858944161155, −4.39497490970996373980727478713, −3.87416039637715735620895897879, −2.90053962886287912153544307598, −1.85017929530678248274301471379, −0.56023213289141030148195971025, 0.56023213289141030148195971025, 1.85017929530678248274301471379, 2.90053962886287912153544307598, 3.87416039637715735620895897879, 4.39497490970996373980727478713, 6.17470739182867370858944161155, 6.32110421439244311670709599002, 6.62464073526742028746026769065, 8.141806120930525218885359744977, 8.995443231769473771126914036239

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