Properties

Label 2-2160-1.1-c3-0-9
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 − 24.4·7-s + 28.9·11-s − 65.3·13-s + 68.1·17-s − 104.·19-s + 154.·23-s + 25·25-s − 205.·29-s + 18.2·31-s + 122.·35-s − 337.·37-s − 195.·41-s − 334.·43-s + 5.00·47-s + 252.·49-s − 319.·53-s − 144.·55-s − 430.·59-s + 594.·61-s + 326.·65-s − 195.·67-s − 425.·71-s + 929.·73-s − 707.·77-s − 24.4·79-s + 545.·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.31·7-s + 0.794·11-s − 1.39·13-s + 0.972·17-s − 1.26·19-s + 1.40·23-s + 0.200·25-s − 1.31·29-s + 0.105·31-s + 0.589·35-s − 1.50·37-s − 0.746·41-s − 1.18·43-s + 0.0155·47-s + 0.735·49-s − 0.829·53-s − 0.355·55-s − 0.950·59-s + 1.24·61-s + 0.623·65-s − 0.357·67-s − 0.711·71-s + 1.48·73-s − 1.04·77-s − 0.0347·79-s + 0.721·83-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

\(L(2)\) \(\approx\) \(0.8409352236\)
\(L(\frac12)\) \(\approx\) \(0.8409352236\)
\(L(\frac{5}{2})\) 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 \( 1 + 24.4T + 343T^{2} \)
11 \( 1 - 28.9T + 1.33e3T^{2} \)
13 \( 1 + 65.3T + 2.19e3T^{2} \)
17 \( 1 - 68.1T + 4.91e3T^{2} \)
19 \( 1 + 104.T + 6.85e3T^{2} \)
23 \( 1 - 154.T + 1.21e4T^{2} \)
29 \( 1 + 205.T + 2.43e4T^{2} \)
31 \( 1 - 18.2T + 2.97e4T^{2} \)
37 \( 1 + 337.T + 5.06e4T^{2} \)
41 \( 1 + 195.T + 6.89e4T^{2} \)
43 \( 1 + 334.T + 7.95e4T^{2} \)
47 \( 1 - 5.00T + 1.03e5T^{2} \)
53 \( 1 + 319.T + 1.48e5T^{2} \)
59 \( 1 + 430.T + 2.05e5T^{2} \)
61 \( 1 - 594.T + 2.26e5T^{2} \)
67 \( 1 + 195.T + 3.00e5T^{2} \)
71 \( 1 + 425.T + 3.57e5T^{2} \)
73 \( 1 - 929.T + 3.89e5T^{2} \)
79 \( 1 + 24.4T + 4.93e5T^{2} \)
83 \( 1 - 545.T + 5.71e5T^{2} \)
89 \( 1 + 84.1T + 7.04e5T^{2} \)
97 \( 1 - 827.T + 9.12e5T^{2} \)
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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

−8.863019401797787895741063399410, −7.892774677245693912004497915990, −6.94393319936921917783122333924, −6.67132495402933254129537036523, −5.53699321859943522663305933679, −4.69416957685383916777915939491, −3.61944691246452682243252028938, −3.08150735387415345350785728250, −1.83643310731236234038105935501, −0.40034842746650343121253446444, 0.40034842746650343121253446444, 1.83643310731236234038105935501, 3.08150735387415345350785728250, 3.61944691246452682243252028938, 4.69416957685383916777915939491, 5.53699321859943522663305933679, 6.67132495402933254129537036523, 6.94393319936921917783122333924, 7.892774677245693912004497915990, 8.863019401797787895741063399410

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