Properties

Label 2-1080-1.1-c3-0-9
Degree $2$
Conductor $1080$
Sign $1$
Analytic cond. $63.7220$
Root an. cond. $7.98261$
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 − 24.3·7-s + 26.8·11-s − 68.2·13-s + 61.8·17-s − 75.1·19-s + 43.3·23-s + 25·25-s − 174.·29-s + 222.·31-s − 121.·35-s + 67.1·37-s + 22.2·41-s − 84.7·43-s + 585.·47-s + 251.·49-s − 38.3·53-s + 134.·55-s − 92·59-s − 226.·61-s − 341.·65-s + 858.·67-s − 116.·71-s + 911.·73-s − 655.·77-s − 285.·79-s + 999.·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.31·7-s + 0.737·11-s − 1.45·13-s + 0.882·17-s − 0.907·19-s + 0.393·23-s + 0.200·25-s − 1.12·29-s + 1.29·31-s − 0.588·35-s + 0.298·37-s + 0.0845·41-s − 0.300·43-s + 1.81·47-s + 0.734·49-s − 0.0994·53-s + 0.329·55-s − 0.203·59-s − 0.474·61-s − 0.651·65-s + 1.56·67-s − 0.195·71-s + 1.46·73-s − 0.970·77-s − 0.406·79-s + 1.32·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $1$
Analytic conductor: \(63.7220\)
Root analytic conductor: \(7.98261\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.623527244\)
\(L(\frac12)\) \(\approx\) \(1.623527244\)
\(L(\frac{5}{2})\) 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 \( 1 + 24.3T + 343T^{2} \)
11 \( 1 - 26.8T + 1.33e3T^{2} \)
13 \( 1 + 68.2T + 2.19e3T^{2} \)
17 \( 1 - 61.8T + 4.91e3T^{2} \)
19 \( 1 + 75.1T + 6.85e3T^{2} \)
23 \( 1 - 43.3T + 1.21e4T^{2} \)
29 \( 1 + 174.T + 2.43e4T^{2} \)
31 \( 1 - 222.T + 2.97e4T^{2} \)
37 \( 1 - 67.1T + 5.06e4T^{2} \)
41 \( 1 - 22.2T + 6.89e4T^{2} \)
43 \( 1 + 84.7T + 7.95e4T^{2} \)
47 \( 1 - 585.T + 1.03e5T^{2} \)
53 \( 1 + 38.3T + 1.48e5T^{2} \)
59 \( 1 + 92T + 2.05e5T^{2} \)
61 \( 1 + 226.T + 2.26e5T^{2} \)
67 \( 1 - 858.T + 3.00e5T^{2} \)
71 \( 1 + 116.T + 3.57e5T^{2} \)
73 \( 1 - 911.T + 3.89e5T^{2} \)
79 \( 1 + 285.T + 4.93e5T^{2} \)
83 \( 1 - 999.T + 5.71e5T^{2} \)
89 \( 1 - 374.T + 7.04e5T^{2} \)
97 \( 1 + 227.T + 9.12e5T^{2} \)
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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

−9.634419074208269226033389174043, −8.915345840884309770810789723294, −7.73513346585348863211355120918, −6.86236280160104781324037033621, −6.22321186652569168025407021730, −5.29278563125356040144039520509, −4.17011298447007846164859932477, −3.13280884238573437061904901019, −2.18279923682525301531473074184, −0.64878227737191441963608796588, 0.64878227737191441963608796588, 2.18279923682525301531473074184, 3.13280884238573437061904901019, 4.17011298447007846164859932477, 5.29278563125356040144039520509, 6.22321186652569168025407021730, 6.86236280160104781324037033621, 7.73513346585348863211355120918, 8.915345840884309770810789723294, 9.634419074208269226033389174043

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