Properties

Label 2-1080-1.1-c3-0-33
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s − 30.2·7-s + 67.3·11-s − 60.7·13-s + 61.1·17-s − 27.8·19-s − 40.4·23-s + 25·25-s + 212.·29-s + 167.·31-s − 151.·35-s − 366.·37-s − 363.·41-s + 153.·43-s − 434.·47-s + 570.·49-s + 79.6·53-s + 336.·55-s + 339.·59-s − 525.·61-s − 303.·65-s + 131.·67-s − 296.·71-s − 1.23e3·73-s − 2.03e3·77-s − 621.·79-s − 76.3·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.63·7-s + 1.84·11-s − 1.29·13-s + 0.872·17-s − 0.336·19-s − 0.366·23-s + 0.200·25-s + 1.35·29-s + 0.969·31-s − 0.729·35-s − 1.62·37-s − 1.38·41-s + 0.545·43-s − 1.34·47-s + 1.66·49-s + 0.206·53-s + 0.825·55-s + 0.748·59-s − 1.10·61-s − 0.580·65-s + 0.240·67-s − 0.495·71-s − 1.97·73-s − 3.01·77-s − 0.885·79-s − 0.100·83-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1080,\ (\ :3/2),\ -1)\)

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 \( 1 + 30.2T + 343T^{2} \)
11 \( 1 - 67.3T + 1.33e3T^{2} \)
13 \( 1 + 60.7T + 2.19e3T^{2} \)
17 \( 1 - 61.1T + 4.91e3T^{2} \)
19 \( 1 + 27.8T + 6.85e3T^{2} \)
23 \( 1 + 40.4T + 1.21e4T^{2} \)
29 \( 1 - 212.T + 2.43e4T^{2} \)
31 \( 1 - 167.T + 2.97e4T^{2} \)
37 \( 1 + 366.T + 5.06e4T^{2} \)
41 \( 1 + 363.T + 6.89e4T^{2} \)
43 \( 1 - 153.T + 7.95e4T^{2} \)
47 \( 1 + 434.T + 1.03e5T^{2} \)
53 \( 1 - 79.6T + 1.48e5T^{2} \)
59 \( 1 - 339.T + 2.05e5T^{2} \)
61 \( 1 + 525.T + 2.26e5T^{2} \)
67 \( 1 - 131.T + 3.00e5T^{2} \)
71 \( 1 + 296.T + 3.57e5T^{2} \)
73 \( 1 + 1.23e3T + 3.89e5T^{2} \)
79 \( 1 + 621.T + 4.93e5T^{2} \)
83 \( 1 + 76.3T + 5.71e5T^{2} \)
89 \( 1 + 192.T + 7.04e5T^{2} \)
97 \( 1 + 874.T + 9.12e5T^{2} \)
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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

−9.248306744627400999054419147543, −8.461115227342521391028209893421, −7.06398477621539594801221273145, −6.62656477442392427031115104198, −5.87865217134129348481622521468, −4.67647807200302959928775533320, −3.59741262856072851520863259276, −2.78189170324357191535722802232, −1.38965178304697007375410027181, 0, 1.38965178304697007375410027181, 2.78189170324357191535722802232, 3.59741262856072851520863259276, 4.67647807200302959928775533320, 5.87865217134129348481622521468, 6.62656477442392427031115104198, 7.06398477621539594801221273145, 8.461115227342521391028209893421, 9.248306744627400999054419147543

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