Properties

Label 2-1080-1.1-c3-0-30
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 − 7.95·7-s + 37.1·11-s − 35.1·13-s + 99.1·17-s − 44.6·19-s − 102.·23-s + 25·25-s − 285.·29-s + 238.·31-s + 39.7·35-s + 339.·37-s + 423.·41-s + 144.·43-s − 418.·47-s − 279.·49-s − 186.·53-s − 185.·55-s − 293.·59-s − 701.·61-s + 175.·65-s − 292.·67-s − 738.·71-s + 453.·73-s − 295.·77-s + 892.·79-s + 66.4·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.429·7-s + 1.01·11-s − 0.749·13-s + 1.41·17-s − 0.539·19-s − 0.930·23-s + 0.200·25-s − 1.82·29-s + 1.38·31-s + 0.192·35-s + 1.50·37-s + 1.61·41-s + 0.511·43-s − 1.29·47-s − 0.815·49-s − 0.484·53-s − 0.455·55-s − 0.648·59-s − 1.47·61-s + 0.335·65-s − 0.533·67-s − 1.23·71-s + 0.726·73-s − 0.436·77-s + 1.27·79-s + 0.0878·83-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 \( 1 + 7.95T + 343T^{2} \)
11 \( 1 - 37.1T + 1.33e3T^{2} \)
13 \( 1 + 35.1T + 2.19e3T^{2} \)
17 \( 1 - 99.1T + 4.91e3T^{2} \)
19 \( 1 + 44.6T + 6.85e3T^{2} \)
23 \( 1 + 102.T + 1.21e4T^{2} \)
29 \( 1 + 285.T + 2.43e4T^{2} \)
31 \( 1 - 238.T + 2.97e4T^{2} \)
37 \( 1 - 339.T + 5.06e4T^{2} \)
41 \( 1 - 423.T + 6.89e4T^{2} \)
43 \( 1 - 144.T + 7.95e4T^{2} \)
47 \( 1 + 418.T + 1.03e5T^{2} \)
53 \( 1 + 186.T + 1.48e5T^{2} \)
59 \( 1 + 293.T + 2.05e5T^{2} \)
61 \( 1 + 701.T + 2.26e5T^{2} \)
67 \( 1 + 292.T + 3.00e5T^{2} \)
71 \( 1 + 738.T + 3.57e5T^{2} \)
73 \( 1 - 453.T + 3.89e5T^{2} \)
79 \( 1 - 892.T + 4.93e5T^{2} \)
83 \( 1 - 66.4T + 5.71e5T^{2} \)
89 \( 1 + 868.T + 7.04e5T^{2} \)
97 \( 1 - 112.T + 9.12e5T^{2} \)
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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

−9.407422353460296216390507586856, −8.037881906616622396619564626332, −7.60223600228903689059165489150, −6.47827803710107923763731921809, −5.81608789978201845783401510059, −4.55460112202936376234151788747, −3.77875790195348105766777160890, −2.74381412182205848640981571588, −1.34616444004392114695325295248, 0, 1.34616444004392114695325295248, 2.74381412182205848640981571588, 3.77875790195348105766777160890, 4.55460112202936376234151788747, 5.81608789978201845783401510059, 6.47827803710107923763731921809, 7.60223600228903689059165489150, 8.037881906616622396619564626332, 9.407422353460296216390507586856

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