Properties

Label 2-1080-1.1-c3-0-10
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 − 2.40·7-s − 17.3·11-s + 71.2·13-s − 111.·17-s − 86.2·19-s + 148.·23-s + 25·25-s − 71.1·29-s + 212.·31-s + 12.0·35-s − 132.·37-s + 29.6·41-s − 126.·43-s + 103.·47-s − 337.·49-s + 298.·53-s + 86.6·55-s − 893.·59-s + 752.·61-s − 356.·65-s + 477.·67-s + 58.7·71-s + 873.·73-s + 41.6·77-s + 819.·79-s + 959.·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.129·7-s − 0.474·11-s + 1.52·13-s − 1.58·17-s − 1.04·19-s + 1.34·23-s + 0.200·25-s − 0.455·29-s + 1.22·31-s + 0.0580·35-s − 0.590·37-s + 0.113·41-s − 0.449·43-s + 0.320·47-s − 0.983·49-s + 0.773·53-s + 0.212·55-s − 1.97·59-s + 1.57·61-s − 0.680·65-s + 0.870·67-s + 0.0981·71-s + 1.40·73-s + 0.0616·77-s + 1.16·79-s + 1.26·83-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

\(L(2)\) \(\approx\) \(1.601496291\)
\(L(\frac12)\) \(\approx\) \(1.601496291\)
\(L(\frac{5}{2})\) 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 \( 1 + 2.40T + 343T^{2} \)
11 \( 1 + 17.3T + 1.33e3T^{2} \)
13 \( 1 - 71.2T + 2.19e3T^{2} \)
17 \( 1 + 111.T + 4.91e3T^{2} \)
19 \( 1 + 86.2T + 6.85e3T^{2} \)
23 \( 1 - 148.T + 1.21e4T^{2} \)
29 \( 1 + 71.1T + 2.43e4T^{2} \)
31 \( 1 - 212.T + 2.97e4T^{2} \)
37 \( 1 + 132.T + 5.06e4T^{2} \)
41 \( 1 - 29.6T + 6.89e4T^{2} \)
43 \( 1 + 126.T + 7.95e4T^{2} \)
47 \( 1 - 103.T + 1.03e5T^{2} \)
53 \( 1 - 298.T + 1.48e5T^{2} \)
59 \( 1 + 893.T + 2.05e5T^{2} \)
61 \( 1 - 752.T + 2.26e5T^{2} \)
67 \( 1 - 477.T + 3.00e5T^{2} \)
71 \( 1 - 58.7T + 3.57e5T^{2} \)
73 \( 1 - 873.T + 3.89e5T^{2} \)
79 \( 1 - 819.T + 4.93e5T^{2} \)
83 \( 1 - 959.T + 5.71e5T^{2} \)
89 \( 1 - 406.T + 7.04e5T^{2} \)
97 \( 1 - 623.T + 9.12e5T^{2} \)
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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

−9.312715432532111529702028692990, −8.627373843852302311354781034310, −8.033772040827097540714613895120, −6.79649452922043283608233435485, −6.34361887533998246677225635459, −5.09186055269416203694297526297, −4.21183733859880638453836429849, −3.27218381271347477918585687449, −2.07815457721748687394197852869, −0.65516536917955004113726460931, 0.65516536917955004113726460931, 2.07815457721748687394197852869, 3.27218381271347477918585687449, 4.21183733859880638453836429849, 5.09186055269416203694297526297, 6.34361887533998246677225635459, 6.79649452922043283608233435485, 8.033772040827097540714613895120, 8.627373843852302311354781034310, 9.312715432532111529702028692990

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