Properties

Label 2-1080-1.1-c3-0-36
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 − 19.1·7-s + 18.3·11-s + 14.0·13-s − 63.8·17-s + 101.·19-s − 155.·23-s + 25·25-s + 58.2·29-s + 136.·31-s − 95.9·35-s − 117.·37-s + 276.·41-s − 451.·43-s + 15.2·47-s + 25.0·49-s + 266.·53-s + 91.9·55-s − 585.·59-s − 392.·61-s + 70.4·65-s − 1.03e3·67-s + 657.·71-s + 125.·73-s − 352.·77-s − 962.·79-s − 721.·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.03·7-s + 0.503·11-s + 0.300·13-s − 0.911·17-s + 1.22·19-s − 1.40·23-s + 0.200·25-s + 0.372·29-s + 0.790·31-s − 0.463·35-s − 0.523·37-s + 1.05·41-s − 1.60·43-s + 0.0474·47-s + 0.0729·49-s + 0.691·53-s + 0.225·55-s − 1.29·59-s − 0.823·61-s + 0.134·65-s − 1.88·67-s + 1.09·71-s + 0.201·73-s − 0.522·77-s − 1.37·79-s − 0.953·83-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 \( 1 + 19.1T + 343T^{2} \)
11 \( 1 - 18.3T + 1.33e3T^{2} \)
13 \( 1 - 14.0T + 2.19e3T^{2} \)
17 \( 1 + 63.8T + 4.91e3T^{2} \)
19 \( 1 - 101.T + 6.85e3T^{2} \)
23 \( 1 + 155.T + 1.21e4T^{2} \)
29 \( 1 - 58.2T + 2.43e4T^{2} \)
31 \( 1 - 136.T + 2.97e4T^{2} \)
37 \( 1 + 117.T + 5.06e4T^{2} \)
41 \( 1 - 276.T + 6.89e4T^{2} \)
43 \( 1 + 451.T + 7.95e4T^{2} \)
47 \( 1 - 15.2T + 1.03e5T^{2} \)
53 \( 1 - 266.T + 1.48e5T^{2} \)
59 \( 1 + 585.T + 2.05e5T^{2} \)
61 \( 1 + 392.T + 2.26e5T^{2} \)
67 \( 1 + 1.03e3T + 3.00e5T^{2} \)
71 \( 1 - 657.T + 3.57e5T^{2} \)
73 \( 1 - 125.T + 3.89e5T^{2} \)
79 \( 1 + 962.T + 4.93e5T^{2} \)
83 \( 1 + 721.T + 5.71e5T^{2} \)
89 \( 1 + 122.T + 7.04e5T^{2} \)
97 \( 1 - 153.T + 9.12e5T^{2} \)
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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

−9.231789095628073223867389409261, −8.378880810674521467222831904237, −7.29787297828388830971350481095, −6.41928275435902914827362905465, −5.90637681931119857788960111012, −4.69041698136533586428660614100, −3.64841282267826420412917611486, −2.70724192828852963877299952642, −1.42832356651378652296097868180, 0, 1.42832356651378652296097868180, 2.70724192828852963877299952642, 3.64841282267826420412917611486, 4.69041698136533586428660614100, 5.90637681931119857788960111012, 6.41928275435902914827362905465, 7.29787297828388830971350481095, 8.378880810674521467222831904237, 9.231789095628073223867389409261

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