Properties

Label 2-1080-1.1-c1-0-7
Degree $2$
Conductor $1080$
Sign $1$
Analytic cond. $8.62384$
Root an. cond. $2.93663$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2·7-s + 11-s + 13-s − 17-s + 4·19-s + 23-s + 25-s − 5·29-s + 31-s + 2·35-s + 6·37-s + 7·43-s + 7·47-s − 3·49-s − 12·53-s + 55-s − 4·59-s + 10·61-s + 65-s − 4·67-s + 12·71-s + 6·73-s + 2·77-s + 15·79-s + 2·83-s − 85-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s + 0.301·11-s + 0.277·13-s − 0.242·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.928·29-s + 0.179·31-s + 0.338·35-s + 0.986·37-s + 1.06·43-s + 1.02·47-s − 3/7·49-s − 1.64·53-s + 0.134·55-s − 0.520·59-s + 1.28·61-s + 0.124·65-s − 0.488·67-s + 1.42·71-s + 0.702·73-s + 0.227·77-s + 1.68·79-s + 0.219·83-s − 0.108·85-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $1$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.998267417\)
\(L(\frac12)\) \(\approx\) \(1.998267417\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−9.694646206897372706324681250720, −9.201687504911266492925537450023, −8.182813863608274578505426423650, −7.47257375177341578947598789177, −6.45999712647659261274458561586, −5.57949756423079328703738239574, −4.73906236640337502706427348642, −3.66996181321809556284576522508, −2.38970903369427825261950932994, −1.19988789118915244209474093338, 1.19988789118915244209474093338, 2.38970903369427825261950932994, 3.66996181321809556284576522508, 4.73906236640337502706427348642, 5.57949756423079328703738239574, 6.45999712647659261274458561586, 7.47257375177341578947598789177, 8.182813863608274578505426423650, 9.201687504911266492925537450023, 9.694646206897372706324681250720

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