Properties

Label 2-1080-1.1-c3-0-21
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 + 9.46·7-s + 49.0·11-s + 55.4·13-s − 27.9·17-s + 26.3·19-s + 9.53·23-s + 25·25-s + 218.·29-s − 158.·31-s + 47.3·35-s + 189.·37-s − 246.·41-s + 40.2·43-s − 213.·47-s − 253.·49-s − 283.·53-s + 245.·55-s − 92·59-s + 370.·61-s + 277.·65-s + 1.08e3·67-s + 333.·71-s − 606.·73-s + 464.·77-s − 44.3·79-s − 107.·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.511·7-s + 1.34·11-s + 1.18·13-s − 0.399·17-s + 0.318·19-s + 0.0864·23-s + 0.200·25-s + 1.39·29-s − 0.920·31-s + 0.228·35-s + 0.842·37-s − 0.937·41-s + 0.142·43-s − 0.664·47-s − 0.738·49-s − 0.734·53-s + 0.601·55-s − 0.203·59-s + 0.776·61-s + 0.528·65-s + 1.98·67-s + 0.557·71-s − 0.972·73-s + 0.687·77-s − 0.0631·79-s − 0.142·83-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

\(L(2)\) \(\approx\) \(3.003853978\)
\(L(\frac12)\) \(\approx\) \(3.003853978\)
\(L(\frac{5}{2})\) 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 \( 1 - 9.46T + 343T^{2} \)
11 \( 1 - 49.0T + 1.33e3T^{2} \)
13 \( 1 - 55.4T + 2.19e3T^{2} \)
17 \( 1 + 27.9T + 4.91e3T^{2} \)
19 \( 1 - 26.3T + 6.85e3T^{2} \)
23 \( 1 - 9.53T + 1.21e4T^{2} \)
29 \( 1 - 218.T + 2.43e4T^{2} \)
31 \( 1 + 158.T + 2.97e4T^{2} \)
37 \( 1 - 189.T + 5.06e4T^{2} \)
41 \( 1 + 246.T + 6.89e4T^{2} \)
43 \( 1 - 40.2T + 7.95e4T^{2} \)
47 \( 1 + 213.T + 1.03e5T^{2} \)
53 \( 1 + 283.T + 1.48e5T^{2} \)
59 \( 1 + 92T + 2.05e5T^{2} \)
61 \( 1 - 370.T + 2.26e5T^{2} \)
67 \( 1 - 1.08e3T + 3.00e5T^{2} \)
71 \( 1 - 333.T + 3.57e5T^{2} \)
73 \( 1 + 606.T + 3.89e5T^{2} \)
79 \( 1 + 44.3T + 4.93e5T^{2} \)
83 \( 1 + 107.T + 5.71e5T^{2} \)
89 \( 1 - 257.T + 7.04e5T^{2} \)
97 \( 1 - 1.31e3T + 9.12e5T^{2} \)
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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

−9.400941184565015510979159469290, −8.734278928703078546158023486228, −7.995159477919477882055771280876, −6.74915740501309006672081450449, −6.27689104271426081196683858938, −5.19677863524265578883424733608, −4.21839909669030073962912585538, −3.27860737584968292133941066063, −1.85855150732230445912457256292, −0.992124289239819836623910230322, 0.992124289239819836623910230322, 1.85855150732230445912457256292, 3.27860737584968292133941066063, 4.21839909669030073962912585538, 5.19677863524265578883424733608, 6.27689104271426081196683858938, 6.74915740501309006672081450449, 7.995159477919477882055771280876, 8.734278928703078546158023486228, 9.400941184565015510979159469290

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