Properties

Label 2-1080-1.1-c5-0-39
Degree $2$
Conductor $1080$
Sign $-1$
Analytic cond. $173.214$
Root an. cond. $13.1610$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 25·5-s − 234·7-s − 347·11-s − 33·13-s − 237·17-s + 1.49e3·19-s + 2.81e3·23-s + 625·25-s + 5.51e3·29-s + 2.91e3·31-s + 5.85e3·35-s + 5.60e3·37-s − 4.71e3·41-s + 1.04e4·43-s − 5.96e3·47-s + 3.79e4·49-s + 1.79e4·53-s + 8.67e3·55-s − 3.03e4·59-s − 3.55e4·61-s + 825·65-s − 1.24e4·67-s − 7.52e3·71-s + 3.63e4·73-s + 8.11e4·77-s − 2.27e4·79-s − 4.62e4·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.80·7-s − 0.864·11-s − 0.0541·13-s − 0.198·17-s + 0.950·19-s + 1.10·23-s + 1/5·25-s + 1.21·29-s + 0.544·31-s + 0.807·35-s + 0.672·37-s − 0.438·41-s + 0.864·43-s − 0.393·47-s + 2.25·49-s + 0.878·53-s + 0.386·55-s − 1.13·59-s − 1.22·61-s + 0.0242·65-s − 0.339·67-s − 0.177·71-s + 0.798·73-s + 1.56·77-s − 0.409·79-s − 0.736·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-1$
Analytic conductor: \(173.214\)
Root analytic conductor: \(13.1610\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1080,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + p^{2} T \)
good7 \( 1 + 234 T + p^{5} T^{2} \)
11 \( 1 + 347 T + p^{5} T^{2} \)
13 \( 1 + 33 T + p^{5} T^{2} \)
17 \( 1 + 237 T + p^{5} T^{2} \)
19 \( 1 - 1496 T + p^{5} T^{2} \)
23 \( 1 - 2811 T + p^{5} T^{2} \)
29 \( 1 - 5513 T + p^{5} T^{2} \)
31 \( 1 - 2911 T + p^{5} T^{2} \)
37 \( 1 - 5602 T + p^{5} T^{2} \)
41 \( 1 + 4716 T + p^{5} T^{2} \)
43 \( 1 - 10479 T + p^{5} T^{2} \)
47 \( 1 + 5963 T + p^{5} T^{2} \)
53 \( 1 - 17964 T + p^{5} T^{2} \)
59 \( 1 + 30372 T + p^{5} T^{2} \)
61 \( 1 + 35530 T + p^{5} T^{2} \)
67 \( 1 + 12476 T + p^{5} T^{2} \)
71 \( 1 + 7520 T + p^{5} T^{2} \)
73 \( 1 - 36378 T + p^{5} T^{2} \)
79 \( 1 + 22727 T + p^{5} T^{2} \)
83 \( 1 + 46254 T + p^{5} T^{2} \)
89 \( 1 - 58832 T + p^{5} T^{2} \)
97 \( 1 + 145906 T + p^{5} T^{2} \)
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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

−8.853218869444449899469013893927, −7.82376024050458221218317361550, −7.02330401256223642574302852309, −6.31294749050252843708013545393, −5.35086728361804737599079760369, −4.30668784068042087692575527875, −3.13056088078942728070718450126, −2.75801093001353744115640120373, −0.932335653233575684596226576497, 0, 0.932335653233575684596226576497, 2.75801093001353744115640120373, 3.13056088078942728070718450126, 4.30668784068042087692575527875, 5.35086728361804737599079760369, 6.31294749050252843708013545393, 7.02330401256223642574302852309, 7.82376024050458221218317361550, 8.853218869444449899469013893927

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