Properties

Label 2-1080-1.1-c3-0-42
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 + 24.7·7-s − 8.16·11-s − 46.7·13-s + 60.3·17-s − 111.·19-s + 36.9·23-s + 25·25-s + 33.2·29-s − 124.·31-s − 123.·35-s − 438.·37-s + 508.·41-s − 48.5·43-s − 248.·47-s + 271.·49-s − 320.·53-s + 40.8·55-s − 652.·59-s + 693.·61-s + 233.·65-s − 12.0·67-s + 1.16e3·71-s − 122.·73-s − 202.·77-s − 441.·79-s + 428.·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.33·7-s − 0.223·11-s − 0.997·13-s + 0.860·17-s − 1.34·19-s + 0.334·23-s + 0.200·25-s + 0.212·29-s − 0.720·31-s − 0.598·35-s − 1.94·37-s + 1.93·41-s − 0.172·43-s − 0.769·47-s + 0.791·49-s − 0.830·53-s + 0.100·55-s − 1.43·59-s + 1.45·61-s + 0.446·65-s − 0.0219·67-s + 1.95·71-s − 0.195·73-s − 0.299·77-s − 0.629·79-s + 0.566·83-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 \( 1 - 24.7T + 343T^{2} \)
11 \( 1 + 8.16T + 1.33e3T^{2} \)
13 \( 1 + 46.7T + 2.19e3T^{2} \)
17 \( 1 - 60.3T + 4.91e3T^{2} \)
19 \( 1 + 111.T + 6.85e3T^{2} \)
23 \( 1 - 36.9T + 1.21e4T^{2} \)
29 \( 1 - 33.2T + 2.43e4T^{2} \)
31 \( 1 + 124.T + 2.97e4T^{2} \)
37 \( 1 + 438.T + 5.06e4T^{2} \)
41 \( 1 - 508.T + 6.89e4T^{2} \)
43 \( 1 + 48.5T + 7.95e4T^{2} \)
47 \( 1 + 248.T + 1.03e5T^{2} \)
53 \( 1 + 320.T + 1.48e5T^{2} \)
59 \( 1 + 652.T + 2.05e5T^{2} \)
61 \( 1 - 693.T + 2.26e5T^{2} \)
67 \( 1 + 12.0T + 3.00e5T^{2} \)
71 \( 1 - 1.16e3T + 3.57e5T^{2} \)
73 \( 1 + 122.T + 3.89e5T^{2} \)
79 \( 1 + 441.T + 4.93e5T^{2} \)
83 \( 1 - 428.T + 5.71e5T^{2} \)
89 \( 1 + 1.54e3T + 7.04e5T^{2} \)
97 \( 1 + 500.T + 9.12e5T^{2} \)
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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

−8.958302060001968415123582014030, −8.131224180690229464624917330075, −7.61014114423984114972701984111, −6.69527127558629306174959252850, −5.40826933796911370235317525215, −4.79787542421900943883923591816, −3.85424453797724763450498610233, −2.54203006092817928821594324137, −1.47578136262274374908855908286, 0, 1.47578136262274374908855908286, 2.54203006092817928821594324137, 3.85424453797724763450498610233, 4.79787542421900943883923591816, 5.40826933796911370235317525215, 6.69527127558629306174959252850, 7.61014114423984114972701984111, 8.131224180690229464624917330075, 8.958302060001968415123582014030

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