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Properties

Label 2-1080-9.4-c3-0-17
Degree $2$
Conductor $1080$
Sign $0.504 - 0.863i$
Analytic cond. $63.7220$
Root an. cond. $7.98261$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.5 − 4.33i)5-s + (16.9 + 29.2i)7-s + (12.0 + 20.8i)11-s + (4.52 − 7.83i)13-s + 59.6·17-s + 114.·19-s + (11.4 − 19.8i)23-s + (−12.5 − 21.6i)25-s + (−26.5 − 45.9i)29-s + (−55.5 + 96.2i)31-s + 169.·35-s − 303.·37-s + (174. − 302. i)41-s + (257. + 445. i)43-s + (−57.2 − 99.1i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (0.912 + 1.58i)7-s + (0.330 + 0.572i)11-s + (0.0964 − 0.167i)13-s + 0.850·17-s + 1.38·19-s + (0.103 − 0.179i)23-s + (−0.100 − 0.173i)25-s + (−0.169 − 0.294i)29-s + (−0.322 + 0.557i)31-s + 0.816·35-s − 1.34·37-s + (0.665 − 1.15i)41-s + (0.912 + 1.58i)43-s + (−0.177 − 0.307i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.504 - 0.863i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.504 - 0.863i$
Analytic conductor: \(63.7220\)
Root analytic conductor: \(7.98261\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :3/2),\ 0.504 - 0.863i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.691204081\)
\(L(\frac12)\) \(\approx\) \(2.691204081\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.5 + 4.33i)T \)
good7 \( 1 + (-16.9 - 29.2i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-12.0 - 20.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-4.52 + 7.83i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 59.6T + 4.91e3T^{2} \)
19 \( 1 - 114.T + 6.85e3T^{2} \)
23 \( 1 + (-11.4 + 19.8i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (26.5 + 45.9i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (55.5 - 96.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 303.T + 5.06e4T^{2} \)
41 \( 1 + (-174. + 302. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-257. - 445. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (57.2 + 99.1i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 296.T + 1.48e5T^{2} \)
59 \( 1 + (341. - 590. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (324. + 562. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (185. - 321. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 881.T + 3.57e5T^{2} \)
73 \( 1 + 545.T + 3.89e5T^{2} \)
79 \( 1 + (323. + 560. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (453. + 785. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 813.T + 7.04e5T^{2} \)
97 \( 1 + (-667. - 1.15e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.373283478611528653006079798989, −8.961836614324284038696278844384, −8.036476758130764852708617685127, −7.30248660761930451101590475234, −5.95821884274016334502024125897, −5.39115278172622972831017745690, −4.65428512228086595798176057031, −3.25347895741274569381214979945, −2.14092960071551631979770436035, −1.20211403675780708226863211099, 0.74839865723743780885002828413, 1.61211397728884793589311818428, 3.22466062411570383603266579415, 3.96053614074734946544490223996, 5.03710806755993910896219050461, 5.92447502290695342613121323408, 7.15608162725607389869283731776, 7.48358066980548457094203282305, 8.431584094030375707405022938655, 9.508837318611355038061568277390

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