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Properties

Label 2-180-12.11-c5-0-12
Degree $2$
Conductor $180$
Sign $-0.664 - 0.747i$
Analytic cond. $28.8690$
Root an. cond. $5.37299$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.61 + 0.646i)2-s + (31.1 + 7.26i)4-s + 25i·5-s + 234. i·7-s + (170. + 60.9i)8-s + (−16.1 + 140. i)10-s − 267.·11-s − 1.06e3·13-s + (−151. + 1.31e3i)14-s + (918. + 452. i)16-s − 664. i·17-s − 1.28e3i·19-s + (−181. + 779. i)20-s + (−1.50e3 − 172. i)22-s − 1.25e3·23-s + ⋯
L(s)  = 1  + (0.993 + 0.114i)2-s + (0.973 + 0.226i)4-s + 0.447i·5-s + 1.80i·7-s + (0.941 + 0.336i)8-s + (−0.0510 + 0.444i)10-s − 0.665·11-s − 1.74·13-s + (−0.206 + 1.79i)14-s + (0.897 + 0.441i)16-s − 0.557i·17-s − 0.818i·19-s + (−0.101 + 0.435i)20-s + (−0.661 − 0.0760i)22-s − 0.492·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.664 - 0.747i$
Analytic conductor: \(28.8690\)
Root analytic conductor: \(5.37299\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :5/2),\ -0.664 - 0.747i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.749098448\)
\(L(\frac12)\) \(\approx\) \(2.749098448\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.61 - 0.646i)T \)
3 \( 1 \)
5 \( 1 - 25iT \)
good7 \( 1 - 234. iT - 1.68e4T^{2} \)
11 \( 1 + 267.T + 1.61e5T^{2} \)
13 \( 1 + 1.06e3T + 3.71e5T^{2} \)
17 \( 1 + 664. iT - 1.41e6T^{2} \)
19 \( 1 + 1.28e3iT - 2.47e6T^{2} \)
23 \( 1 + 1.25e3T + 6.43e6T^{2} \)
29 \( 1 - 3.16e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.31e3iT - 2.86e7T^{2} \)
37 \( 1 - 7.45e3T + 6.93e7T^{2} \)
41 \( 1 - 1.68e4iT - 1.15e8T^{2} \)
43 \( 1 + 9.06e3iT - 1.47e8T^{2} \)
47 \( 1 - 2.47e4T + 2.29e8T^{2} \)
53 \( 1 - 9.85e3iT - 4.18e8T^{2} \)
59 \( 1 - 3.69e4T + 7.14e8T^{2} \)
61 \( 1 + 9.08e3T + 8.44e8T^{2} \)
67 \( 1 + 1.05e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.12e4T + 1.80e9T^{2} \)
73 \( 1 + 8.41e4T + 2.07e9T^{2} \)
79 \( 1 - 1.84e4iT - 3.07e9T^{2} \)
83 \( 1 - 4.86e4T + 3.93e9T^{2} \)
89 \( 1 - 1.07e5iT - 5.58e9T^{2} \)
97 \( 1 - 1.62e5T + 8.58e9T^{2} \)
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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

−12.18786462089879091732494623637, −11.55162004971045464890036765564, −10.31311017392561142708666537084, −9.110936643904827850061265407046, −7.77796822677736009428563205554, −6.74615518509125028021875728411, −5.51506606804201264710056693803, −4.83878878061731209455261660648, −2.86675525592129454281361201849, −2.34596424193647920006262683948, 0.56622259929484830563270529443, 2.20633878333229780789099433671, 3.86871355653809587777317769919, 4.62089281040208406522893222502, 5.88024990571960586606665258763, 7.31271384584122654003773332828, 7.79881244108149253114503174727, 9.921104103719466002771737519727, 10.38554417584028942277093061684, 11.57623723693854235275511482686

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