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Properties

Label 2-180-12.11-c5-0-27
Degree $2$
Conductor $180$
Sign $-0.625 + 0.780i$
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  + (−3.87 − 4.11i)2-s + (−1.92 + 31.9i)4-s − 25i·5-s − 220. i·7-s + (139. − 115. i)8-s + (−102. + 96.9i)10-s + 530.·11-s + 536.·13-s + (−907. + 854. i)14-s + (−1.01e3 − 123. i)16-s + 2.15e3i·17-s − 741. i·19-s + (798. + 48.2i)20-s + (−2.05e3 − 2.18e3i)22-s + 2.37e3·23-s + ⋯
L(s)  = 1  + (−0.685 − 0.728i)2-s + (−0.0602 + 0.998i)4-s − 0.447i·5-s − 1.69i·7-s + (0.768 − 0.640i)8-s + (−0.325 + 0.306i)10-s + 1.32·11-s + 0.879·13-s + (−1.23 + 1.16i)14-s + (−0.992 − 0.120i)16-s + 1.80i·17-s − 0.471i·19-s + (0.446 + 0.0269i)20-s + (−0.906 − 0.962i)22-s + 0.938·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.625 + 0.780i$
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: \((2,\ 180,\ (\ :5/2),\ -0.625 + 0.780i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.444803564\)
\(L(\frac12)\) \(\approx\) \(1.444803564\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.87 + 4.11i)T \)
3 \( 1 \)
5 \( 1 + 25iT \)
good7 \( 1 + 220. iT - 1.68e4T^{2} \)
11 \( 1 - 530.T + 1.61e5T^{2} \)
13 \( 1 - 536.T + 3.71e5T^{2} \)
17 \( 1 - 2.15e3iT - 1.41e6T^{2} \)
19 \( 1 + 741. iT - 2.47e6T^{2} \)
23 \( 1 - 2.37e3T + 6.43e6T^{2} \)
29 \( 1 + 1.16e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.27e3iT - 2.86e7T^{2} \)
37 \( 1 - 9.87e3T + 6.93e7T^{2} \)
41 \( 1 + 2.07e4iT - 1.15e8T^{2} \)
43 \( 1 + 1.58e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.21e4T + 2.29e8T^{2} \)
53 \( 1 - 3.26e4iT - 4.18e8T^{2} \)
59 \( 1 - 4.89e3T + 7.14e8T^{2} \)
61 \( 1 + 2.04e4T + 8.44e8T^{2} \)
67 \( 1 + 6.40e4iT - 1.35e9T^{2} \)
71 \( 1 + 6.26e3T + 1.80e9T^{2} \)
73 \( 1 + 1.02e4T + 2.07e9T^{2} \)
79 \( 1 - 4.97e4iT - 3.07e9T^{2} \)
83 \( 1 - 6.45e4T + 3.93e9T^{2} \)
89 \( 1 + 6.53e3iT - 5.58e9T^{2} \)
97 \( 1 + 1.05e5T + 8.58e9T^{2} \)
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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

−11.08860664751712283375582852936, −10.60134393368980082967588363376, −9.433114243823205917756805936926, −8.545892351057595433242554764397, −7.46990950657964809320299513698, −6.40515351746543178470366638733, −4.22226679284088820139118130371, −3.70456034628504878170731755340, −1.55044391024195080406278479970, −0.68685051438740608505562777913, 1.31048885267992453936818229039, 2.89154934636837461580449242156, 4.91774192877248305364296163357, 6.07059883148706034338150031411, 6.79857323301603637791458348649, 8.190318310032999433259219227188, 9.118760008747769909955815633432, 9.655496944970582790032923409588, 11.27532850448286479921738427416, 11.72717657731574797841965016900

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