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Properties

Label 2-180-180.139-c0-0-1
Degree $2$
Conductor $180$
Sign $0.766 - 0.642i$
Analytic cond. $0.0898317$
Root an. cond. $0.299719$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + 0.999·6-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s − 0.999·10-s + (0.499 + 0.866i)12-s + (0.499 − 0.866i)14-s + (0.499 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.499 − 0.866i)18-s + (−0.499 − 0.866i)20-s − 0.999·21-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + 0.999·6-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s − 0.999·10-s + (0.499 + 0.866i)12-s + (0.499 − 0.866i)14-s + (0.499 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.499 − 0.866i)18-s + (−0.499 − 0.866i)20-s − 0.999·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(0.0898317\)
Root analytic conductor: \(0.299719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :0),\ 0.766 - 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8169931246\)
\(L(\frac12)\) \(\approx\) \(0.8169931246\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−13.25266003726459056934228925380, −12.34233518978514723443508067396, −11.32059066798460248258732799065, −9.877856881384228442551797346298, −8.548958346639384954526037770529, −7.48473382694494253624140028139, −6.99564495359185224065559830264, −5.96797693585416330644713210577, −4.02505199161513633549089042307, −3.01554335708622184459064059850, 2.52796415098278775375283093093, 3.87921310680456161343166166782, 4.86178503697550767939988741389, 5.96541110648486450872095668393, 8.175583683876575953362057897275, 9.052683058898347171661999093782, 9.755306957471470115556646605831, 10.87563652432827529726951676893, 11.93483636824218202361140453035, 12.63618427465138715563032354413

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