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Properties

Label 2-180-180.139-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.4308082118\)
\(L(\frac12)\) \(\approx\) \(0.4308082118\)
\(L(1)\) 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

−12.42240255709205703010474074280, −11.81595518153823601415363561867, −10.91868286944090228996622822922, −10.36180860361938804868043823832, −9.144102951616797963774964463606, −8.310140351924579553413081198973, −6.85589382162551048678061698581, −5.24204552749825469170490668341, −3.96139403068010698317325041094, −2.69175475848050787785423695874, 1.22605061796298853307315365905, 4.43443877732487047943431891381, 5.44771769571650859231311693028, 6.73967792274990793223112557608, 7.72987037568855243508650243723, 8.286011076617848398893161331147, 9.612274076556058093848870083957, 10.90274568229085661893260839840, 11.73600259250111435577981040744, 13.04264661200903517342027650101

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