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Properties

Label 2-180-20.19-c0-0-1
Degree $2$
Conductor $180$
Sign $i$
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  − i·2-s − 4-s − i·5-s + i·8-s − 10-s + 16-s + 2i·17-s + i·20-s − 25-s − i·32-s + 2·34-s + 40-s − 49-s + i·50-s − 2i·53-s + ⋯
L(s)  = 1  − i·2-s − 4-s − i·5-s + i·8-s − 10-s + 16-s + 2i·17-s + i·20-s − 25-s − i·32-s + 2·34-s + 40-s − 49-s + i·50-s − 2i·53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $i$
Analytic conductor: \(0.0898317\)
Root analytic conductor: \(0.299719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6199840944\)
\(L(\frac12)\) \(\approx\) \(0.6199840944\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−12.68773874131481316135968972332, −11.77201509376005350156308605147, −10.70856199700068145277697392096, −9.768152071362591591297198359402, −8.731949602901659574138655814886, −8.002705591916857497167492234651, −6.00544160009276436343889004620, −4.78133270384470936549343138359, −3.66834307526012773911986557907, −1.72822679618767786561111164747, 3.07697035633363634318314595901, 4.65105749693461699670374546515, 5.94207158086714460524284264874, 7.01138108804813254061843343097, 7.70984593560360609597004373017, 9.098082573711693085294217039423, 9.951066944493501011919944634999, 11.12243734817267578579130886343, 12.24786515957334543450342310498, 13.61416047276907475538878389672

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