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Properties

Label 2-180-4.3-c2-0-2
Degree $2$
Conductor $180$
Sign $0.904 - 0.427i$
Analytic cond. $4.90464$
Root an. cond. $2.21464$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.438 − 1.95i)2-s + (−3.61 + 1.71i)4-s − 2.23·5-s + 6.33i·7-s + (4.92 + 6.30i)8-s + (0.979 + 4.36i)10-s + 9.27i·11-s + 18.5·13-s + (12.3 − 2.77i)14-s + (10.1 − 12.3i)16-s − 13.9·17-s + 17.2i·19-s + (8.08 − 3.82i)20-s + (18.1 − 4.06i)22-s + 33.7i·23-s + ⋯
L(s)  = 1  + (−0.219 − 0.975i)2-s + (−0.904 + 0.427i)4-s − 0.447·5-s + 0.904i·7-s + (0.615 + 0.788i)8-s + (0.0979 + 0.436i)10-s + 0.843i·11-s + 1.42·13-s + (0.882 − 0.198i)14-s + (0.634 − 0.772i)16-s − 0.818·17-s + 0.907i·19-s + (0.404 − 0.191i)20-s + (0.823 − 0.184i)22-s + 1.46i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.427i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.904 - 0.427i$
Analytic conductor: \(4.90464\)
Root analytic conductor: \(2.21464\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1),\ 0.904 - 0.427i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.958654 + 0.215257i\)
\(L(\frac12)\) \(\approx\) \(0.958654 + 0.215257i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.438 + 1.95i)T \)
3 \( 1 \)
5 \( 1 + 2.23T \)
good7 \( 1 - 6.33iT - 49T^{2} \)
11 \( 1 - 9.27iT - 121T^{2} \)
13 \( 1 - 18.5T + 169T^{2} \)
17 \( 1 + 13.9T + 289T^{2} \)
19 \( 1 - 17.2iT - 361T^{2} \)
23 \( 1 - 33.7iT - 529T^{2} \)
29 \( 1 - 28.6T + 841T^{2} \)
31 \( 1 + 23.4iT - 961T^{2} \)
37 \( 1 + 67.3T + 1.36e3T^{2} \)
41 \( 1 - 44.0T + 1.68e3T^{2} \)
43 \( 1 - 50.2iT - 1.84e3T^{2} \)
47 \( 1 - 31.1iT - 2.20e3T^{2} \)
53 \( 1 + 81.6T + 2.80e3T^{2} \)
59 \( 1 - 19.2iT - 3.48e3T^{2} \)
61 \( 1 + 53.1T + 3.72e3T^{2} \)
67 \( 1 + 4.49iT - 4.48e3T^{2} \)
71 \( 1 - 13.3iT - 5.04e3T^{2} \)
73 \( 1 - 40.8T + 5.32e3T^{2} \)
79 \( 1 + 141. iT - 6.24e3T^{2} \)
83 \( 1 + 69.8iT - 6.88e3T^{2} \)
89 \( 1 - 46.3T + 7.92e3T^{2} \)
97 \( 1 - 68.5T + 9.40e3T^{2} \)
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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

−12.30429729498732600532523429757, −11.55081995494566015310266382841, −10.67931421171999839188448011688, −9.493935353289226263970550826894, −8.676153372231630212860844422861, −7.68837365014332595257856693740, −5.98929693905611625061371189835, −4.54983053900970408592468708474, −3.31992461228834141145714139882, −1.72291709474541861075746628187, 0.68319979180481208243229548186, 3.66093448115693733894819019188, 4.78061260824715246557973185895, 6.30271657742776224024032381548, 7.04045355821527838605828308116, 8.383409083161970752604601809342, 8.856405908464438570321914326592, 10.49169951896916819603529023870, 11.01539132170928538096630182505, 12.62388284921956488083219953779

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