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Properties

Label 2-180-4.3-c2-0-0
Degree $2$
Conductor $180$
Sign $-0.410 - 0.911i$
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  + (−1.67 − 1.08i)2-s + (1.64 + 3.64i)4-s − 2.23·5-s − 0.596i·7-s + (1.19 − 7.91i)8-s + (3.75 + 2.42i)10-s + 9.27i·11-s − 23.5·13-s + (−0.647 + 1.00i)14-s + (−10.5 + 11.9i)16-s − 3.97·17-s + 7.04i·19-s + (−3.67 − 8.15i)20-s + (10.0 − 15.5i)22-s + 32.0i·23-s + ⋯
L(s)  = 1  + (−0.839 − 0.542i)2-s + (0.410 + 0.911i)4-s − 0.447·5-s − 0.0852i·7-s + (0.149 − 0.988i)8-s + (0.375 + 0.242i)10-s + 0.843i·11-s − 1.80·13-s + (−0.0462 + 0.0715i)14-s + (−0.662 + 0.749i)16-s − 0.233·17-s + 0.370i·19-s + (−0.183 − 0.407i)20-s + (0.457 − 0.708i)22-s + 1.39i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.410 - 0.911i$
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: \((2,\ 180,\ (\ :1),\ -0.410 - 0.911i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.180547 + 0.279435i\)
\(L(\frac12)\) \(\approx\) \(0.180547 + 0.279435i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.67 + 1.08i)T \)
3 \( 1 \)
5 \( 1 + 2.23T \)
good7 \( 1 + 0.596iT - 49T^{2} \)
11 \( 1 - 9.27iT - 121T^{2} \)
13 \( 1 + 23.5T + 169T^{2} \)
17 \( 1 + 3.97T + 289T^{2} \)
19 \( 1 - 7.04iT - 361T^{2} \)
23 \( 1 - 32.0iT - 529T^{2} \)
29 \( 1 + 35.6T + 841T^{2} \)
31 \( 1 - 59.2iT - 961T^{2} \)
37 \( 1 + 5.38T + 1.36e3T^{2} \)
41 \( 1 + 40.0T + 1.68e3T^{2} \)
43 \( 1 + 36.1iT - 1.84e3T^{2} \)
47 \( 1 + 74.0iT - 2.20e3T^{2} \)
53 \( 1 - 2.55T + 2.80e3T^{2} \)
59 \( 1 - 36.4iT - 3.48e3T^{2} \)
61 \( 1 + 8.73T + 3.72e3T^{2} \)
67 \( 1 + 69.7iT - 4.48e3T^{2} \)
71 \( 1 + 59.2iT - 5.04e3T^{2} \)
73 \( 1 + 83.0T + 5.32e3T^{2} \)
79 \( 1 + 65.8iT - 6.24e3T^{2} \)
83 \( 1 - 129. iT - 6.88e3T^{2} \)
89 \( 1 - 130.T + 7.92e3T^{2} \)
97 \( 1 - 93.1T + 9.40e3T^{2} \)
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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

−12.24025680787957231150186381017, −11.92223508625670613423245369168, −10.58861565621585930955048536362, −9.817393991016349550440770345999, −8.875426540942175638419064790763, −7.54193632869861059858036695727, −7.07095429303324635446849392966, −5.00893913766092648658088574882, −3.54782087952661178316993144190, −1.98087852772968817053311207708, 0.24453724710631501383165316715, 2.51314763976849669746618219432, 4.61976572653188327045386594410, 5.90936276707316527056008046950, 7.12674417469998976446187302626, 7.967366650599665252134623567034, 9.023056110810549338122917222209, 9.944396893432809388545431696827, 11.02515470538445147886998938511, 11.85255815962459564206017461389

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