Properties

Label 2-180-180.59-c1-0-5
Degree $2$
Conductor $180$
Sign $-0.922 - 0.386i$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.442 + 1.34i)2-s + (0.112 + 1.72i)3-s + (−1.60 − 1.18i)4-s + (0.989 + 2.00i)5-s + (−2.37 − 0.613i)6-s + (0.573 + 0.993i)7-s + (2.30 − 1.63i)8-s + (−2.97 + 0.388i)9-s + (−3.13 + 0.442i)10-s + (0.629 + 1.09i)11-s + (1.87 − 2.91i)12-s + (−4.39 − 2.53i)13-s + (−1.58 + 0.331i)14-s + (−3.35 + 1.93i)15-s + (1.17 + 3.82i)16-s + 3.29·17-s + ⋯
L(s)  = 1  + (−0.312 + 0.949i)2-s + (0.0649 + 0.997i)3-s + (−0.804 − 0.594i)4-s + (0.442 + 0.896i)5-s + (−0.968 − 0.250i)6-s + (0.216 + 0.375i)7-s + (0.815 − 0.578i)8-s + (−0.991 + 0.129i)9-s + (−0.990 + 0.140i)10-s + (0.189 + 0.328i)11-s + (0.540 − 0.841i)12-s + (−1.21 − 0.704i)13-s + (−0.424 + 0.0885i)14-s + (−0.866 + 0.499i)15-s + (0.294 + 0.955i)16-s + 0.799·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 - 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.922 - 0.386i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ -0.922 - 0.386i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.189176 + 0.941363i\)
\(L(\frac12)\) \(\approx\) \(0.189176 + 0.941363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.442 - 1.34i)T \)
3 \( 1 + (-0.112 - 1.72i)T \)
5 \( 1 + (-0.989 - 2.00i)T \)
good7 \( 1 + (-0.573 - 0.993i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.629 - 1.09i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.39 + 2.53i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.29T + 17T^{2} \)
19 \( 1 + 3.62iT - 19T^{2} \)
23 \( 1 + (-3.09 - 1.78i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.184 - 0.106i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-9.12 - 5.26i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.72iT - 37T^{2} \)
41 \( 1 + (-5.81 - 3.35i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.41 + 2.45i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.925 + 0.534i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.12T + 53T^{2} \)
59 \( 1 + (4.87 - 8.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.24 + 9.08i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.42 + 12.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.68T + 71T^{2} \)
73 \( 1 + 9.08iT - 73T^{2} \)
79 \( 1 + (5.78 - 3.34i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.93 - 2.84i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.26iT - 89T^{2} \)
97 \( 1 + (-6.69 + 3.86i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.53428670643115608516464792135, −11.99425681596288190610299297022, −10.65425781987320249158182573246, −9.968178511829532300565454961168, −9.209387782635874057283547459730, −7.977110309117070711417671677285, −6.86614091305233950137376707707, −5.60755477460206801828722761562, −4.74009054221091262253690801494, −2.94712121341270404416632699279, 1.07058242699742850647923908483, 2.48033662855235977892859771406, 4.34005335068441628815309184142, 5.69984362720324907343711363777, 7.35194470770274334184277362543, 8.276263317746711946472408265963, 9.241293805286978451401131381588, 10.18811326628392881645667950987, 11.56693811955718479870437256671, 12.22594600286932757786849375111

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