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Properties

Label 2-180-180.119-c1-0-30
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: $\chi_{180} (119, \cdot )$
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

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

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