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Properties

Label 2-180-180.59-c1-0-6
Degree $2$
Conductor $180$
Sign $0.425 - 0.904i$
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  + (−1.41 + 0.0449i)2-s + (1.62 + 0.589i)3-s + (1.99 − 0.126i)4-s + (−1.91 + 1.14i)5-s + (−2.32 − 0.759i)6-s + (1.44 + 2.50i)7-s + (−2.81 + 0.269i)8-s + (2.30 + 1.91i)9-s + (2.65 − 1.71i)10-s + (0.395 + 0.684i)11-s + (3.32 + 0.968i)12-s + (−4.04 − 2.33i)13-s + (−2.15 − 3.47i)14-s + (−3.80 + 0.741i)15-s + (3.96 − 0.506i)16-s + 5.89·17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0317i)2-s + (0.940 + 0.340i)3-s + (0.997 − 0.0634i)4-s + (−0.857 + 0.513i)5-s + (−0.950 − 0.310i)6-s + (0.546 + 0.945i)7-s + (−0.995 + 0.0951i)8-s + (0.768 + 0.639i)9-s + (0.841 − 0.540i)10-s + (0.119 + 0.206i)11-s + (0.960 + 0.279i)12-s + (−1.12 − 0.647i)13-s + (−0.575 − 0.927i)14-s + (−0.981 + 0.191i)15-s + (0.991 − 0.126i)16-s + 1.42·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.425 - 0.904i$
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: \((2,\ 180,\ (\ :1/2),\ 0.425 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.794056 + 0.503907i\)
\(L(\frac12)\) \(\approx\) \(0.794056 + 0.503907i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0449i)T \)
3 \( 1 + (-1.62 - 0.589i)T \)
5 \( 1 + (1.91 - 1.14i)T \)
good7 \( 1 + (-1.44 - 2.50i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.395 - 0.684i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.04 + 2.33i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.89T + 17T^{2} \)
19 \( 1 - 4.55iT - 19T^{2} \)
23 \( 1 + (1.15 + 0.666i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.29 + 1.32i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.26 - 1.30i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.44iT - 37T^{2} \)
41 \( 1 + (5.91 + 3.41i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.95 + 10.3i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.60 + 1.50i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.70T + 53T^{2} \)
59 \( 1 + (-5.29 + 9.17i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.869 + 1.50i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.76 - 6.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.88T + 71T^{2} \)
73 \( 1 - 6.50iT - 73T^{2} \)
79 \( 1 + (3.30 - 1.91i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.33 + 4.81i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.00741iT - 89T^{2} \)
97 \( 1 + (-5.74 + 3.31i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.32222270792457606378409870167, −11.92900133235607087839293634444, −10.45697245472564537872574949742, −9.897161056321724500175007451660, −8.598206747484369852399451108449, −7.964383631274631849006811624506, −7.17436235490910705290046473946, −5.38572859516115004677031423898, −3.49764994740936489982964475045, −2.26833924343648726717138402032, 1.21131908273403614076764301140, 3.11036622336906424408782077999, 4.59041943455217253375015253196, 6.84599978288650296862065791700, 7.64176297404431360706955142773, 8.257115923978148467295083286447, 9.351067012127990030983958262608, 10.23216744312606676333509505685, 11.58460153269414888845277696686, 12.19654378033463995204534321731

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