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Properties

Label 2-180-20.19-c2-0-17
Degree $2$
Conductor $180$
Sign $-0.866 + 0.499i$
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  + (−1.73 − i)2-s + (1.99 + 3.46i)4-s + 5i·5-s − 10.3·7-s − 7.99i·8-s + (5 − 8.66i)10-s − 10.3i·11-s − 18i·13-s + (18 + 10.3i)14-s + (−8 + 13.8i)16-s − 10i·17-s − 13.8i·19-s + (−17.3 + 9.99i)20-s + (−10.3 + 18i)22-s − 6.92·23-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + i·5-s − 1.48·7-s − 0.999i·8-s + (0.5 − 0.866i)10-s − 0.944i·11-s − 1.38i·13-s + (1.28 + 0.742i)14-s + (−0.5 + 0.866i)16-s − 0.588i·17-s − 0.729i·19-s + (−0.866 + 0.499i)20-s + (−0.472 + 0.818i)22-s − 0.301·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.866 + 0.499i$
Analytic conductor: \(4.90464\)
Root analytic conductor: \(2.21464\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1),\ -0.866 + 0.499i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0798908 - 0.298156i\)
\(L(\frac12)\) \(\approx\) \(0.0798908 - 0.298156i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 + i)T \)
3 \( 1 \)
5 \( 1 - 5iT \)
good7 \( 1 + 10.3T + 49T^{2} \)
11 \( 1 + 10.3iT - 121T^{2} \)
13 \( 1 + 18iT - 169T^{2} \)
17 \( 1 + 10iT - 289T^{2} \)
19 \( 1 + 13.8iT - 361T^{2} \)
23 \( 1 + 6.92T + 529T^{2} \)
29 \( 1 + 36T + 841T^{2} \)
31 \( 1 + 6.92iT - 961T^{2} \)
37 \( 1 - 54iT - 1.36e3T^{2} \)
41 \( 1 + 18T + 1.68e3T^{2} \)
43 \( 1 - 20.7T + 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 + 26iT - 2.80e3T^{2} \)
59 \( 1 + 31.1iT - 3.48e3T^{2} \)
61 \( 1 + 74T + 3.72e3T^{2} \)
67 \( 1 + 41.5T + 4.48e3T^{2} \)
71 \( 1 - 103. iT - 5.04e3T^{2} \)
73 \( 1 + 36iT - 5.32e3T^{2} \)
79 \( 1 + 90.0iT - 6.24e3T^{2} \)
83 \( 1 - 90.0T + 6.88e3T^{2} \)
89 \( 1 + 18T + 7.92e3T^{2} \)
97 \( 1 + 72iT - 9.40e3T^{2} \)
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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

−11.76143892800781633694666327312, −10.83048397936829485987821569630, −10.07249853246143828095928621085, −9.241871890729569829726270294011, −7.951778713441264346579436727135, −6.91695597816170132434497235598, −5.96799338933057284703192938681, −3.43742046502072454336179897635, −2.82795598849732284461674779715, −0.23041075168654342416447235850, 1.83729652023832807816969641358, 4.12165445256262683031701901796, 5.68791583125425206057988374036, 6.67821230283248354552895136562, 7.73417020219251135542917084716, 9.153951718552026362492184599664, 9.423906121838463718494241824908, 10.50410950317689437193491373447, 11.96003155587327508918423080915, 12.69725135637644525078902807793

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