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Properties

Label 2-180-36.11-c1-0-18
Degree $2$
Conductor $180$
Sign $-0.574 + 0.818i$
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  + (−1.18 + 0.774i)2-s + (−0.427 − 1.67i)3-s + (0.801 − 1.83i)4-s + (0.866 − 0.5i)5-s + (1.80 + 1.65i)6-s + (−4.06 − 2.34i)7-s + (0.469 + 2.78i)8-s + (−2.63 + 1.43i)9-s + (−0.638 + 1.26i)10-s + (−1.34 + 2.32i)11-s + (−3.41 − 0.562i)12-s + (−1.04 − 1.80i)13-s + (6.63 − 0.368i)14-s + (−1.20 − 1.23i)15-s + (−2.71 − 2.93i)16-s − 3.78i·17-s + ⋯
L(s)  = 1  + (−0.836 + 0.547i)2-s + (−0.246 − 0.969i)3-s + (0.400 − 0.916i)4-s + (0.387 − 0.223i)5-s + (0.737 + 0.675i)6-s + (−1.53 − 0.888i)7-s + (0.165 + 0.986i)8-s + (−0.878 + 0.478i)9-s + (−0.201 + 0.399i)10-s + (−0.404 + 0.701i)11-s + (−0.986 − 0.162i)12-s + (−0.289 − 0.501i)13-s + (1.77 − 0.0985i)14-s + (−0.312 − 0.320i)15-s + (−0.678 − 0.734i)16-s − 0.918i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.574 + 0.818i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ -0.574 + 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.209915 - 0.403626i\)
\(L(\frac12)\) \(\approx\) \(0.209915 - 0.403626i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.18 - 0.774i)T \)
3 \( 1 + (0.427 + 1.67i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
good7 \( 1 + (4.06 + 2.34i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.34 - 2.32i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.04 + 1.80i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.78iT - 17T^{2} \)
19 \( 1 + 3.78iT - 19T^{2} \)
23 \( 1 + (-0.163 - 0.282i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.75 + 1.01i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.70 + 4.44i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.76T + 37T^{2} \)
41 \( 1 + (2.34 - 1.35i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.18 - 3.57i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.465 - 0.805i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.41iT - 53T^{2} \)
59 \( 1 + (1.58 + 2.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.65 + 11.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.62 - 2.66i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 - 13.7T + 73T^{2} \)
79 \( 1 + (7.94 + 4.58i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.11 - 5.39i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.99iT - 89T^{2} \)
97 \( 1 + (-0.0157 + 0.0272i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.48761753632296356396431010958, −11.17086006745840426810859274014, −10.03547243293135836083682252936, −9.424016397337048816795263961460, −7.979182881065531491986307134054, −7.04228099828402575224903839541, −6.43331851857184955302812869550, −5.13424476166893197144422173486, −2.60515140937478483168296527648, −0.52664822139388529254322621514, 2.70589677838884046272856103260, 3.72757120698191175273620512692, 5.72603963609357570015573612912, 6.62788472403437899372849308454, 8.450558837131843335878472694621, 9.209819178434445008692748012452, 10.07054066846902146744505695971, 10.64957097443296607157043584940, 11.93206879370022599960754686236, 12.59994152381245681348925348347

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