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Properties

Label 2-180-36.11-c1-0-16
Degree $2$
Conductor $180$
Sign $0.956 - 0.290i$
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  + (1.04 + 0.957i)2-s + (0.561 − 1.63i)3-s + (0.165 + 1.99i)4-s + (0.866 − 0.5i)5-s + (2.15 − 1.16i)6-s + (0.811 + 0.468i)7-s + (−1.73 + 2.23i)8-s + (−2.36 − 1.84i)9-s + (1.37 + 0.309i)10-s + (0.291 − 0.505i)11-s + (3.35 + 0.849i)12-s + (−0.579 − 1.00i)13-s + (0.395 + 1.26i)14-s + (−0.332 − 1.69i)15-s + (−3.94 + 0.658i)16-s + 0.238i·17-s + ⋯
L(s)  = 1  + (0.735 + 0.677i)2-s + (0.324 − 0.945i)3-s + (0.0826 + 0.996i)4-s + (0.387 − 0.223i)5-s + (0.879 − 0.476i)6-s + (0.306 + 0.177i)7-s + (−0.614 + 0.789i)8-s + (−0.789 − 0.613i)9-s + (0.436 + 0.0977i)10-s + (0.0880 − 0.152i)11-s + (0.969 + 0.245i)12-s + (−0.160 − 0.278i)13-s + (0.105 + 0.338i)14-s + (−0.0858 − 0.438i)15-s + (−0.986 + 0.164i)16-s + 0.0577i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.956 - 0.290i$
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: \((2,\ 180,\ (\ :1/2),\ 0.956 - 0.290i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83014 + 0.271504i\)
\(L(\frac12)\) \(\approx\) \(1.83014 + 0.271504i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.04 - 0.957i)T \)
3 \( 1 + (-0.561 + 1.63i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
good7 \( 1 + (-0.811 - 0.468i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.291 + 0.505i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.579 + 1.00i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.238iT - 17T^{2} \)
19 \( 1 - 7.00iT - 19T^{2} \)
23 \( 1 + (2.53 + 4.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.91 + 4.57i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.43 - 3.13i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.50T + 37T^{2} \)
41 \( 1 + (-2.86 + 1.65i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.35 - 5.39i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.27 + 7.40i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.9iT - 53T^{2} \)
59 \( 1 + (4.94 + 8.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.33 - 5.77i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.69 + 4.44i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.95T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 + (-4.37 - 2.52i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.36 - 4.09i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.36iT - 89T^{2} \)
97 \( 1 + (2.26 - 3.92i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.67419362305525309044423096176, −12.30661839512586628862051738083, −11.06700514593571929798044088269, −9.357335801396874675865945879584, −8.244373578328888215240070069776, −7.56386647312631414444170362082, −6.25738046928056749577051238204, −5.51908010161870928840260743885, −3.82075132286162402348399742432, −2.20017881679341014963979225303, 2.28442375608842967423958286124, 3.67153297163463443843355509034, 4.79176702022117076995945201824, 5.80425565969095219567613339755, 7.35302926449921187601364133777, 9.163785300870838938482806835477, 9.630099035331298282247358263627, 10.98219187072219501341365537716, 11.23047719002118016937050469790, 12.72187601787970203663197536169

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