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Properties

Label 2-180-36.11-c1-0-21
Degree $2$
Conductor $180$
Sign $-0.586 + 0.809i$
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  + (−0.145 − 1.40i)2-s + (1.30 − 1.13i)3-s + (−1.95 + 0.409i)4-s + (0.866 − 0.5i)5-s + (−1.79 − 1.66i)6-s + (−2.51 − 1.45i)7-s + (0.860 + 2.69i)8-s + (0.403 − 2.97i)9-s + (−0.829 − 1.14i)10-s + (1.36 − 2.36i)11-s + (−2.08 + 2.76i)12-s + (0.923 + 1.60i)13-s + (−1.67 + 3.75i)14-s + (0.560 − 1.63i)15-s + (3.66 − 1.60i)16-s + 6.64i·17-s + ⋯
L(s)  = 1  + (−0.102 − 0.994i)2-s + (0.753 − 0.657i)3-s + (−0.978 + 0.204i)4-s + (0.387 − 0.223i)5-s + (−0.731 − 0.681i)6-s + (−0.950 − 0.549i)7-s + (0.304 + 0.952i)8-s + (0.134 − 0.990i)9-s + (−0.262 − 0.362i)10-s + (0.411 − 0.712i)11-s + (−0.602 + 0.798i)12-s + (0.256 + 0.443i)13-s + (−0.448 + 1.00i)14-s + (0.144 − 0.423i)15-s + (0.916 − 0.400i)16-s + 1.61i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.586 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.586 + 0.809i$
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: \((2,\ 180,\ (\ :1/2),\ -0.586 + 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.566909 - 1.11053i\)
\(L(\frac12)\) \(\approx\) \(0.566909 - 1.11053i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.145 + 1.40i)T \)
3 \( 1 + (-1.30 + 1.13i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
good7 \( 1 + (2.51 + 1.45i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.36 + 2.36i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.923 - 1.60i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.64iT - 17T^{2} \)
19 \( 1 + 4.90iT - 19T^{2} \)
23 \( 1 + (-3.31 - 5.74i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.89 - 2.82i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.52 - 2.61i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.280T + 37T^{2} \)
41 \( 1 + (-7.31 + 4.22i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.19 - 3.57i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.743 - 1.28i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4.62iT - 53T^{2} \)
59 \( 1 + (-2.56 - 4.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.61 - 9.72i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.48 - 0.856i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.25T + 71T^{2} \)
73 \( 1 + 9.36T + 73T^{2} \)
79 \( 1 + (7.08 + 4.09i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.57 - 7.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 14.8iT - 89T^{2} \)
97 \( 1 + (-5.94 + 10.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.60358550600056699274366064205, −11.35518649026704305464202346108, −10.30601488511714068821772062033, −9.170622603598438302615314033612, −8.706458704495006193403337443142, −7.24384680158574181890253029057, −5.99511228307581576229095273843, −4.02348737774111116862987011957, −3.00772660658292345237896945499, −1.31409602163060777881370688650, 2.88282216336681065721057925349, 4.36348968548442621999846290932, 5.63753264532484714955225081108, 6.79627940061884787286661459606, 7.919900717686806659283217807480, 9.167961553396277588946321804328, 9.579382431662826505565201131364, 10.54779935339916528474536775937, 12.37483437340793230525188805922, 13.28224405212850916321099907928

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