↑

Properties

Label 2-180-60.59-c1-0-5
Degree $2$
Conductor $180$
Sign $0.499 - 0.866i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 1.11i)2-s + (−0.500 + 1.93i)4-s + (1.73 − 1.41i)5-s + 3.16·7-s + (−2.59 + 1.11i)8-s + (3.08 + 0.711i)10-s − 5.47·11-s + 2.44i·13-s + (2.73 + 3.53i)14-s + (−3.5 − 1.93i)16-s + (1.87 + 4.06i)20-s + (−4.74 − 6.12i)22-s − 4.47i·23-s + (0.999 − 4.89i)25-s + (−2.73 + 2.12i)26-s + ⋯
L(s)  = 1  + (0.612 + 0.790i)2-s + (−0.250 + 0.968i)4-s + (0.774 − 0.632i)5-s + 1.19·7-s + (−0.918 + 0.395i)8-s + (0.974 + 0.225i)10-s − 1.65·11-s + 0.679i·13-s + (0.731 + 0.944i)14-s + (−0.875 − 0.484i)16-s + (0.418 + 0.908i)20-s + (−1.01 − 1.30i)22-s − 0.932i·23-s + (0.199 − 0.979i)25-s + (−0.537 + 0.416i)26-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.499 - 0.866i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.499 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46351 + 0.845275i\)
\(L(\frac12)\) \(\approx\) \(1.46351 + 0.845275i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 1.11i)T \)
3 \( 1 \)
5 \( 1 + (-1.73 + 1.41i)T \)
good7 \( 1 - 3.16T + 7T^{2} \)
11 \( 1 + 5.47T + 11T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 4.47iT - 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + 7.74iT - 31T^{2} \)
37 \( 1 - 7.34iT - 37T^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 + 6.32T + 43T^{2} \)
47 \( 1 - 8.94iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 5.47T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 14.6iT - 73T^{2} \)
79 \( 1 - 7.74iT - 79T^{2} \)
83 \( 1 - 8.94iT - 83T^{2} \)
89 \( 1 - 9.89iT - 89T^{2} \)
97 \( 1 - 4.89iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.12714920929866534258086062789, −12.10829941751916243385531544012, −11.00168744759728323270489290114, −9.667951009340410295860301946365, −8.392650730955273847458226455067, −7.82886941387015147042408449619, −6.32413766985827940452884498117, −5.19260398622656913622101029000, −4.53623790421284494261846647315, −2.37845468342244169491368772862, 1.92629348770758220656886073997, 3.18472832852416684506109623616, 5.04991459409789181282576901385, 5.59838183917113591182611844429, 7.24499990862314861725767434184, 8.546291008938029300613294612552, 9.991413698512133858996077050297, 10.64036655437187397317027882975, 11.33618677716138634861811915111, 12.61840783426833584445118656102

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