↑

Properties

Label 2-180-20.3-c3-0-20
Degree $2$
Conductor $180$
Sign $0.455 + 0.890i$
Analytic cond. $10.6203$
Root an. cond. $3.25888$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.70 − 0.813i)2-s + (6.67 + 4.40i)4-s + (0.435 + 11.1i)5-s + (−17.7 − 17.7i)7-s + (−14.4 − 17.3i)8-s + (7.91 − 30.6i)10-s − 7.37i·11-s + (−2.68 − 2.68i)13-s + (33.6 + 62.6i)14-s + (25.1 + 58.8i)16-s + (20.2 − 20.2i)17-s + 135.·19-s + (−46.3 + 76.4i)20-s + (−6.00 + 19.9i)22-s + (71.0 − 71.0i)23-s + ⋯
L(s)  = 1  + (−0.957 − 0.287i)2-s + (0.834 + 0.551i)4-s + (0.0389 + 0.999i)5-s + (−0.959 − 0.959i)7-s + (−0.640 − 0.767i)8-s + (0.250 − 0.968i)10-s − 0.202i·11-s + (−0.0572 − 0.0572i)13-s + (0.643 + 1.19i)14-s + (0.392 + 0.919i)16-s + (0.288 − 0.288i)17-s + 1.63·19-s + (−0.518 + 0.855i)20-s + (−0.0581 + 0.193i)22-s + (0.644 − 0.644i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.455 + 0.890i$
Analytic conductor: \(10.6203\)
Root analytic conductor: \(3.25888\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :3/2),\ 0.455 + 0.890i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.746703 - 0.456914i\)
\(L(\frac12)\) \(\approx\) \(0.746703 - 0.456914i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.70 + 0.813i)T \)
3 \( 1 \)
5 \( 1 + (-0.435 - 11.1i)T \)
good7 \( 1 + (17.7 + 17.7i)T + 343iT^{2} \)
11 \( 1 + 7.37iT - 1.33e3T^{2} \)
13 \( 1 + (2.68 + 2.68i)T + 2.19e3iT^{2} \)
17 \( 1 + (-20.2 + 20.2i)T - 4.91e3iT^{2} \)
19 \( 1 - 135.T + 6.85e3T^{2} \)
23 \( 1 + (-71.0 + 71.0i)T - 1.21e4iT^{2} \)
29 \( 1 + 34.2iT - 2.43e4T^{2} \)
31 \( 1 + 187. iT - 2.97e4T^{2} \)
37 \( 1 + (-250. + 250. i)T - 5.06e4iT^{2} \)
41 \( 1 - 211.T + 6.89e4T^{2} \)
43 \( 1 + (-46.7 + 46.7i)T - 7.95e4iT^{2} \)
47 \( 1 + (189. + 189. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-74.5 - 74.5i)T + 1.48e5iT^{2} \)
59 \( 1 + 101.T + 2.05e5T^{2} \)
61 \( 1 - 232.T + 2.26e5T^{2} \)
67 \( 1 + (34.7 + 34.7i)T + 3.00e5iT^{2} \)
71 \( 1 + 614. iT - 3.57e5T^{2} \)
73 \( 1 + (37.4 + 37.4i)T + 3.89e5iT^{2} \)
79 \( 1 + 1.00e3T + 4.93e5T^{2} \)
83 \( 1 + (-423. + 423. i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.04e3iT - 7.04e5T^{2} \)
97 \( 1 + (536. - 536. i)T - 9.12e5iT^{2} \)
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

−11.67719349097749723609835417517, −10.86455978618609006237156049027, −9.987264017757222182136350902491, −9.357535002953016805000992203941, −7.71159818047494749745990905975, −7.10832023047965947757255040426, −6.05790346906341806459008706374, −3.71333006078073232492712894277, −2.72587740330261801329146070847, −0.62086139859664446584339775164, 1.21450569063159624181578801281, 2.98186846075398722290006816655, 5.17030127436919343475963706356, 6.06976085006662542409690081194, 7.37345060731740307555708394793, 8.503701720092380482719429405643, 9.382493202793180344366414411651, 9.880182228132555757697691542805, 11.44116894654339269510669581200, 12.23365349741359740969069113080

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