Properties

Label 2-180-5.4-c1-0-1
Degree $2$
Conductor $180$
Sign $0.447 + 0.894i$
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 about

Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 2i)5-s − 4i·7-s + 4·11-s + 4i·17-s − 4i·23-s + (−3 + 4i)25-s − 6·29-s + 4·31-s + (−8 + 4i)35-s + 8i·37-s + 10·41-s + 4i·43-s − 4i·47-s − 9·49-s + 12i·53-s + ⋯
L(s)  = 1  + (−0.447 − 0.894i)5-s − 1.51i·7-s + 1.20·11-s + 0.970i·17-s − 0.834i·23-s + (−0.600 + 0.800i)25-s − 1.11·29-s + 0.718·31-s + (−1.35 + 0.676i)35-s + 1.31i·37-s + 1.56·41-s + 0.609i·43-s − 0.583i·47-s − 1.28·49-s + 1.64i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.934842 - 0.577764i\)
\(L(\frac12)\) \(\approx\) \(0.934842 - 0.577764i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (1 + 2i)T \)
good7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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

−12.54147526348877983719269710054, −11.52645430686467146683683686669, −10.55163908123743082372848797502, −9.458454479382057127776008723656, −8.382882220249915634657648452507, −7.38886104694848865058450009257, −6.23353108455923494477092678570, −4.51725979962021635841320920932, −3.81805801479417206425639861289, −1.15980437545413181859700964876, 2.41444319834448939407476003740, 3.78070346781804821885924275094, 5.47840540371397839884954858337, 6.54357012282433405503891675422, 7.64201655135141947135104127120, 8.949372517253268092969552758100, 9.658139086457238696235799681109, 11.21489140015531959037072666364, 11.69464799482469199651062459216, 12.60025063249490897346720021931

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