↑

Properties

Label 2-180-12.11-c1-0-3
Degree $2$
Conductor $180$
Sign $0.614 + 0.789i$
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  + (−1.39 − 0.207i)2-s + (1.91 + 0.579i)4-s − i·5-s − 1.63i·7-s + (−2.55 − 1.20i)8-s + (−0.207 + 1.39i)10-s + 3.95·11-s + 0.585·13-s + (−0.339 + 2.29i)14-s + (3.32 + 2.21i)16-s − 4i·17-s − 7.91i·19-s + (0.579 − 1.91i)20-s + (−5.53 − 0.819i)22-s + 5.59·23-s + ⋯
L(s)  = 1  + (−0.989 − 0.146i)2-s + (0.957 + 0.289i)4-s − 0.447i·5-s − 0.619i·7-s + (−0.904 − 0.426i)8-s + (−0.0654 + 0.442i)10-s + 1.19·11-s + 0.162·13-s + (−0.0907 + 0.612i)14-s + (0.832 + 0.554i)16-s − 0.970i·17-s − 1.81i·19-s + (0.129 − 0.428i)20-s + (−1.18 − 0.174i)22-s + 1.16·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 + 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.614 + 0.789i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.614 + 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.716135 - 0.350107i\)
\(L(\frac12)\) \(\approx\) \(0.716135 - 0.350107i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.207i)T \)
3 \( 1 \)
5 \( 1 + iT \)
good7 \( 1 + 1.63iT - 7T^{2} \)
11 \( 1 - 3.95T + 11T^{2} \)
13 \( 1 - 0.585T + 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 + 7.91iT - 19T^{2} \)
23 \( 1 - 5.59T + 23T^{2} \)
29 \( 1 - 7.65iT - 29T^{2} \)
31 \( 1 - 5.59iT - 31T^{2} \)
37 \( 1 + 9.07T + 37T^{2} \)
41 \( 1 + 1.41iT - 41T^{2} \)
43 \( 1 - 7.91iT - 43T^{2} \)
47 \( 1 + 3.27T + 47T^{2} \)
53 \( 1 - 5.17iT - 53T^{2} \)
59 \( 1 + 11.8T + 59T^{2} \)
61 \( 1 + 0.828T + 61T^{2} \)
67 \( 1 - 11.1iT - 67T^{2} \)
71 \( 1 + 4.63T + 71T^{2} \)
73 \( 1 - 8.82T + 73T^{2} \)
79 \( 1 + 10.2iT - 79T^{2} \)
83 \( 1 - 3.27T + 83T^{2} \)
89 \( 1 + 5.41iT - 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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

−12.26794760378907491896751633977, −11.34849439540397646977344620482, −10.55926419509153212450266277031, −9.165128439402976126694550249523, −8.903499165070859218992798543628, −7.29178956865751735355123087331, −6.68982924932277756461423271998, −4.87157939910257338634966895829, −3.18268434641946101065646405870, −1.14288406890099305048925118572, 1.84722318949126075872138500483, 3.62139934538957270480617976242, 5.79543572949825276710885226399, 6.57548399375315767373379310439, 7.84901952847535757726754210753, 8.785826929078435336472656114944, 9.737605110136993622613958058043, 10.68128669427732999983970768287, 11.70055583076137140646065700885, 12.41412145007416249141313045561

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