↑

Properties

Label 2-180-20.3-c1-0-8
Degree $2$
Conductor $180$
Sign $0.998 + 0.0512i$
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  + (1.41 + 0.0912i)2-s + (1.98 + 0.257i)4-s + (−1.32 − 1.80i)5-s + (1.86 + 1.86i)7-s + (2.77 + 0.544i)8-s + (−1.69 − 2.66i)10-s + 0.728i·11-s + (−3.12 − 3.12i)13-s + (2.46 + 2.80i)14-s + (3.86 + 1.02i)16-s + (−1.12 + 1.12i)17-s − 3.73·19-s + (−2.15 − 3.91i)20-s + (−0.0664 + 1.02i)22-s + (−5.83 + 5.83i)23-s + ⋯
L(s)  = 1  + (0.997 + 0.0645i)2-s + (0.991 + 0.128i)4-s + (−0.590 − 0.807i)5-s + (0.705 + 0.705i)7-s + (0.981 + 0.192i)8-s + (−0.537 − 0.843i)10-s + 0.219i·11-s + (−0.866 − 0.866i)13-s + (0.658 + 0.749i)14-s + (0.966 + 0.255i)16-s + (−0.272 + 0.272i)17-s − 0.856·19-s + (−0.481 − 0.876i)20-s + (−0.0141 + 0.219i)22-s + (−1.21 + 1.21i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0512i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.998 + 0.0512i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.998 + 0.0512i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93828 - 0.0496827i\)
\(L(\frac12)\) \(\approx\) \(1.93828 - 0.0496827i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0912i)T \)
3 \( 1 \)
5 \( 1 + (1.32 + 1.80i)T \)
good7 \( 1 + (-1.86 - 1.86i)T + 7iT^{2} \)
11 \( 1 - 0.728iT - 11T^{2} \)
13 \( 1 + (3.12 + 3.12i)T + 13iT^{2} \)
17 \( 1 + (1.12 - 1.12i)T - 17iT^{2} \)
19 \( 1 + 3.73T + 19T^{2} \)
23 \( 1 + (5.83 - 5.83i)T - 23iT^{2} \)
29 \( 1 + 2.64iT - 29T^{2} \)
31 \( 1 + 6.01iT - 31T^{2} \)
37 \( 1 + (-3.12 + 3.12i)T - 37iT^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 + (5.10 - 5.10i)T - 43iT^{2} \)
47 \( 1 + (-2.09 - 2.09i)T + 47iT^{2} \)
53 \( 1 + (0.484 + 0.484i)T + 53iT^{2} \)
59 \( 1 - 4.92T + 59T^{2} \)
61 \( 1 - 2.31T + 61T^{2} \)
67 \( 1 + (-5.10 - 5.10i)T + 67iT^{2} \)
71 \( 1 + 13.1iT - 71T^{2} \)
73 \( 1 + (-3.96 - 3.96i)T + 73iT^{2} \)
79 \( 1 - 7.11T + 79T^{2} \)
83 \( 1 + (-3.55 + 3.55i)T - 83iT^{2} \)
89 \( 1 + 1.03iT - 89T^{2} \)
97 \( 1 + (12.5 - 12.5i)T - 97iT^{2} \)
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

−12.60993257483592000548847710858, −11.90910526153328669827138533682, −11.12403668005177785230213400021, −9.724682180101360149784428552930, −8.227479134280960949018496229558, −7.61352567039829008345124090322, −5.94422416262137492129452570502, −5.01428979381997125171840118729, −3.98254460810815760008969542840, −2.17871864739234445901901550988, 2.33760090207469763837043206504, 3.92015861617883938914466347429, 4.77160137102534546130446456921, 6.46473292737030013914438989169, 7.20021691418377848911954538824, 8.282126506889047358234550063528, 10.17122677980944192928602929555, 10.91459179135278564828365639316, 11.73602490804612558314336005518, 12.57032310466843061926133700477

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