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Properties

Label 2-180-20.7-c1-0-9
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

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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: $\chi_{180} (127, \cdot )$
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 \( 1 + (12.5 + 12.5i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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