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Properties

Label 2-180-20.3-c1-0-12
Degree $2$
Conductor $180$
Sign $-0.978 + 0.205i$
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  + (−0.0912 − 1.41i)2-s + (−1.98 + 0.257i)4-s + (−1.32 − 1.80i)5-s + (−1.86 − 1.86i)7-s + (0.544 + 2.77i)8-s + (−2.42 + 2.02i)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 + (3.08 + 3.23i)20-s + (−1.02 + 0.0664i)22-s + (5.83 − 5.83i)23-s + ⋯
L(s)  = 1  + (−0.0645 − 0.997i)2-s + (−0.991 + 0.128i)4-s + (−0.590 − 0.807i)5-s + (−0.705 − 0.705i)7-s + (0.192 + 0.981i)8-s + (−0.767 + 0.641i)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 + (0.689 + 0.724i)20-s + (−0.219 + 0.0141i)22-s + (1.21 − 1.21i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.978 + 0.205i$
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: \((2,\ 180,\ (\ :1/2),\ -0.978 + 0.205i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0726603 - 0.699411i\)
\(L(\frac12)\) \(\approx\) \(0.0726603 - 0.699411i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0912 + 1.41i)T \)
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

−12.40616092988916719663254309510, −11.19829980129983180148243087435, −10.29759895932539885088357145145, −9.344576239551261681548908079635, −8.348677355793103056156372753699, −7.23679948210377898602444765306, −5.33702183434064462039015071196, −4.22899816440755952426924798154, −2.99302165522768223192597578994, −0.67517895084308304558528426940, 3.06389277382583427528938240666, 4.57438379151832218267006157986, 5.93855102787290637114701054931, 7.01052847308758286418228332216, 7.67640558345182012781087959668, 9.190473998434783199614375015988, 9.706399715879164230349585604873, 11.22175058770864163864903536292, 12.21760364072289159651895443648, 13.28399365473598260849610706547

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