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Properties

Label 2-180-20.3-c1-0-9
Degree $2$
Conductor $180$
Sign $-0.881 + 0.471i$
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  + (−1.28 − 0.581i)2-s + (1.32 + 1.50i)4-s + (−2.12 + 0.707i)5-s + (−2.64 − 2.64i)7-s + (−0.832 − 2.70i)8-s + (3.14 + 0.322i)10-s − 3.74i·11-s + (−2 − 2i)13-s + (1.87 + 4.94i)14-s + (−0.5 + 3.96i)16-s + (1.41 − 1.41i)17-s − 5.29·19-s + (−3.86 − 2.24i)20-s + (−2.17 + 4.82i)22-s + (−3.74 + 3.74i)23-s + ⋯
L(s)  = 1  + (−0.911 − 0.411i)2-s + (0.661 + 0.750i)4-s + (−0.948 + 0.316i)5-s + (−0.999 − 0.999i)7-s + (−0.294 − 0.955i)8-s + (0.994 + 0.102i)10-s − 1.12i·11-s + (−0.554 − 0.554i)13-s + (0.500 + 1.32i)14-s + (−0.125 + 0.992i)16-s + (0.342 − 0.342i)17-s − 1.21·19-s + (−0.864 − 0.502i)20-s + (−0.464 + 1.02i)22-s + (−0.780 + 0.780i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.881 + 0.471i$
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: \((2,\ 180,\ (\ :1/2),\ -0.881 + 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0749571 - 0.299222i\)
\(L(\frac12)\) \(\approx\) \(0.0749571 - 0.299222i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 + 0.581i)T \)
3 \( 1 \)
5 \( 1 + (2.12 - 0.707i)T \)
good7 \( 1 + (2.64 + 2.64i)T + 7iT^{2} \)
11 \( 1 + 3.74iT - 11T^{2} \)
13 \( 1 + (2 + 2i)T + 13iT^{2} \)
17 \( 1 + (-1.41 + 1.41i)T - 17iT^{2} \)
19 \( 1 + 5.29T + 19T^{2} \)
23 \( 1 + (3.74 - 3.74i)T - 23iT^{2} \)
29 \( 1 - 1.41iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (2 - 2i)T - 37iT^{2} \)
41 \( 1 - 8.48T + 41T^{2} \)
43 \( 1 + (-5.29 + 5.29i)T - 43iT^{2} \)
47 \( 1 + (-3.74 - 3.74i)T + 47iT^{2} \)
53 \( 1 + (5.65 + 5.65i)T + 53iT^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + (-5.29 - 5.29i)T + 67iT^{2} \)
71 \( 1 + 14.9iT - 71T^{2} \)
73 \( 1 + (5 + 5i)T + 73iT^{2} \)
79 \( 1 - 10.5T + 79T^{2} \)
83 \( 1 + (3.74 - 3.74i)T - 83iT^{2} \)
89 \( 1 + 11.3iT - 89T^{2} \)
97 \( 1 + (5 - 5i)T - 97iT^{2} \)
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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

−12.10351564184669093883336791299, −10.96489268208283633487789402131, −10.42946900132684512380116580335, −9.316094683694316881591435428381, −8.092372609678963815865016348454, −7.34225035273634881170436828115, −6.26769062990250629850421912920, −3.95187781906609273716814352604, −3.03700706431383673837337320104, −0.35459451897503113628042179627, 2.39029074187163900059197050616, 4.41736591055411149798935492936, 5.98137251596742340687402733522, 7.00203048948520959036336589966, 8.064383430718241516549510333459, 9.056038712858371866915026692860, 9.808632405847132265342941242524, 10.97009081326445768135198180857, 12.33082745296594668130960851913, 12.46624226486342823469409488568

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