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Properties

Label 2-2160-180.139-c0-0-0
Degree $2$
Conductor $2160$
Sign $-0.173 - 0.984i$
Analytic cond. $1.07798$
Root an. cond. $1.03825$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)5-s + (0.866 + 1.5i)7-s + (−0.866 + 1.5i)23-s + (−0.499 − 0.866i)25-s + (−0.5 − 0.866i)29-s − 1.73·35-s + (0.5 − 0.866i)41-s + (0.866 + 1.5i)47-s + (−1 + 1.73i)49-s + (−0.5 − 0.866i)61-s + (−0.866 + 1.5i)67-s + (−0.866 − 1.5i)83-s + 89-s + (1 + 1.73i)101-s + 1.73·107-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s + (0.866 + 1.5i)7-s + (−0.866 + 1.5i)23-s + (−0.499 − 0.866i)25-s + (−0.5 − 0.866i)29-s − 1.73·35-s + (0.5 − 0.866i)41-s + (0.866 + 1.5i)47-s + (−1 + 1.73i)49-s + (−0.5 − 0.866i)61-s + (−0.866 + 1.5i)67-s + (−0.866 − 1.5i)83-s + 89-s + (1 + 1.73i)101-s + 1.73·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :0),\ -0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.064931203\)
\(L(\frac12)\) \(\approx\) \(1.064931203\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.368157754533704897246694691815, −8.697566198040515126857762338451, −7.78655611514018007652620768231, −7.43865526607474839635750959135, −6.10152030812417905127427122256, −5.73698672350256743905285147228, −4.66258351014817769349426701055, −3.68691545613280419740794780920, −2.66039398739313116975375322202, −1.85507581649513027495424109356, 0.76524831502332005118360907554, 1.88191527200502079658863358174, 3.47721463976858957705235265307, 4.35368246118398338684239370523, 4.72765627252375934363096693239, 5.80882538162803068541835881723, 6.94358138324772474405549435297, 7.56958714984671079574676366834, 8.250418633241100987435494834928, 8.844349915268557763908733644449

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