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Properties

Label 2-2160-80.19-c0-0-1
Degree $2$
Conductor $2160$
Sign $0.991 + 0.130i$
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  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.707 + 0.707i)5-s + (0.707 + 0.707i)8-s + (0.500 − 0.866i)10-s + (0.500 − 0.866i)16-s + 0.517i·17-s + (0.366 − 0.366i)19-s + (−0.965 − 0.258i)20-s + 1.93i·23-s + 1.00i·25-s − i·31-s + (−0.965 − 0.258i)32-s + (0.499 − 0.133i)34-s + (−0.448 − 0.258i)38-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.707 + 0.707i)5-s + (0.707 + 0.707i)8-s + (0.500 − 0.866i)10-s + (0.500 − 0.866i)16-s + 0.517i·17-s + (0.366 − 0.366i)19-s + (−0.965 − 0.258i)20-s + 1.93i·23-s + 1.00i·25-s − i·31-s + (−0.965 − 0.258i)32-s + (0.499 − 0.133i)34-s + (−0.448 − 0.258i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.991 + 0.130i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :0),\ 0.991 + 0.130i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.058186506\)
\(L(\frac12)\) \(\approx\) \(1.058186506\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 \)
5 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 - T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - 0.517iT - T^{2} \)
19 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
23 \( 1 - 1.93iT - T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + iT - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 1.73iT - T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.465591008867530802270386246057, −8.744756167842171413550132224518, −7.72351082650728388862039573242, −7.11826312605787133549555476701, −5.92598205582598653706588661687, −5.29130392602542830934279151855, −4.09366226022981265413710188631, −3.28000817800384115058100961368, −2.38473862226755546348882993558, −1.41713149911821548035486776677, 0.919124983430756210206107214694, 2.27227434816082316708205065682, 3.77480847209916392236115232554, 4.87155574257639302693703292957, 5.24217325605760824922170987117, 6.29616645277790475707216594370, 6.78344835226290518970763787746, 7.82218486636041281990185478058, 8.590629216642488251148534661719, 8.996328543344348896531593738902

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