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Properties

Label 2-2160-540.259-c0-0-1
Degree $2$
Conductor $2160$
Sign $0.286 + 0.957i$
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  + (0.342 − 0.939i)3-s + (0.939 + 0.342i)5-s + (0.118 − 0.673i)7-s + (−0.766 − 0.642i)9-s + (0.642 − 0.766i)15-s + (−0.592 − 0.342i)21-s + (−0.342 − 1.93i)23-s + (0.766 + 0.642i)25-s + (−0.866 + 0.500i)27-s + (0.266 + 0.223i)29-s + (0.342 − 0.592i)35-s + (−1.17 + 0.984i)41-s + (1.62 − 0.592i)43-s + (−0.500 − 0.866i)45-s + (−0.223 + 1.26i)47-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)3-s + (0.939 + 0.342i)5-s + (0.118 − 0.673i)7-s + (−0.766 − 0.642i)9-s + (0.642 − 0.766i)15-s + (−0.592 − 0.342i)21-s + (−0.342 − 1.93i)23-s + (0.766 + 0.642i)25-s + (−0.866 + 0.500i)27-s + (0.266 + 0.223i)29-s + (0.342 − 0.592i)35-s + (−1.17 + 0.984i)41-s + (1.62 − 0.592i)43-s + (−0.500 − 0.866i)45-s + (−0.223 + 1.26i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.286 + 0.957i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :0),\ 0.286 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.511364496\)
\(L(\frac12)\) \(\approx\) \(1.511364496\)
\(L(1)\) 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 \( 1 + (-0.342 + 0.939i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
good7 \( 1 + (-0.118 + 0.673i)T + (-0.939 - 0.342i)T^{2} \)
11 \( 1 + (-0.766 + 0.642i)T^{2} \)
13 \( 1 + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.342 + 1.93i)T + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.939 - 0.342i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (-1.62 + 0.592i)T + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.223 - 1.26i)T + (-0.939 - 0.342i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
67 \( 1 + (0.984 - 0.826i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (-0.984 - 0.826i)T + (0.173 + 0.984i)T^{2} \)
89 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.766 + 0.642i)T^{2} \)
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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

−9.026259573232122499224526128195, −8.276354720143162449143999648326, −7.51472759121194087571838583144, −6.64717453063933137376783112143, −6.29734574762364254234866802669, −5.26078429138109061337373744701, −4.17187210833548276002232401630, −2.97287020528930725746174636359, −2.22333283458433175650552721276, −1.09884854478204742840896356890, 1.75122750271444623174506195638, 2.67151709415962196876956070673, 3.67913100951093002033152663407, 4.67518602992274797781365938478, 5.56293892536385545522785837989, 5.83719424232935308661342313424, 7.14662911184352331451192697103, 8.129481832049853057931163426888, 8.906448854542505581136156049073, 9.323925477847070434955511806770

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