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Properties

Label 2-2160-240.83-c0-0-1
Degree $2$
Conductor $2160$
Sign $0.584 + 0.811i$
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  + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (−0.707 − 0.707i)8-s + 1.00·10-s + (0.707 − 0.707i)11-s + i·13-s − 1.00·16-s + (0.707 + 0.707i)17-s + (0.707 − 0.707i)20-s − 1.00i·22-s + (0.707 − 0.707i)23-s + 1.00i·25-s + (0.707 + 0.707i)26-s + (−0.707 − 0.707i)29-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (−0.707 − 0.707i)8-s + 1.00·10-s + (0.707 − 0.707i)11-s + i·13-s − 1.00·16-s + (0.707 + 0.707i)17-s + (0.707 − 0.707i)20-s − 1.00i·22-s + (0.707 − 0.707i)23-s + 1.00i·25-s + (0.707 + 0.707i)26-s + (−0.707 − 0.707i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.584 + 0.811i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :0),\ 0.584 + 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.960942002\)
\(L(\frac12)\) \(\approx\) \(1.960942002\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.354013577683039362561979008138, −8.658507239135255416879567362128, −7.37359938631955290868199707454, −6.38984767765554831714236969744, −6.11291575861926805959168187419, −5.12733775927277385797823846845, −4.06782762135547836546210517358, −3.35839383917449221237263436594, −2.34840303986454687848843743689, −1.41097822543807620976198449082, 1.49683688304972513288255829459, 2.85830229168009217376887100185, 3.75139535317117595261612570909, 4.91329552612959679352130677452, 5.28538032218939531118593246962, 6.10055728291232742287314563741, 7.06118132599108263225895833294, 7.59671826868683794236910766877, 8.646691367825217732399887287720, 9.154028413843418373892626934308

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