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Properties

Label 2-2880-120.29-c0-0-6
Degree $2$
Conductor $2880$
Sign $-0.169 + 0.985i$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
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  − i·5-s − 1.41i·7-s + 1.41·11-s − 1.41·13-s + 2·23-s − 25-s − 1.41·35-s − 1.41·37-s − 1.41i·41-s − 1.00·49-s − 1.41i·55-s − 1.41·59-s + 1.41i·65-s − 2.00i·77-s + 1.41i·89-s + ⋯
L(s)  = 1  − i·5-s − 1.41i·7-s + 1.41·11-s − 1.41·13-s + 2·23-s − 25-s − 1.41·35-s − 1.41·37-s − 1.41i·41-s − 1.00·49-s − 1.41i·55-s − 1.41·59-s + 1.41i·65-s − 2.00i·77-s + 1.41i·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.169 + 0.985i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ -0.169 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.188805409\)
\(L(\frac12)\) \(\approx\) \(1.188805409\)
\(L(1)\) 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 \( 1 + iT \)
good7 \( 1 + 1.41iT - T^{2} \)
11 \( 1 - 1.41T + T^{2} \)
13 \( 1 + 1.41T + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 2T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + 1.41T + T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
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

−8.984184249226157113850803834779, −7.933141933295879751580662789296, −7.09455239120657384030416849449, −6.81091798880029636507686970013, −5.49005516939160698353316654336, −4.73715458753281013145042218267, −4.14435274004206410912863426062, −3.25913280662840655263764674716, −1.75131245771021746880707341781, −0.77878056760116765090875203817, 1.72186417172226829918787137117, 2.72803698089358560208662802568, 3.30611141443052772614680051194, 4.57717646231359259236740012048, 5.34428609002587137181062018068, 6.25528272141656758958509754123, 6.84961717298419026095861458033, 7.49389857904176055939174384184, 8.564879163710126111719837870871, 9.239361849147235296759587120568

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