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Properties

Label 2-2880-45.29-c0-0-1
Degree $2$
Conductor $2880$
Sign $0.984 + 0.173i$
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  + (0.258 − 0.965i)3-s + (−0.866 + 0.5i)5-s + (1.67 + 0.965i)7-s + (−0.866 − 0.499i)9-s + (0.258 + 0.965i)15-s + (1.36 − 1.36i)21-s + (−0.258 − 0.448i)23-s + (0.499 − 0.866i)25-s + (−0.707 + 0.707i)27-s + (1.5 + 0.866i)29-s − 1.93·35-s + (0.866 − 0.5i)41-s + (1.22 + 0.707i)43-s + 45-s + (−0.965 + 1.67i)47-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)3-s + (−0.866 + 0.5i)5-s + (1.67 + 0.965i)7-s + (−0.866 − 0.499i)9-s + (0.258 + 0.965i)15-s + (1.36 − 1.36i)21-s + (−0.258 − 0.448i)23-s + (0.499 − 0.866i)25-s + (−0.707 + 0.707i)27-s + (1.5 + 0.866i)29-s − 1.93·35-s + (0.866 − 0.5i)41-s + (1.22 + 0.707i)43-s + 45-s + (−0.965 + 1.67i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ 0.984 + 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.368562229\)
\(L(\frac12)\) \(\approx\) \(1.368562229\)
\(L(1)\) 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 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
good7 \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.448 + 0.258i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + 1.73iT - T^{2} \)
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

−8.594886601241103349016684013609, −8.082064116233172178826173213938, −7.67535775091383441364023389547, −6.76056915083737118356724785696, −6.01342034822723096558437681280, −5.05104500581828775411935892544, −4.28418658436038878314967673257, −3.00593349830144066190814197488, −2.32642108060598531025016913072, −1.23539794264401250898936645523, 1.04973076338465104559083170965, 2.43552499060173066510676426020, 3.79518292018972579805920352108, 4.20611188671089193841067609478, 4.89413779268764871139573468038, 5.53165492996216806213706788536, 6.93348009251577523429159054199, 7.81268736056672215320597545693, 8.190807188861454492219708848540, 8.780364292160042903100583708976

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