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Properties

Label 2-2880-320.99-c0-0-1
Degree $2$
Conductor $2880$
Sign $0.956 + 0.290i$
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  + (−0.555 − 0.831i)2-s + (−0.382 + 0.923i)4-s + (0.831 + 0.555i)5-s + (0.980 − 0.195i)8-s − i·10-s + (−0.707 − 0.707i)16-s + (0.785 − 0.785i)17-s + (1.08 + 1.63i)19-s + (−0.831 + 0.555i)20-s + (0.425 − 1.02i)23-s + (0.382 + 0.923i)25-s − 1.84·31-s + (−0.195 + 0.980i)32-s + (−1.08 − 0.216i)34-s + (0.750 − 1.81i)38-s + ⋯
L(s)  = 1  + (−0.555 − 0.831i)2-s + (−0.382 + 0.923i)4-s + (0.831 + 0.555i)5-s + (0.980 − 0.195i)8-s − i·10-s + (−0.707 − 0.707i)16-s + (0.785 − 0.785i)17-s + (1.08 + 1.63i)19-s + (−0.831 + 0.555i)20-s + (0.425 − 1.02i)23-s + (0.382 + 0.923i)25-s − 1.84·31-s + (−0.195 + 0.980i)32-s + (−1.08 − 0.216i)34-s + (0.750 − 1.81i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.956 + 0.290i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ 0.956 + 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.096993271\)
\(L(\frac12)\) \(\approx\) \(1.096993271\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.555 + 0.831i)T \)
3 \( 1 \)
5 \( 1 + (-0.831 - 0.555i)T \)
good7 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.923 - 0.382i)T^{2} \)
13 \( 1 + (0.382 - 0.923i)T^{2} \)
17 \( 1 + (-0.785 + 0.785i)T - iT^{2} \)
19 \( 1 + (-1.08 - 1.63i)T + (-0.382 + 0.923i)T^{2} \)
23 \( 1 + (-0.425 + 1.02i)T + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (-0.923 - 0.382i)T^{2} \)
31 \( 1 + 1.84T + T^{2} \)
37 \( 1 + (-0.382 - 0.923i)T^{2} \)
41 \( 1 + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + (0.923 - 0.382i)T^{2} \)
47 \( 1 + (0.275 + 0.275i)T + iT^{2} \)
53 \( 1 + (-1.81 + 0.360i)T + (0.923 - 0.382i)T^{2} \)
59 \( 1 + (0.382 + 0.923i)T^{2} \)
61 \( 1 + (-0.382 - 0.0761i)T + (0.923 + 0.382i)T^{2} \)
67 \( 1 + (0.923 + 0.382i)T^{2} \)
71 \( 1 + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (-1 - i)T + iT^{2} \)
83 \( 1 + (0.636 - 0.425i)T + (0.382 - 0.923i)T^{2} \)
89 \( 1 + (-0.707 + 0.707i)T^{2} \)
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

−9.187007268985429418954631148009, −8.280228951999094702177758604774, −7.48161177408641137996546824485, −6.89309214668764451264765675007, −5.74434808137176570940559483122, −5.10963714131955134779924817468, −3.81468446128978213376199552289, −3.13668647299613061234249361676, −2.22064479210144622642506506996, −1.23564750510362012238186589730, 1.02856902338970504797221619465, 2.01877809426966954457764772114, 3.42331721755121040873229848626, 4.66783197250580897408022619627, 5.41879273326694999529022103190, 5.79789705644748617487203805382, 6.88275463092049135044080805145, 7.40428794381233681797511928937, 8.299687080226392235105559892796, 9.079750722701433986376732766214

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