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Properties

Label 2-2880-320.99-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.8898797456\)
\(L(\frac12)\) \(\approx\) \(0.8898797456\)
\(L(1)\) 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

−9.120061279454497001307069079567, −8.248096255041330974930008528487, −7.76168273562099924702891196145, −7.18218092040545728164728104719, −6.10368536554321625391242157034, −5.52134996969287893972080718926, −4.66885774348180665945702048083, −3.80334333076731815355118757337, −3.35241149126426109739908691116, −1.68344106243174136352646710511, 0.45636217956892889161536858698, 2.13029353185086076312920816065, 2.97198200672937566586808348675, 3.69563600687430576954593187819, 4.65915660577112355432479692751, 5.17118632181013832685252246984, 6.39867956036978824188095497587, 6.96860325396398945600967748216, 7.80305430429244160462080134388, 8.896660550302004382422769191927

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