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Properties

Label 2-2880-320.59-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  + (−0.980 + 0.195i)2-s + (0.923 − 0.382i)4-s + (0.195 − 0.980i)5-s + (−0.831 + 0.555i)8-s + i·10-s + (0.707 − 0.707i)16-s + (−1.38 − 1.38i)17-s + (−1.63 + 0.324i)19-s + (−0.195 − 0.980i)20-s + (−1.81 + 0.750i)23-s + (−0.923 − 0.382i)25-s − 0.765·31-s + (−0.555 + 0.831i)32-s + (1.63 + 1.08i)34-s + (1.53 − 0.636i)38-s + ⋯
L(s)  = 1  + (−0.980 + 0.195i)2-s + (0.923 − 0.382i)4-s + (0.195 − 0.980i)5-s + (−0.831 + 0.555i)8-s + i·10-s + (0.707 − 0.707i)16-s + (−1.38 − 1.38i)17-s + (−1.63 + 0.324i)19-s + (−0.195 − 0.980i)20-s + (−1.81 + 0.750i)23-s + (−0.923 − 0.382i)25-s − 0.765·31-s + (−0.555 + 0.831i)32-s + (1.63 + 1.08i)34-s + (1.53 − 0.636i)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: $\chi_{2880} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ -0.956 + 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2510725751\)
\(L(\frac12)\) \(\approx\) \(0.2510725751\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.980 - 0.195i)T \)
3 \( 1 \)
5 \( 1 + (-0.195 + 0.980i)T \)
good7 \( 1 + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.382 - 0.923i)T^{2} \)
13 \( 1 + (-0.923 + 0.382i)T^{2} \)
17 \( 1 + (1.38 + 1.38i)T + iT^{2} \)
19 \( 1 + (1.63 - 0.324i)T + (0.923 - 0.382i)T^{2} \)
23 \( 1 + (1.81 - 0.750i)T + (0.707 - 0.707i)T^{2} \)
29 \( 1 + (-0.382 - 0.923i)T^{2} \)
31 \( 1 + 0.765T + T^{2} \)
37 \( 1 + (0.923 + 0.382i)T^{2} \)
41 \( 1 + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (0.382 - 0.923i)T^{2} \)
47 \( 1 + (-0.785 + 0.785i)T - iT^{2} \)
53 \( 1 + (0.636 - 0.425i)T + (0.382 - 0.923i)T^{2} \)
59 \( 1 + (-0.923 - 0.382i)T^{2} \)
61 \( 1 + (0.923 + 0.617i)T + (0.382 + 0.923i)T^{2} \)
67 \( 1 + (0.382 + 0.923i)T^{2} \)
71 \( 1 + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (-1 + i)T - iT^{2} \)
83 \( 1 + (-0.360 - 1.81i)T + (-0.923 + 0.382i)T^{2} \)
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

−8.706833646625033485676786435140, −8.016773441974026595377771227722, −7.28219957256163459342340140484, −6.36642996655401902859539308389, −5.75809785151939016560554204977, −4.79802277365491126531669023996, −3.92756373171342407493432071748, −2.41001776749189815301762316469, −1.74727881184879437866192008087, −0.19256821388186976275676793529, 1.99279056669789436085029395049, 2.32204895712051937381912871088, 3.63939954760991403304588403943, 4.32666912536351148561091522083, 6.07646964577384163642969965157, 6.26652561047752240714496427532, 7.06830612864318044989871913352, 7.940605323123596313997278384531, 8.572870680365869875938389054308, 9.213102320561193652159557762284

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