Properties

Label 2-2880-3.2-c2-0-7
Degree $2$
Conductor $2880$
Sign $-0.816 + 0.577i$
Analytic cond. $78.4743$
Root an. cond. $8.85857$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.23i·5-s − 13.4·7-s + 17.6i·11-s + 7.48·13-s + 16.9i·17-s + 10.9·19-s − 21.9i·23-s − 5.00·25-s + 47.3i·29-s − 16.9·31-s − 30.1i·35-s + 5.53·37-s + 66.3i·41-s − 38.9·43-s + 32.5i·47-s + ⋯
L(s)  = 1  + 0.447i·5-s − 1.92·7-s + 1.60i·11-s + 0.575·13-s + 0.998i·17-s + 0.577·19-s − 0.952i·23-s − 0.200·25-s + 1.63i·29-s − 0.547·31-s − 0.861i·35-s + 0.149·37-s + 1.61i·41-s − 0.906·43-s + 0.692i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.816 + 0.577i$
Analytic conductor: \(78.4743\)
Root analytic conductor: \(8.85857\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1),\ -0.816 + 0.577i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4781759506\)
\(L(\frac12)\) \(\approx\) \(0.4781759506\)
\(L(2)\) 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 \( 1 - 2.23iT \)
good7 \( 1 + 13.4T + 49T^{2} \)
11 \( 1 - 17.6iT - 121T^{2} \)
13 \( 1 - 7.48T + 169T^{2} \)
17 \( 1 - 16.9iT - 289T^{2} \)
19 \( 1 - 10.9T + 361T^{2} \)
23 \( 1 + 21.9iT - 529T^{2} \)
29 \( 1 - 47.3iT - 841T^{2} \)
31 \( 1 + 16.9T + 961T^{2} \)
37 \( 1 - 5.53T + 1.36e3T^{2} \)
41 \( 1 - 66.3iT - 1.68e3T^{2} \)
43 \( 1 + 38.9T + 1.84e3T^{2} \)
47 \( 1 - 32.5iT - 2.20e3T^{2} \)
53 \( 1 + 11.2iT - 2.80e3T^{2} \)
59 \( 1 - 31.8iT - 3.48e3T^{2} \)
61 \( 1 + 46.9T + 3.72e3T^{2} \)
67 \( 1 - 76T + 4.48e3T^{2} \)
71 \( 1 + 77.7iT - 5.04e3T^{2} \)
73 \( 1 - 94.9T + 5.32e3T^{2} \)
79 \( 1 + 6.92T + 6.24e3T^{2} \)
83 \( 1 - 62.1iT - 6.88e3T^{2} \)
89 \( 1 + 62.2iT - 7.92e3T^{2} \)
97 \( 1 + 124.T + 9.40e3T^{2} \)
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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

−9.298658193573259243930269974566, −8.303708137102356964219672897989, −7.33851732743563765815095085400, −6.63883814994343128113529167742, −6.33484792732960975005714064830, −5.25080577518498545709925877548, −4.17974991680375483144716163003, −3.40813934341592492468609471910, −2.67861688137078239359607768648, −1.47579951102444598491255350780, 0.13796071161197140231592230668, 0.866539520986737301996674551145, 2.51666465656612974725774855539, 3.45366884666049976908480489175, 3.79802390209566812433652428655, 5.31458991174551035590850879045, 5.84244726416050522283063949889, 6.52879153488440707173402838998, 7.32253354180164477801126009831, 8.260268460823991593950116774178

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