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Properties

Label 2-1584-3.2-c2-0-16
Degree $2$
Conductor $1584$
Sign $0.816 + 0.577i$
Analytic cond. $43.1608$
Root an. cond. $6.56969$
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  + 6.21i·5-s − 11.1·7-s + 3.31i·11-s − 17.7·13-s − 8.53i·17-s − 16.4·19-s − 25.0i·23-s − 13.5·25-s − 9.46i·29-s + 46.1·31-s − 69.2i·35-s − 37.4·37-s + 51.8i·41-s + 55.6·43-s + 37.8i·47-s + ⋯
L(s)  = 1  + 1.24i·5-s − 1.59·7-s + 0.301i·11-s − 1.36·13-s − 0.502i·17-s − 0.866·19-s − 1.09i·23-s − 0.543·25-s − 0.326i·29-s + 1.49·31-s − 1.97i·35-s − 1.01·37-s + 1.26i·41-s + 1.29·43-s + 0.805i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1584\)    =    \(2^{4} \cdot 3^{2} \cdot 11\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(43.1608\)
Root analytic conductor: \(6.56969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1584} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1584,\ (\ :1),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7733425825\)
\(L(\frac12)\) \(\approx\) \(0.7733425825\)
\(L(2)\) 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 \( 1 \)
11 \( 1 - 3.31iT \)
good5 \( 1 - 6.21iT - 25T^{2} \)
7 \( 1 + 11.1T + 49T^{2} \)
13 \( 1 + 17.7T + 169T^{2} \)
17 \( 1 + 8.53iT - 289T^{2} \)
19 \( 1 + 16.4T + 361T^{2} \)
23 \( 1 + 25.0iT - 529T^{2} \)
29 \( 1 + 9.46iT - 841T^{2} \)
31 \( 1 - 46.1T + 961T^{2} \)
37 \( 1 + 37.4T + 1.36e3T^{2} \)
41 \( 1 - 51.8iT - 1.68e3T^{2} \)
43 \( 1 - 55.6T + 1.84e3T^{2} \)
47 \( 1 - 37.8iT - 2.20e3T^{2} \)
53 \( 1 + 58.5iT - 2.80e3T^{2} \)
59 \( 1 + 44.6iT - 3.48e3T^{2} \)
61 \( 1 + 10.1T + 3.72e3T^{2} \)
67 \( 1 - 58.3T + 4.48e3T^{2} \)
71 \( 1 - 75.1iT - 5.04e3T^{2} \)
73 \( 1 - 112.T + 5.32e3T^{2} \)
79 \( 1 + 52.2T + 6.24e3T^{2} \)
83 \( 1 - 18.0iT - 6.88e3T^{2} \)
89 \( 1 - 136. iT - 7.92e3T^{2} \)
97 \( 1 - 127.T + 9.40e3T^{2} \)
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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

−9.493130227638597920479631937317, −8.308125946667552224643009202784, −7.31004272689267568618826788161, −6.61218814888368044512507471901, −6.33991699056392629071811364894, −4.98145230238658235548876748717, −3.95163067774874393461827198437, −2.80833909071347147290039768949, −2.50791107135214422548089288100, −0.29866670447237979631722448516, 0.73565631840000546129069396716, 2.22565777960903220277608826347, 3.34902549949287365991359387023, 4.29102049954717682691779719099, 5.21534795257621470443754870656, 6.01343249433722302395360695420, 6.87116204901053044445405627487, 7.72930212002958631076269217594, 8.779744736507970962647021342506, 9.192909658320878921291721518526

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