Properties

Label 2-1584-1.1-c3-0-14
Degree $2$
Conductor $1584$
Sign $1$
Analytic cond. $93.4590$
Root an. cond. $9.66742$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 11·11-s − 88·13-s + 66·17-s + 40·19-s + 6·23-s − 125·25-s + 54·29-s − 8·31-s − 106·37-s − 354·41-s + 124·43-s + 546·47-s − 339·49-s + 408·53-s + 552·59-s + 404·61-s + 4·67-s + 126·71-s − 166·73-s + 22·77-s + 874·79-s + 444·83-s − 1.00e3·89-s + 176·91-s − 802·97-s + 1.71e3·101-s + ⋯
L(s)  = 1  − 0.107·7-s − 0.301·11-s − 1.87·13-s + 0.941·17-s + 0.482·19-s + 0.0543·23-s − 25-s + 0.345·29-s − 0.0463·31-s − 0.470·37-s − 1.34·41-s + 0.439·43-s + 1.69·47-s − 0.988·49-s + 1.05·53-s + 1.21·59-s + 0.847·61-s + 0.00729·67-s + 0.210·71-s − 0.266·73-s + 0.0325·77-s + 1.24·79-s + 0.587·83-s − 1.19·89-s + 0.202·91-s − 0.839·97-s + 1.68·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1584\)    =    \(2^{4} \cdot 3^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(93.4590\)
Root analytic conductor: \(9.66742\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1584,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.550745463\)
\(L(\frac12)\) \(\approx\) \(1.550745463\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
good5 \( 1 + p^{3} T^{2} \)
7 \( 1 + 2 T + p^{3} T^{2} \)
13 \( 1 + 88 T + p^{3} T^{2} \)
17 \( 1 - 66 T + p^{3} T^{2} \)
19 \( 1 - 40 T + p^{3} T^{2} \)
23 \( 1 - 6 T + p^{3} T^{2} \)
29 \( 1 - 54 T + p^{3} T^{2} \)
31 \( 1 + 8 T + p^{3} T^{2} \)
37 \( 1 + 106 T + p^{3} T^{2} \)
41 \( 1 + 354 T + p^{3} T^{2} \)
43 \( 1 - 124 T + p^{3} T^{2} \)
47 \( 1 - 546 T + p^{3} T^{2} \)
53 \( 1 - 408 T + p^{3} T^{2} \)
59 \( 1 - 552 T + p^{3} T^{2} \)
61 \( 1 - 404 T + p^{3} T^{2} \)
67 \( 1 - 4 T + p^{3} T^{2} \)
71 \( 1 - 126 T + p^{3} T^{2} \)
73 \( 1 + 166 T + p^{3} T^{2} \)
79 \( 1 - 874 T + p^{3} T^{2} \)
83 \( 1 - 444 T + p^{3} T^{2} \)
89 \( 1 + 1002 T + p^{3} T^{2} \)
97 \( 1 + 802 T + p^{3} T^{2} \)
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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

−9.168166762156785688734842454821, −8.126583848949444245837767817889, −7.46585567437421522353954709563, −6.80774901067437801673882038337, −5.57700398232703899787414335873, −5.08430646054821277136859387495, −3.97675481699700471609695543513, −2.93031583173743060384752106683, −2.02406840555751295424658957424, −0.58294907090338085918178071776, 0.58294907090338085918178071776, 2.02406840555751295424658957424, 2.93031583173743060384752106683, 3.97675481699700471609695543513, 5.08430646054821277136859387495, 5.57700398232703899787414335873, 6.80774901067437801673882038337, 7.46585567437421522353954709563, 8.126583848949444245837767817889, 9.168166762156785688734842454821

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