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Properties

Label 2-27584-1.1-c1-0-29
Degree $2$
Conductor $27584$
Sign $-1$
Analytic cond. $220.259$
Root an. cond. $14.8411$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $3$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 3·5-s − 4·7-s + 6·9-s − 5·11-s − 4·13-s + 9·15-s − 6·17-s − 7·19-s + 12·21-s + 3·23-s + 4·25-s − 9·27-s − 3·29-s − 4·31-s + 15·33-s + 12·35-s − 8·37-s + 12·39-s − 6·41-s − 8·43-s − 18·45-s − 6·47-s + 9·49-s + 18·51-s − 53-s + 15·55-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.34·5-s − 1.51·7-s + 2·9-s − 1.50·11-s − 1.10·13-s + 2.32·15-s − 1.45·17-s − 1.60·19-s + 2.61·21-s + 0.625·23-s + 4/5·25-s − 1.73·27-s − 0.557·29-s − 0.718·31-s + 2.61·33-s + 2.02·35-s − 1.31·37-s + 1.92·39-s − 0.937·41-s − 1.21·43-s − 2.68·45-s − 0.875·47-s + 9/7·49-s + 2.52·51-s − 0.137·53-s + 2.02·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27584\)    =    \(2^{6} \cdot 431\)
Sign: $-1$
Analytic conductor: \(220.259\)
Root analytic conductor: \(14.8411\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((2,\ 27584,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
431 \( 1 - T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−15.99959877968722, −15.56115539228788, −15.36627443446851, −14.80978929640748, −13.58729608371114, −12.98123984188194, −12.69922070167028, −12.46947215282012, −11.66552156276745, −11.22512334930725, −10.79663081194918, −10.15984485865329, −9.936033696520783, −8.897069541842533, −8.395912241224259, −7.481703975526001, −6.998081870837675, −6.731992002723620, −6.015191499655305, −5.320239109892303, −4.733091673538975, −4.314934684182771, −3.475735290570383, −2.763855965713960, −1.806843103914617, 0, 0, 0, 1.806843103914617, 2.763855965713960, 3.475735290570383, 4.314934684182771, 4.733091673538975, 5.320239109892303, 6.015191499655305, 6.731992002723620, 6.998081870837675, 7.481703975526001, 8.395912241224259, 8.897069541842533, 9.936033696520783, 10.15984485865329, 10.79663081194918, 11.22512334930725, 11.66552156276745, 12.46947215282012, 12.69922070167028, 12.98123984188194, 13.58729608371114, 14.80978929640748, 15.36627443446851, 15.56115539228788, 15.99959877968722

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