Properties

Label 2-3584-1.1-c1-0-0
Degree $2$
Conductor $3584$
Sign $1$
Analytic cond. $28.6183$
Root an. cond. $5.34961$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.47·3-s − 2.59·5-s − 7-s + 3.12·9-s − 4.33·11-s + 4.86·13-s + 6.42·15-s − 5.17·17-s − 7.11·19-s + 2.47·21-s − 2.29·23-s + 1.74·25-s − 0.314·27-s − 8.20·29-s + 1.04·31-s + 10.7·33-s + 2.59·35-s − 4.77·37-s − 12.0·39-s + 7.95·41-s − 8.63·43-s − 8.12·45-s − 6.55·47-s + 49-s + 12.8·51-s − 10.3·53-s + 11.2·55-s + ⋯
L(s)  = 1  − 1.42·3-s − 1.16·5-s − 0.377·7-s + 1.04·9-s − 1.30·11-s + 1.34·13-s + 1.65·15-s − 1.25·17-s − 1.63·19-s + 0.540·21-s − 0.479·23-s + 0.348·25-s − 0.0606·27-s − 1.52·29-s + 0.187·31-s + 1.86·33-s + 0.438·35-s − 0.784·37-s − 1.92·39-s + 1.24·41-s − 1.31·43-s − 1.21·45-s − 0.956·47-s + 0.142·49-s + 1.79·51-s − 1.41·53-s + 1.51·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $1$
Analytic conductor: \(28.6183\)
Root analytic conductor: \(5.34961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.03022217805\)
\(L(\frac12)\) \(\approx\) \(0.03022217805\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
good3 \( 1 + 2.47T + 3T^{2} \)
5 \( 1 + 2.59T + 5T^{2} \)
11 \( 1 + 4.33T + 11T^{2} \)
13 \( 1 - 4.86T + 13T^{2} \)
17 \( 1 + 5.17T + 17T^{2} \)
19 \( 1 + 7.11T + 19T^{2} \)
23 \( 1 + 2.29T + 23T^{2} \)
29 \( 1 + 8.20T + 29T^{2} \)
31 \( 1 - 1.04T + 31T^{2} \)
37 \( 1 + 4.77T + 37T^{2} \)
41 \( 1 - 7.95T + 41T^{2} \)
43 \( 1 + 8.63T + 43T^{2} \)
47 \( 1 + 6.55T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + 12.7T + 59T^{2} \)
61 \( 1 + 3.05T + 61T^{2} \)
67 \( 1 - 2.21T + 67T^{2} \)
71 \( 1 + 8.38T + 71T^{2} \)
73 \( 1 - 2.61T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 - 3.18T + 83T^{2} \)
89 \( 1 - 8.12T + 89T^{2} \)
97 \( 1 - 1.91T + 97T^{2} \)
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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

−8.372526743890463899938540209026, −7.85326331301076442630516258043, −6.87869177111716610751821922526, −6.27381677141906970681605995476, −5.68151647282273241059860789033, −4.67949917908282837544267123415, −4.15215178487223966392233453577, −3.19403896906198433049446712565, −1.82319913526552768991724787840, −0.10474674767365683762810363620, 0.10474674767365683762810363620, 1.82319913526552768991724787840, 3.19403896906198433049446712565, 4.15215178487223966392233453577, 4.67949917908282837544267123415, 5.68151647282273241059860789033, 6.27381677141906970681605995476, 6.87869177111716610751821922526, 7.85326331301076442630516258043, 8.372526743890463899938540209026

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