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Properties

Label 2-3584-16.13-c1-0-62
Degree $2$
Conductor $3584$
Sign $0.923 - 0.382i$
Analytic cond. $28.6183$
Root an. cond. $5.34961$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 + 1.73i)3-s + (0.732 − 0.732i)5-s − i·7-s + 2.99i·9-s + (2 − 2i)11-s + (1.26 + 1.26i)13-s + 2.53·15-s + 3.46·17-s + (1.73 + 1.73i)19-s + (1.73 − 1.73i)21-s − 2.53i·23-s + 3.92i·25-s + (−4.46 − 4.46i)29-s − 0.535·31-s + 6.92·33-s + ⋯
L(s)  = 1  + (0.999 + 0.999i)3-s + (0.327 − 0.327i)5-s − 0.377i·7-s + 0.999i·9-s + (0.603 − 0.603i)11-s + (0.351 + 0.351i)13-s + 0.654·15-s + 0.840·17-s + (0.397 + 0.397i)19-s + (0.377 − 0.377i)21-s − 0.528i·23-s + 0.785i·25-s + (−0.828 − 0.828i)29-s − 0.0962·31-s + 1.20·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(28.6183\)
Root analytic conductor: \(5.34961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (2689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :1/2),\ 0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.247628923\)
\(L(\frac12)\) \(\approx\) \(3.247628923\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good3 \( 1 + (-1.73 - 1.73i)T + 3iT^{2} \)
5 \( 1 + (-0.732 + 0.732i)T - 5iT^{2} \)
11 \( 1 + (-2 + 2i)T - 11iT^{2} \)
13 \( 1 + (-1.26 - 1.26i)T + 13iT^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + (-1.73 - 1.73i)T + 19iT^{2} \)
23 \( 1 + 2.53iT - 23T^{2} \)
29 \( 1 + (4.46 + 4.46i)T + 29iT^{2} \)
31 \( 1 + 0.535T + 31T^{2} \)
37 \( 1 + (-6.46 + 6.46i)T - 37iT^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 + (0.535 - 0.535i)T - 43iT^{2} \)
47 \( 1 - 4.53T + 47T^{2} \)
53 \( 1 + (1 - i)T - 53iT^{2} \)
59 \( 1 + (5.73 - 5.73i)T - 59iT^{2} \)
61 \( 1 + (8.73 + 8.73i)T + 61iT^{2} \)
67 \( 1 + (-10.9 - 10.9i)T + 67iT^{2} \)
71 \( 1 - 4iT - 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + (5.73 + 5.73i)T + 83iT^{2} \)
89 \( 1 - 4.92iT - 89T^{2} \)
97 \( 1 - 18.3T + 97T^{2} \)
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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

−8.827791769960048595138783659865, −7.993801101230309543845394821894, −7.37081584239676867129118342381, −6.20796938951244665122144440338, −5.57278041637856831076628239000, −4.55404713996549990672456122295, −3.81865984675934158102528128554, −3.34566521854026752998123201384, −2.22292425794912675678658146523, −1.02070509121374482114457910290, 1.15927683176533795144552595686, 1.95285918034023277677628342213, 2.86727706459602687816742225816, 3.48098352390004185953318540282, 4.68524693099458368402991029854, 5.69246043021082417419977764650, 6.45424858552572516540979719618, 7.13509066608405432544008565744, 7.79793484305089595130888011562, 8.348723358277213939969767752612

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