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Properties

Label 2-3584-16.13-c1-0-84
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)3-s − i·7-s − i·9-s + (−2 + 2i)11-s + (2 + 2i)13-s − 6·17-s + (−3 − 3i)19-s + (1 − i)21-s + 5i·25-s + (4 − 4i)27-s + (1 + i)29-s − 10·31-s − 4·33-s + (−3 + 3i)37-s + 4i·39-s + ⋯
L(s)  = 1  + (0.577 + 0.577i)3-s − 0.377i·7-s − 0.333i·9-s + (−0.603 + 0.603i)11-s + (0.554 + 0.554i)13-s − 1.45·17-s + (−0.688 − 0.688i)19-s + (0.218 − 0.218i)21-s + i·25-s + (0.769 − 0.769i)27-s + (0.185 + 0.185i)29-s − 1.79·31-s − 0.696·33-s + (−0.493 + 0.493i)37-s + 0.640i·39-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: \(1\)
Selberg data: \((2,\ 3584,\ (\ :1/2),\ -0.923 + 0.382i)\)

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 \( 1 \)
7 \( 1 + iT \)
good3 \( 1 + (-1 - i)T + 3iT^{2} \)
5 \( 1 - 5iT^{2} \)
11 \( 1 + (2 - 2i)T - 11iT^{2} \)
13 \( 1 + (-2 - 2i)T + 13iT^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + (3 + 3i)T + 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-1 - i)T + 29iT^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 + 10iT - 41T^{2} \)
43 \( 1 + (-6 + 6i)T - 43iT^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 + (9 - 9i)T - 53iT^{2} \)
59 \( 1 + (1 - i)T - 59iT^{2} \)
61 \( 1 + 61iT^{2} \)
67 \( 1 + (8 + 8i)T + 67iT^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + (5 + 5i)T + 83iT^{2} \)
89 \( 1 - 2iT - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−8.507871975840782453690850853511, −7.34922678383216768506795839539, −6.93786836489810613847079537759, −6.02752393820995617136026947341, −4.99012670020190997956704837157, −4.24378161615368493495217603535, −3.67268291627944659813031966978, −2.63640686141437127103802973184, −1.70503560200793821722600108697, 0, 1.61684759377345009731048383277, 2.42199510024285480985843967912, 3.18415701380702655979221065690, 4.24906535165083519928443101487, 5.14542754358432177937005458236, 6.02977766227982988445550899149, 6.60551680947599689258316124031, 7.64390093900598107795477986980, 8.156363640030683973322346271502

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