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Properties

Label 2-3584-8.5-c1-0-75
Degree $2$
Conductor $3584$
Sign $i$
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  + 1.81i·3-s − 2.66i·5-s − 7-s − 0.295·9-s − 4.22i·11-s + 1.10i·13-s + 4.84·15-s + 5.14·17-s − 1.59i·19-s − 1.81i·21-s − 4.83·23-s − 2.12·25-s + 4.90i·27-s − 4.03i·29-s − 2.30·31-s + ⋯
L(s)  = 1  + 1.04i·3-s − 1.19i·5-s − 0.377·7-s − 0.0984·9-s − 1.27i·11-s + 0.306i·13-s + 1.25·15-s + 1.24·17-s − 0.364i·19-s − 0.396i·21-s − 1.00·23-s − 0.425·25-s + 0.944i·27-s − 0.749i·29-s − 0.414·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $i$
Analytic conductor: \(28.6183\)
Root analytic conductor: \(5.34961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (1793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.275177889\)
\(L(\frac12)\) \(\approx\) \(1.275177889\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.81iT - 3T^{2} \)
5 \( 1 + 2.66iT - 5T^{2} \)
11 \( 1 + 4.22iT - 11T^{2} \)
13 \( 1 - 1.10iT - 13T^{2} \)
17 \( 1 - 5.14T + 17T^{2} \)
19 \( 1 + 1.59iT - 19T^{2} \)
23 \( 1 + 4.83T + 23T^{2} \)
29 \( 1 + 4.03iT - 29T^{2} \)
31 \( 1 + 2.30T + 31T^{2} \)
37 \( 1 + 0.485iT - 37T^{2} \)
41 \( 1 + 4.14T + 41T^{2} \)
43 \( 1 + 4.91iT - 43T^{2} \)
47 \( 1 + 6.97T + 47T^{2} \)
53 \( 1 - 6.45iT - 53T^{2} \)
59 \( 1 + 0.825iT - 59T^{2} \)
61 \( 1 + 15.3iT - 61T^{2} \)
67 \( 1 - 1.98iT - 67T^{2} \)
71 \( 1 + 7.69T + 71T^{2} \)
73 \( 1 - 16.9T + 73T^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 - 3.82iT - 83T^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−8.449592984380181413105596645062, −7.950468933640385528457116238094, −6.82011348791505596335534705001, −5.80666008915318389141100862018, −5.33862874818878621492892293401, −4.49726356703313016387530643308, −3.79439753417172263465509180328, −3.09294571680672973065899396696, −1.57840833887417553677322955399, −0.38752071470393420487497864610, 1.33959147903078549405144409670, 2.20517703576733931542535115405, 3.09301747757577985332047208879, 3.91707064404315846104236623586, 5.07914899724366044788352821077, 6.06862306137403500329793732747, 6.62635727294480019056684881488, 7.34335364399030892912250642447, 7.62280950654166029715340442506, 8.502779176000951346990022729547

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