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Properties

Label 2-3584-8.5-c1-0-3
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  + 2.47i·3-s + 2.59i·5-s − 7-s − 3.12·9-s − 4.33i·11-s + 4.86i·13-s − 6.42·15-s − 5.17·17-s + 7.11i·19-s − 2.47i·21-s − 2.29·23-s − 1.74·25-s − 0.314i·27-s − 8.20i·29-s − 1.04·31-s + ⋯
L(s)  = 1  + 1.42i·3-s + 1.16i·5-s − 0.377·7-s − 1.04·9-s − 1.30i·11-s + 1.34i·13-s − 1.65·15-s − 1.25·17-s + 1.63i·19-s − 0.540i·21-s − 0.479·23-s − 0.348·25-s − 0.0606i·27-s − 1.52i·29-s − 0.187·31-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

\(L(1)\) \(\approx\) \(0.5034347637\)
\(L(\frac12)\) \(\approx\) \(0.5034347637\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.47iT - 3T^{2} \)
5 \( 1 - 2.59iT - 5T^{2} \)
11 \( 1 + 4.33iT - 11T^{2} \)
13 \( 1 - 4.86iT - 13T^{2} \)
17 \( 1 + 5.17T + 17T^{2} \)
19 \( 1 - 7.11iT - 19T^{2} \)
23 \( 1 + 2.29T + 23T^{2} \)
29 \( 1 + 8.20iT - 29T^{2} \)
31 \( 1 + 1.04T + 31T^{2} \)
37 \( 1 - 4.77iT - 37T^{2} \)
41 \( 1 + 7.95T + 41T^{2} \)
43 \( 1 + 8.63iT - 43T^{2} \)
47 \( 1 - 6.55T + 47T^{2} \)
53 \( 1 - 10.3iT - 53T^{2} \)
59 \( 1 + 12.7iT - 59T^{2} \)
61 \( 1 + 3.05iT - 61T^{2} \)
67 \( 1 + 2.21iT - 67T^{2} \)
71 \( 1 + 8.38T + 71T^{2} \)
73 \( 1 + 2.61T + 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 + 3.18iT - 83T^{2} \)
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

−9.164484636036445823382285960899, −8.613352800897442334633536016533, −7.69396836784365478790608206809, −6.57012593010470954769706173201, −6.26179276426409791408182653112, −5.34298562688183934317668887101, −4.20262145030889763528954116323, −3.81294304433113342604463190535, −3.04797717829915490530808336559, −2.01888326608766007658636151483, 0.15374387963723536597159976569, 1.14424656594456727257199034130, 2.08803680899737126211406137665, 2.92071881761363701322711528813, 4.33634152412097530453804154749, 5.00437847471253542971404480090, 5.77546756107545450624959522037, 6.88872726269089309216751542881, 7.05641726568068208239770265019, 7.956547751405096939247699303470

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