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Properties

Label 2-3584-16.13-c1-0-55
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 + (−2 + 2i)5-s + i·7-s − i·9-s + (−2 + 2i)11-s + 4·15-s − 2·17-s + (3 + 3i)19-s + (1 − i)21-s − 4i·23-s − 3i·25-s + (−4 + 4i)27-s + (7 + 7i)29-s + 6·31-s + 4·33-s + ⋯
L(s)  = 1  + (−0.577 − 0.577i)3-s + (−0.894 + 0.894i)5-s + 0.377i·7-s − 0.333i·9-s + (−0.603 + 0.603i)11-s + 1.03·15-s − 0.485·17-s + (0.688 + 0.688i)19-s + (0.218 − 0.218i)21-s − 0.834i·23-s − 0.600i·25-s + (−0.769 + 0.769i)27-s + (1.29 + 1.29i)29-s + 1.07·31-s + 0.696·33-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: \(1\)
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 \( 1 + (2 - 2i)T - 5iT^{2} \)
11 \( 1 + (2 - 2i)T - 11iT^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + (-3 - 3i)T + 19iT^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + (-7 - 7i)T + 29iT^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 - 2iT - 41T^{2} \)
43 \( 1 + (6 - 6i)T - 43iT^{2} \)
47 \( 1 + 10T + 47T^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 + (7 - 7i)T - 59iT^{2} \)
61 \( 1 + (-2 - 2i)T + 61iT^{2} \)
67 \( 1 + (8 + 8i)T + 67iT^{2} \)
71 \( 1 - 4iT - 71T^{2} \)
73 \( 1 + 14iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + (3 + 3i)T + 83iT^{2} \)
89 \( 1 - 10iT - 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−8.011448880203797533884299140563, −7.45285611450903001395411246290, −6.62336589275315757970408163728, −6.34434427454256381569107007670, −5.20984902525311716493396149347, −4.45743586498327452295321344360, −3.34932312375998979076903337646, −2.72124377519638254412480312830, −1.40816621546170426044088423240, 0, 0.997819047486142157950637454039, 2.57356930524888605710413905810, 3.62707727887976064415127950913, 4.55845557930761765672675216511, 4.85158628959594320109486251729, 5.69442957324611895193295124170, 6.60504228364565900026710850533, 7.61911714559719413092592010498, 8.145520862771066887239374850421

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