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Properties

Label 2-3584-7.6-c0-0-2
Degree $2$
Conductor $3584$
Sign $-i$
Analytic cond. $1.78864$
Root an. cond. $1.33740$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·5-s − i·7-s + 9-s − 1.41·11-s + 1.41i·13-s − 1.00·25-s + 2i·31-s + 1.41·35-s + 1.41·43-s + 1.41i·45-s + 2i·47-s − 49-s − 2.00i·55-s − 1.41i·61-s − i·63-s + ⋯
L(s)  = 1  + 1.41i·5-s − i·7-s + 9-s − 1.41·11-s + 1.41i·13-s − 1.00·25-s + 2i·31-s + 1.41·35-s + 1.41·43-s + 1.41i·45-s + 2i·47-s − 49-s − 2.00i·55-s − 1.41i·61-s − i·63-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $-i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (2561, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.127407360\)
\(L(\frac12)\) \(\approx\) \(1.127407360\)
\(L(1)\) 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 \( 1 - T^{2} \)
5 \( 1 - 1.41iT - T^{2} \)
11 \( 1 + 1.41T + T^{2} \)
13 \( 1 - 1.41iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 2iT - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.41T + T^{2} \)
47 \( 1 - 2iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.41iT - T^{2} \)
67 \( 1 + 1.41T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.023033473704469315051684180277, −7.79631770301545098106018998575, −7.42009246807009111000160565402, −6.79000529182725527899313953551, −6.25704832894590949948994758262, −4.96326697214951717021633596758, −4.29941116920293469845474378640, −3.41641202176248070030045285624, −2.58700198493936614235213626002, −1.50872690361613829391944336909, 0.66440081984730776364344582529, 1.98852164419987799898172843405, 2.84720679346905027335178056338, 4.08279796690732106460286932667, 4.88588452162000120400617885349, 5.50601390622913120500388026647, 5.95063896988352383909471638250, 7.37780300002046136379866558539, 7.889855822329043200921034708805, 8.520025146913641682011753170563

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