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Properties

Label 2-3584-224.181-c0-0-0
Degree $2$
Conductor $3584$
Sign $0.980 - 0.195i$
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  + (0.707 + 0.707i)7-s + (0.707 − 0.707i)9-s + (1.70 + 0.707i)11-s + (1 − i)23-s + (−0.707 − 0.707i)25-s + (−1.70 + 0.707i)29-s + (−0.292 + 0.707i)37-s + (−0.707 − 0.292i)43-s + 1.00i·49-s + (−0.707 − 0.292i)53-s + 1.00·63-s + (0.707 − 0.292i)67-s + (0.707 + 1.70i)77-s + 1.41i·79-s − 1.00i·81-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)7-s + (0.707 − 0.707i)9-s + (1.70 + 0.707i)11-s + (1 − i)23-s + (−0.707 − 0.707i)25-s + (−1.70 + 0.707i)29-s + (−0.292 + 0.707i)37-s + (−0.707 − 0.292i)43-s + 1.00i·49-s + (−0.707 − 0.292i)53-s + 1.00·63-s + (0.707 − 0.292i)67-s + (0.707 + 1.70i)77-s + 1.41i·79-s − 1.00i·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.980 - 0.195i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (1217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ 0.980 - 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.622381044\)
\(L(\frac12)\) \(\approx\) \(1.622381044\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (0.707 - 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (-1 + i)T - iT^{2} \)
29 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (-0.707 + 0.707i)T^{2} \)
67 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
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

−8.855873323106419223645913347163, −8.123119352432503886818050778331, −7.05629492401487632456523253250, −6.69946146839539796290721903209, −5.82334399049606068110052591746, −4.84090532637585864881173315109, −4.19410894358792032749728721280, −3.38597114131175002917153142418, −2.05716030755713228218105476889, −1.31980013440966420702442776858, 1.24491026419306281339092878023, 1.90039386967745192039866151846, 3.52202474950726525188352021695, 3.95926916136639188772365098370, 4.87983044166724496821516152477, 5.66895450037369079482594448878, 6.58375744163703846495940197755, 7.40544893255494694252710352106, 7.74119283722102375461676324619, 8.796044459468968114966148288013

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