Properties

Label 2-3584-224.13-c0-0-0
Degree $2$
Conductor $3584$
Sign $0.195 - 0.980i$
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 + (0.292 + 0.707i)11-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (−0.292 + 0.707i)29-s + (−1.70 + 0.707i)37-s + (0.707 + 1.70i)43-s − 1.00i·49-s + (0.707 + 1.70i)53-s + 1.00·63-s + (−0.707 + 1.70i)67-s + (−0.707 − 0.292i)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 + (0.292 + 0.707i)11-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (−0.292 + 0.707i)29-s + (−1.70 + 0.707i)37-s + (0.707 + 1.70i)43-s − 1.00i·49-s + (0.707 + 1.70i)53-s + 1.00·63-s + (−0.707 + 1.70i)67-s + (−0.707 − 0.292i)77-s + 1.41i·79-s + 1.00i·81-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.195 - 0.980i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (2113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ 0.195 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9417996379\)
\(L(\frac12)\) \(\approx\) \(0.9417996379\)
\(L(1)\) 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 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
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 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.707 + 0.707i)T^{2} \)
67 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
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

−8.973140837725068775369346172277, −8.367536446704372837571398415693, −7.20804451460797334147010443232, −6.73332731206263269836249060569, −5.89318797626734875488982857651, −5.26601461054313002469683555138, −4.26690851059388906974143169980, −3.23205804989832047987685988740, −2.70305810637319658797469385740, −1.33818556008429459999772826554, 0.57184627605194205377383849911, 2.09369277184077846606514369400, 3.14739466895736509218870991284, 3.74922575953610651330772104277, 4.85305527769848657169341325760, 5.54905046090371344498692225880, 6.41308578932996328607184793168, 7.07700923915932512084401364385, 7.77752627077723301187017365798, 8.826664949840606702659203918090

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