Properties

Label 2-3584-3584.2533-c0-0-0
Degree $2$
Conductor $3584$
Sign $-0.824 + 0.565i$
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  + (0.0490 − 0.998i)2-s + (−0.995 − 0.0980i)4-s + (0.595 − 0.803i)7-s + (−0.146 + 0.989i)8-s + (−0.242 − 0.970i)9-s + (1.19 − 1.38i)11-s + (−0.773 − 0.634i)14-s + (0.980 + 0.195i)16-s + (−0.980 + 0.195i)18-s + (−1.32 − 1.26i)22-s + (0.0841 + 1.71i)23-s + (0.427 − 0.903i)25-s + (−0.671 + 0.740i)28-s + (0.416 − 0.137i)29-s + (0.242 − 0.970i)32-s + ⋯
L(s)  = 1  + (0.0490 − 0.998i)2-s + (−0.995 − 0.0980i)4-s + (0.595 − 0.803i)7-s + (−0.146 + 0.989i)8-s + (−0.242 − 0.970i)9-s + (1.19 − 1.38i)11-s + (−0.773 − 0.634i)14-s + (0.980 + 0.195i)16-s + (−0.980 + 0.195i)18-s + (−1.32 − 1.26i)22-s + (0.0841 + 1.71i)23-s + (0.427 − 0.903i)25-s + (−0.671 + 0.740i)28-s + (0.416 − 0.137i)29-s + (0.242 − 0.970i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $-0.824 + 0.565i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (2533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ -0.824 + 0.565i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.274902723\)
\(L(\frac12)\) \(\approx\) \(1.274902723\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0490 + 0.998i)T \)
7 \( 1 + (-0.595 + 0.803i)T \)
good3 \( 1 + (0.242 + 0.970i)T^{2} \)
5 \( 1 + (-0.427 + 0.903i)T^{2} \)
11 \( 1 + (-1.19 + 1.38i)T + (-0.146 - 0.989i)T^{2} \)
13 \( 1 + (-0.941 - 0.336i)T^{2} \)
17 \( 1 + (-0.980 + 0.195i)T^{2} \)
19 \( 1 + (-0.671 + 0.740i)T^{2} \)
23 \( 1 + (-0.0841 - 1.71i)T + (-0.995 + 0.0980i)T^{2} \)
29 \( 1 + (-0.416 + 0.137i)T + (0.803 - 0.595i)T^{2} \)
31 \( 1 + (0.923 + 0.382i)T^{2} \)
37 \( 1 + (0.0320 - 1.30i)T + (-0.998 - 0.0490i)T^{2} \)
41 \( 1 + (0.634 - 0.773i)T^{2} \)
43 \( 1 + (0.892 + 0.110i)T + (0.970 + 0.242i)T^{2} \)
47 \( 1 + (-0.831 - 0.555i)T^{2} \)
53 \( 1 + (0.936 + 0.309i)T + (0.803 + 0.595i)T^{2} \)
59 \( 1 + (0.941 - 0.336i)T^{2} \)
61 \( 1 + (-0.857 - 0.514i)T^{2} \)
67 \( 1 + (0.120 + 0.213i)T + (-0.514 + 0.857i)T^{2} \)
71 \( 1 + (-1.51 + 0.906i)T + (0.471 - 0.881i)T^{2} \)
73 \( 1 + (0.290 - 0.956i)T^{2} \)
79 \( 1 + (0.317 - 0.594i)T + (-0.555 - 0.831i)T^{2} \)
83 \( 1 + (0.998 - 0.0490i)T^{2} \)
89 \( 1 + (0.995 + 0.0980i)T^{2} \)
97 \( 1 + (0.382 - 0.923i)T^{2} \)
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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

−8.475505000989090787523452835868, −8.083932227969937577331471216828, −6.83683698191582498899865369650, −6.16740505689325483665693221689, −5.26052140253291227909851008042, −4.33550170329187956351812308604, −3.57748157197829710207729014039, −3.12102284541281755666609040041, −1.57587737621980811923607115982, −0.831846796716841781983216487102, 1.56805431416053129016135617233, 2.59415175963627749062485799261, 3.95628713451865247749407220528, 4.76254204143102645021806461715, 5.14903420149775742122918492066, 6.14700919025491995076493023404, 6.83642697220030421738116131881, 7.51068798496274728897717068934, 8.269791028660640309737178944322, 8.871847565292255299693771236244

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