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Properties

Label 2-3584-3584.2645-c0-0-0
Degree $2$
Conductor $3584$
Sign $-0.278 + 0.960i$
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  + (0.857 − 0.514i)2-s + (0.471 − 0.881i)4-s + (0.740 − 0.671i)7-s + (−0.0490 − 0.998i)8-s + (0.903 − 0.427i)9-s + (−1.08 − 1.14i)11-s + (0.290 − 0.956i)14-s + (−0.555 − 0.831i)16-s + (0.555 − 0.831i)18-s + (−1.51 − 0.420i)22-s + (0.577 + 0.346i)23-s + (0.146 + 0.989i)25-s + (−0.242 − 0.970i)28-s + (−1.82 + 0.808i)29-s + (−0.903 − 0.427i)32-s + ⋯
L(s)  = 1  + (0.857 − 0.514i)2-s + (0.471 − 0.881i)4-s + (0.740 − 0.671i)7-s + (−0.0490 − 0.998i)8-s + (0.903 − 0.427i)9-s + (−1.08 − 1.14i)11-s + (0.290 − 0.956i)14-s + (−0.555 − 0.831i)16-s + (0.555 − 0.831i)18-s + (−1.51 − 0.420i)22-s + (0.577 + 0.346i)23-s + (0.146 + 0.989i)25-s + (−0.242 − 0.970i)28-s + (−1.82 + 0.808i)29-s + (−0.903 − 0.427i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $-0.278 + 0.960i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (2645, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ -0.278 + 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.261155759\)
\(L(\frac12)\) \(\approx\) \(2.261155759\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.857 + 0.514i)T \)
7 \( 1 + (-0.740 + 0.671i)T \)
good3 \( 1 + (-0.903 + 0.427i)T^{2} \)
5 \( 1 + (-0.146 - 0.989i)T^{2} \)
11 \( 1 + (1.08 + 1.14i)T + (-0.0490 + 0.998i)T^{2} \)
13 \( 1 + (-0.595 - 0.803i)T^{2} \)
17 \( 1 + (0.555 - 0.831i)T^{2} \)
19 \( 1 + (-0.242 - 0.970i)T^{2} \)
23 \( 1 + (-0.577 - 0.346i)T + (0.471 + 0.881i)T^{2} \)
29 \( 1 + (1.82 - 0.808i)T + (0.671 - 0.740i)T^{2} \)
31 \( 1 + (-0.382 + 0.923i)T^{2} \)
37 \( 1 + (0.679 - 1.19i)T + (-0.514 - 0.857i)T^{2} \)
41 \( 1 + (0.956 + 0.290i)T^{2} \)
43 \( 1 + (0.608 - 0.385i)T + (0.427 - 0.903i)T^{2} \)
47 \( 1 + (-0.980 + 0.195i)T^{2} \)
53 \( 1 + (-1.80 - 0.798i)T + (0.671 + 0.740i)T^{2} \)
59 \( 1 + (0.595 - 0.803i)T^{2} \)
61 \( 1 + (-0.336 + 0.941i)T^{2} \)
67 \( 1 + (-1.05 - 0.182i)T + (0.941 + 0.336i)T^{2} \)
71 \( 1 + (0.427 + 1.19i)T + (-0.773 + 0.634i)T^{2} \)
73 \( 1 + (-0.0980 - 0.995i)T^{2} \)
79 \( 1 + (-1.24 + 1.01i)T + (0.195 - 0.980i)T^{2} \)
83 \( 1 + (0.514 - 0.857i)T^{2} \)
89 \( 1 + (-0.471 + 0.881i)T^{2} \)
97 \( 1 + (0.923 + 0.382i)T^{2} \)
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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

−8.544832502039318606729071556076, −7.46923645549576915491553202295, −7.14029954295454924187089720013, −6.08208572693777735495756720192, −5.26169701206936950731353286468, −4.78185318285138065162055101257, −3.68036676558570872369268376774, −3.26587050150046635800302115137, −1.93304978220651135394455757379, −1.02484240350525830235613548046, 2.06229535001242759944938693659, 2.34353937028025979978821598677, 3.78296574303864241347253703799, 4.54269224745769854737852238438, 5.17621890499802498225697353848, 5.69397952151923202541924777845, 6.87575884622963530243749672968, 7.36468382315395686844771150948, 7.999686234415792899073884035301, 8.668409786785635192885273307828

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