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Properties

Label 2-3584-3584.2589-c0-0-0
Degree $2$
Conductor $3584$
Sign $0.876 + 0.482i$
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  + (0.146 − 0.989i)2-s + (−0.956 − 0.290i)4-s + (0.941 + 0.336i)7-s + (−0.427 + 0.903i)8-s + (−0.671 − 0.740i)9-s + (−1.01 + 1.60i)11-s + (0.471 − 0.881i)14-s + (0.831 + 0.555i)16-s + (−0.831 + 0.555i)18-s + (1.43 + 1.24i)22-s + (0.0143 + 0.0970i)23-s + (0.970 − 0.242i)25-s + (−0.803 − 0.595i)28-s + (1.28 + 0.907i)29-s + (0.671 − 0.740i)32-s + ⋯
L(s)  = 1  + (0.146 − 0.989i)2-s + (−0.956 − 0.290i)4-s + (0.941 + 0.336i)7-s + (−0.427 + 0.903i)8-s + (−0.671 − 0.740i)9-s + (−1.01 + 1.60i)11-s + (0.471 − 0.881i)14-s + (0.831 + 0.555i)16-s + (−0.831 + 0.555i)18-s + (1.43 + 1.24i)22-s + (0.0143 + 0.0970i)23-s + (0.970 − 0.242i)25-s + (−0.803 − 0.595i)28-s + (1.28 + 0.907i)29-s + (0.671 − 0.740i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.876 + 0.482i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (2589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ 0.876 + 0.482i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.184715585\)
\(L(\frac12)\) \(\approx\) \(1.184715585\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.146 + 0.989i)T \)
7 \( 1 + (-0.941 - 0.336i)T \)
good3 \( 1 + (0.671 + 0.740i)T^{2} \)
5 \( 1 + (-0.970 + 0.242i)T^{2} \)
11 \( 1 + (1.01 - 1.60i)T + (-0.427 - 0.903i)T^{2} \)
13 \( 1 + (0.514 + 0.857i)T^{2} \)
17 \( 1 + (-0.831 + 0.555i)T^{2} \)
19 \( 1 + (-0.803 - 0.595i)T^{2} \)
23 \( 1 + (-0.0143 - 0.0970i)T + (-0.956 + 0.290i)T^{2} \)
29 \( 1 + (-1.28 - 0.907i)T + (0.336 + 0.941i)T^{2} \)
31 \( 1 + (0.382 + 0.923i)T^{2} \)
37 \( 1 + (-1.06 - 0.0787i)T + (0.989 + 0.146i)T^{2} \)
41 \( 1 + (-0.881 - 0.471i)T^{2} \)
43 \( 1 + (-0.123 + 0.319i)T + (-0.740 - 0.671i)T^{2} \)
47 \( 1 + (0.195 - 0.980i)T^{2} \)
53 \( 1 + (0.0401 - 0.0282i)T + (0.336 - 0.941i)T^{2} \)
59 \( 1 + (-0.514 + 0.857i)T^{2} \)
61 \( 1 + (-0.0490 + 0.998i)T^{2} \)
67 \( 1 + (0.0457 - 1.86i)T + (-0.998 - 0.0490i)T^{2} \)
71 \( 1 + (-0.00961 - 0.195i)T + (-0.995 + 0.0980i)T^{2} \)
73 \( 1 + (-0.773 + 0.634i)T^{2} \)
79 \( 1 + (-1.70 + 0.168i)T + (0.980 - 0.195i)T^{2} \)
83 \( 1 + (-0.989 + 0.146i)T^{2} \)
89 \( 1 + (0.956 + 0.290i)T^{2} \)
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

−8.761811103176881726462128583030, −8.172876714663338609686119838839, −7.33989083517038951014833865360, −6.32017382938896137438884893328, −5.26628134372252742188220297766, −4.87726581450129612506823592248, −4.06951009008829820782393292988, −2.86158885991050434038168529049, −2.33381284633268643189914897923, −1.17447485124913976827740736845, 0.77897634115373715453091433857, 2.54322869438296648136467745184, 3.37563357341378819174933219127, 4.56188136455430973604507584744, 5.07527307394460430239065406828, 5.79457673116683888929519756481, 6.44355479395221887476412062307, 7.54566111820535664666228303115, 8.100388596554196922054187587715, 8.384930701542103513505333133686

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