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Properties

Label 2-3584-3584.1973-c0-0-0
Degree $2$
Conductor $3584$
Sign $-0.482 - 0.876i$
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  + (−0.989 + 0.146i)2-s + (0.956 − 0.290i)4-s + (0.336 + 0.941i)7-s + (−0.903 + 0.427i)8-s + (0.740 + 0.671i)9-s + (−0.197 + 0.877i)11-s + (−0.471 − 0.881i)14-s + (0.831 − 0.555i)16-s + (−0.831 − 0.555i)18-s + (0.0661 − 0.896i)22-s + (−1.97 − 0.293i)23-s + (−0.242 + 0.970i)25-s + (0.595 + 0.803i)28-s + (−0.339 + 1.95i)29-s + (−0.740 + 0.671i)32-s + ⋯
L(s)  = 1  + (−0.989 + 0.146i)2-s + (0.956 − 0.290i)4-s + (0.336 + 0.941i)7-s + (−0.903 + 0.427i)8-s + (0.740 + 0.671i)9-s + (−0.197 + 0.877i)11-s + (−0.471 − 0.881i)14-s + (0.831 − 0.555i)16-s + (−0.831 − 0.555i)18-s + (0.0661 − 0.896i)22-s + (−1.97 − 0.293i)23-s + (−0.242 + 0.970i)25-s + (0.595 + 0.803i)28-s + (−0.339 + 1.95i)29-s + (−0.740 + 0.671i)32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $-0.482 - 0.876i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (1973, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ -0.482 - 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7086835087\)
\(L(\frac12)\) \(\approx\) \(0.7086835087\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.989 - 0.146i)T \)
7 \( 1 + (-0.336 - 0.941i)T \)
good3 \( 1 + (-0.740 - 0.671i)T^{2} \)
5 \( 1 + (0.242 - 0.970i)T^{2} \)
11 \( 1 + (0.197 - 0.877i)T + (-0.903 - 0.427i)T^{2} \)
13 \( 1 + (-0.857 - 0.514i)T^{2} \)
17 \( 1 + (-0.831 - 0.555i)T^{2} \)
19 \( 1 + (0.595 + 0.803i)T^{2} \)
23 \( 1 + (1.97 + 0.293i)T + (0.956 + 0.290i)T^{2} \)
29 \( 1 + (0.339 - 1.95i)T + (-0.941 - 0.336i)T^{2} \)
31 \( 1 + (0.382 - 0.923i)T^{2} \)
37 \( 1 + (-0.331 - 0.286i)T + (0.146 + 0.989i)T^{2} \)
41 \( 1 + (0.881 - 0.471i)T^{2} \)
43 \( 1 + (-0.466 + 1.05i)T + (-0.671 - 0.740i)T^{2} \)
47 \( 1 + (0.195 + 0.980i)T^{2} \)
53 \( 1 + (0.247 + 1.42i)T + (-0.941 + 0.336i)T^{2} \)
59 \( 1 + (0.857 - 0.514i)T^{2} \)
61 \( 1 + (-0.998 + 0.0490i)T^{2} \)
67 \( 1 + (1.32 + 1.26i)T + (0.0490 + 0.998i)T^{2} \)
71 \( 1 + (0.195 + 0.00961i)T + (0.995 + 0.0980i)T^{2} \)
73 \( 1 + (0.773 + 0.634i)T^{2} \)
79 \( 1 + (1.02 + 0.100i)T + (0.980 + 0.195i)T^{2} \)
83 \( 1 + (-0.146 + 0.989i)T^{2} \)
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

−8.908478859615437430615685749020, −8.281419605684460814689622254976, −7.53572089462551633438062631621, −7.07428481634442400919611975022, −6.06548403782554959407064841220, −5.35795643071794237516577484161, −4.56935525363625973546629421771, −3.29901394961623102111208550032, −2.04966582181370535042828017569, −1.74325808952564365839219676177, 0.55475327523535474830886727798, 1.63395222842703739702717334895, 2.72888089962688980607463980059, 3.88539825764811931696880311001, 4.30223231464922094473201397133, 6.01418051197581996790835132347, 6.17890078498808982649173255435, 7.34482382474953488450286250834, 7.78720355912209589332979139675, 8.384405332673998409589601257560

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