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Properties

Label 2-984-984.539-c0-0-0
Degree $2$
Conductor $984$
Sign $0.172 - 0.984i$
Analytic cond. $0.491079$
Root an. cond. $0.700770$
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  + (−0.891 − 0.453i)2-s + (0.309 + 0.951i)3-s + (0.587 + 0.809i)4-s + (0.156 − 0.987i)6-s + (−0.156 − 0.987i)8-s + (−0.809 + 0.587i)9-s + (−0.243 + 1.01i)11-s + (−0.587 + 0.809i)12-s + (−0.309 + 0.951i)16-s + (0.965 + 1.57i)17-s + (0.987 − 0.156i)18-s + (−0.987 − 0.843i)19-s + (0.678 − 0.794i)22-s + (0.891 − 0.453i)24-s + (0.951 + 0.309i)25-s + ⋯
L(s)  = 1  + (−0.891 − 0.453i)2-s + (0.309 + 0.951i)3-s + (0.587 + 0.809i)4-s + (0.156 − 0.987i)6-s + (−0.156 − 0.987i)8-s + (−0.809 + 0.587i)9-s + (−0.243 + 1.01i)11-s + (−0.587 + 0.809i)12-s + (−0.309 + 0.951i)16-s + (0.965 + 1.57i)17-s + (0.987 − 0.156i)18-s + (−0.987 − 0.843i)19-s + (0.678 − 0.794i)22-s + (0.891 − 0.453i)24-s + (0.951 + 0.309i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.172 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.172 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(984\)    =    \(2^{3} \cdot 3 \cdot 41\)
Sign: $0.172 - 0.984i$
Analytic conductor: \(0.491079\)
Root analytic conductor: \(0.700770\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{984} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 984,\ (\ :0),\ 0.172 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6835670034\)
\(L(\frac12)\) \(\approx\) \(0.6835670034\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.891 + 0.453i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (0.987 + 0.156i)T \)
good5 \( 1 + (-0.951 - 0.309i)T^{2} \)
7 \( 1 + (-0.987 + 0.156i)T^{2} \)
11 \( 1 + (0.243 - 1.01i)T + (-0.891 - 0.453i)T^{2} \)
13 \( 1 + (-0.156 + 0.987i)T^{2} \)
17 \( 1 + (-0.965 - 1.57i)T + (-0.453 + 0.891i)T^{2} \)
19 \( 1 + (0.987 + 0.843i)T + (0.156 + 0.987i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (0.453 + 0.891i)T^{2} \)
31 \( 1 + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + (0.863 - 1.69i)T + (-0.587 - 0.809i)T^{2} \)
47 \( 1 + (0.987 + 0.156i)T^{2} \)
53 \( 1 + (-0.453 - 0.891i)T^{2} \)
59 \( 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.587 + 0.809i)T^{2} \)
67 \( 1 + (0.108 + 0.453i)T + (-0.891 + 0.453i)T^{2} \)
71 \( 1 + (-0.891 - 0.453i)T^{2} \)
73 \( 1 + (0.437 + 0.437i)T + iT^{2} \)
79 \( 1 + (-0.707 + 0.707i)T^{2} \)
83 \( 1 - 1.61iT - T^{2} \)
89 \( 1 + (0.152 + 1.93i)T + (-0.987 + 0.156i)T^{2} \)
97 \( 1 + (-1.47 + 0.355i)T + (0.891 - 0.453i)T^{2} \)
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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

−10.26220725760011028804887799308, −9.723625589872419176464221698512, −8.750791358095341734589960100935, −8.256296759210285109877320563024, −7.28240891432548080993332784117, −6.24792502592221420401757378211, −4.92892010231542889980648799105, −3.98247116430882561024793248267, −3.00157811190859520223560280267, −1.86349957019538516759405936273, 0.851669057308698973672044917377, 2.27301522081470766190512000892, 3.31720628577364964262638657138, 5.20722564623695890595917760107, 5.96856998986629964722327157306, 6.87342246502265020859358565491, 7.51295779780671043488199565596, 8.463406370670805391744399736390, 8.780176687815548979284392824980, 9.916538067768579014648007202182

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