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Properties

Label 2-1984-1984.805-c0-0-2
Degree $2$
Conductor $1984$
Sign $-0.634 + 0.773i$
Analytic cond. $0.990144$
Root an. cond. $0.995060$
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  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.0761 + 0.382i)5-s + (0.541 − 1.30i)7-s + (−0.707 − 0.707i)8-s + (−0.382 − 0.923i)9-s + (0.216 + 0.324i)10-s + (−0.541 − 1.30i)14-s − 1.00·16-s + (−0.923 − 0.382i)18-s + (−1.63 + 0.324i)19-s + (0.382 + 0.0761i)20-s + (0.783 + 0.324i)25-s + (−1.30 − 0.541i)28-s + i·31-s + (−0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.0761 + 0.382i)5-s + (0.541 − 1.30i)7-s + (−0.707 − 0.707i)8-s + (−0.382 − 0.923i)9-s + (0.216 + 0.324i)10-s + (−0.541 − 1.30i)14-s − 1.00·16-s + (−0.923 − 0.382i)18-s + (−1.63 + 0.324i)19-s + (0.382 + 0.0761i)20-s + (0.783 + 0.324i)25-s + (−1.30 − 0.541i)28-s + i·31-s + (−0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $-0.634 + 0.773i$
Analytic conductor: \(0.990144\)
Root analytic conductor: \(0.995060\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1984} (805, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1984,\ (\ :0),\ -0.634 + 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.538000164\)
\(L(\frac12)\) \(\approx\) \(1.538000164\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
31 \( 1 - iT \)
good3 \( 1 + (0.382 + 0.923i)T^{2} \)
5 \( 1 + (0.0761 - 0.382i)T + (-0.923 - 0.382i)T^{2} \)
7 \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.382 + 0.923i)T^{2} \)
13 \( 1 + (0.923 - 0.382i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (1.63 - 0.324i)T + (0.923 - 0.382i)T^{2} \)
23 \( 1 + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (-0.382 - 0.923i)T^{2} \)
37 \( 1 + (-0.923 - 0.382i)T^{2} \)
41 \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (0.382 - 0.923i)T^{2} \)
47 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
53 \( 1 + (-0.382 + 0.923i)T^{2} \)
59 \( 1 + (-0.382 + 1.92i)T + (-0.923 - 0.382i)T^{2} \)
61 \( 1 + (0.382 + 0.923i)T^{2} \)
67 \( 1 + (1.08 - 1.63i)T + (-0.382 - 0.923i)T^{2} \)
71 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.707 - 0.707i)T^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 + (-0.923 + 0.382i)T^{2} \)
89 \( 1 + (-0.707 - 0.707i)T^{2} \)
97 \( 1 - 1.84iT - T^{2} \)
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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

−9.170152355526575637548133943227, −8.450122233695874397584509067011, −7.23806906045048770147878387362, −6.66671765731666246768720779022, −5.84282263155001985660382411097, −4.76657993077276397757475466711, −3.99966689746712817424183431874, −3.38285537613082161250820354269, −2.18372756939707383393416092157, −0.892178960139212753480368462383, 2.18305130862488984633103743682, 2.77330886096013433575365194137, 4.33444970316350860550421533334, 4.76260544617283612967526364172, 5.75738614831445316189597746220, 6.13257243324627281895394446481, 7.36383759165892393536862939939, 8.077736060251856419560358670365, 8.715480382174563564443011339474, 9.135437825981540743981604345971

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