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Properties

Label 2-1984-124.87-c0-0-1
Degree $2$
Conductor $1984$
Sign $0.390 + 0.920i$
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  + (0.866 − 0.5i)3-s + (0.5 − 0.866i)5-s + (0.866 − 0.5i)7-s + (−0.866 − 0.5i)11-s + (−0.5 + 0.866i)13-s − 0.999i·15-s + (−0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (0.499 − 0.866i)21-s + i·27-s − i·31-s − 0.999·33-s − 0.999i·35-s + (0.5 + 0.866i)37-s + 0.999i·39-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (0.5 − 0.866i)5-s + (0.866 − 0.5i)7-s + (−0.866 − 0.5i)11-s + (−0.5 + 0.866i)13-s − 0.999i·15-s + (−0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (0.499 − 0.866i)21-s + i·27-s − i·31-s − 0.999·33-s − 0.999i·35-s + (0.5 + 0.866i)37-s + 0.999i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $0.390 + 0.920i$
Analytic conductor: \(0.990144\)
Root analytic conductor: \(0.995060\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1984} (831, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1984,\ (\ :0),\ 0.390 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.691237571\)
\(L(\frac12)\) \(\approx\) \(1.691237571\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 + iT \)
good3 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.138825705308747047387175132157, −8.422494033789079331343160080017, −7.65314199614272044393209711562, −7.25439254932664930900163848393, −5.97440261911471317640928119174, −4.94375045952144991524498094054, −4.59502158495483365140477653730, −3.06061208596065407044108493998, −2.23123348947667574866130524240, −1.22162048872448090732152873889, 1.95544786336614072643194391488, 2.70762004757654679328877032014, 3.46588726751270915960862410420, 4.64277121032283770493315847182, 5.46125830841240864794367092631, 6.26922485789195572822733717201, 7.40932840493053371924014099936, 7.989401062766440563433115833049, 8.719738924250434128485840810752, 9.499844400176862470648837315029

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