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Properties

Label 2-1984-124.67-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: $\chi_{1984} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1984,\ (\ :0),\ -0.390 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5859835599\)
\(L(\frac12)\) \(\approx\) \(0.5859835599\)
\(L(1)\) 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

−9.216013402420924958650044607004, −8.312309290520488767959502215556, −7.16097622180595919136855654497, −6.58277062721921466378865427556, −6.21066078101829189244331727224, −5.41311767434470613029956186916, −4.04920243340125605272452847751, −3.22685080572336038712856775444, −2.09132947034359915894078793871, −0.46337993976328213097538312990, 1.56709422524660298961366664399, 2.76727707435279128502493559731, 4.20099839177430129348746323847, 4.80100663378862606117118451118, 5.51451975976903869881156777510, 6.44890806638460420342135796666, 6.85660914156674188332592560191, 8.289592496790348947584232954493, 9.026934785833289586988119868568, 9.682663775497985750784784097881

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