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Properties

Label 2-1984-124.39-c0-0-1
Degree $2$
Conductor $1984$
Sign $0.938 - 0.344i$
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  + (0.951 − 0.309i)3-s + 0.618·5-s + (−0.587 + 0.809i)7-s + (0.5 + 1.53i)13-s + (0.587 − 0.190i)15-s + (0.809 − 0.587i)17-s + (0.951 + 0.309i)19-s + (−0.309 + 0.951i)21-s + (−0.587 − 0.809i)23-s − 0.618·25-s + (−0.587 + 0.809i)27-s + (0.190 − 0.587i)29-s + (−0.951 − 0.309i)31-s + (−0.363 + 0.500i)35-s + 37-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)3-s + 0.618·5-s + (−0.587 + 0.809i)7-s + (0.5 + 1.53i)13-s + (0.587 − 0.190i)15-s + (0.809 − 0.587i)17-s + (0.951 + 0.309i)19-s + (−0.309 + 0.951i)21-s + (−0.587 − 0.809i)23-s − 0.618·25-s + (−0.587 + 0.809i)27-s + (0.190 − 0.587i)29-s + (−0.951 − 0.309i)31-s + (−0.363 + 0.500i)35-s + 37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $0.938 - 0.344i$
Analytic conductor: \(0.990144\)
Root analytic conductor: \(0.995060\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1984} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1984,\ (\ :0),\ 0.938 - 0.344i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.657870207\)
\(L(\frac12)\) \(\approx\) \(1.657870207\)
\(L(1)\) 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 \( 1 + (0.951 + 0.309i)T \)
good3 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
5 \( 1 - 0.618T + T^{2} \)
7 \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
47 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + iT - T^{2} \)
71 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
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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

−9.397904999627430210743030000091, −8.772944903273448986942880103594, −7.88635667438275206206397428979, −7.18762492766664410185207344013, −6.08029919643724681972828133929, −5.70863258042475344763673631170, −4.38989956192805635676232590720, −3.33450199780905893891616981680, −2.48778799847449776539721405219, −1.72116060258528191578250837782, 1.21935206800908537906641241670, 2.71926357824756536903725371882, 3.41147836899457305430451078416, 4.04943079453308849619638043413, 5.63028497198983850609157688259, 5.83521314651275898936694134630, 7.21871080863734393395745222628, 7.78009202241179534203051045928, 8.582452520458737709524295430086, 9.397890938371335075018436428294

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