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Properties

Label 2-1984-1984.1053-c0-0-0
Degree $2$
Conductor $1984$
Sign $0.995 + 0.0980i$
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 + (−0.617 + 0.923i)5-s + (1.30 − 0.541i)7-s + (0.707 − 0.707i)8-s + (0.923 + 0.382i)9-s + (1.08 − 0.216i)10-s + (−1.30 − 0.541i)14-s − 1.00·16-s + (−0.382 − 0.923i)18-s + (0.324 − 0.216i)19-s + (−0.923 − 0.617i)20-s + (−0.0897 − 0.216i)25-s + (0.541 + 1.30i)28-s − i·31-s + (0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.617 + 0.923i)5-s + (1.30 − 0.541i)7-s + (0.707 − 0.707i)8-s + (0.923 + 0.382i)9-s + (1.08 − 0.216i)10-s + (−1.30 − 0.541i)14-s − 1.00·16-s + (−0.382 − 0.923i)18-s + (0.324 − 0.216i)19-s + (−0.923 − 0.617i)20-s + (−0.0897 − 0.216i)25-s + (0.541 + 1.30i)28-s − i·31-s + (0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $0.995 + 0.0980i$
Analytic conductor: \(0.990144\)
Root analytic conductor: \(0.995060\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1984} (1053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1984,\ (\ :0),\ 0.995 + 0.0980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9125115649\)
\(L(\frac12)\) \(\approx\) \(0.9125115649\)
\(L(1)\) 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 \( 1 + (-0.923 - 0.382i)T^{2} \)
5 \( 1 + (0.617 - 0.923i)T + (-0.382 - 0.923i)T^{2} \)
7 \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.923 - 0.382i)T^{2} \)
13 \( 1 + (0.382 - 0.923i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2} \)
23 \( 1 + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.923 + 0.382i)T^{2} \)
37 \( 1 + (-0.382 - 0.923i)T^{2} \)
41 \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + (-0.923 + 0.382i)T^{2} \)
47 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
53 \( 1 + (0.923 - 0.382i)T^{2} \)
59 \( 1 + (0.923 - 1.38i)T + (-0.382 - 0.923i)T^{2} \)
61 \( 1 + (-0.923 - 0.382i)T^{2} \)
67 \( 1 + (-1.63 - 0.324i)T + (0.923 + 0.382i)T^{2} \)
71 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 + (-0.382 + 0.923i)T^{2} \)
89 \( 1 + (0.707 - 0.707i)T^{2} \)
97 \( 1 + 0.765iT - T^{2} \)
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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

−9.546868586165047902831525941722, −8.418135198204544223647383507925, −7.76670989925709252187810184178, −7.36983897849146028571972063538, −6.59364531178916258142172239598, −4.97748841602872903618199432528, −4.24728786209041054968509552576, −3.44305154075778060702549495398, −2.29524560826845892351682193508, −1.27257035456235441016621518908, 1.07499357010188044969447869996, 1.97811577226503272670195040594, 3.84513983545750180537635236010, 4.90568193877656213398587195256, 5.13278628201327798570186556287, 6.35179238430660434538107029717, 7.23489744388181450009596208530, 7.987997175728063461912335494654, 8.454478268980492484385281095790, 9.133994270623492757992394559183

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