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Properties

Label 1-1984-1984.1267-r0-0-0
Degree $1$
Conductor $1984$
Sign $0.494 + 0.869i$
Analytic cond. $9.21365$
Root an. cond. $9.21365$
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  + (−0.852 + 0.522i)3-s + (0.923 − 0.382i)5-s + (−0.453 − 0.891i)7-s + (0.453 − 0.891i)9-s + (−0.996 + 0.0784i)11-s + (0.522 + 0.852i)13-s + (−0.587 + 0.809i)15-s + (0.951 − 0.309i)17-s + (−0.972 + 0.233i)19-s + (0.852 + 0.522i)21-s + (−0.453 + 0.891i)23-s + (0.707 − 0.707i)25-s + (0.0784 + 0.996i)27-s + (0.233 + 0.972i)29-s + (0.809 − 0.587i)33-s + ⋯
L(s)  = 1  + (−0.852 + 0.522i)3-s + (0.923 − 0.382i)5-s + (−0.453 − 0.891i)7-s + (0.453 − 0.891i)9-s + (−0.996 + 0.0784i)11-s + (0.522 + 0.852i)13-s + (−0.587 + 0.809i)15-s + (0.951 − 0.309i)17-s + (−0.972 + 0.233i)19-s + (0.852 + 0.522i)21-s + (−0.453 + 0.891i)23-s + (0.707 − 0.707i)25-s + (0.0784 + 0.996i)27-s + (0.233 + 0.972i)29-s + (0.809 − 0.587i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $0.494 + 0.869i$
Analytic conductor: \(9.21365\)
Root analytic conductor: \(9.21365\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1984} (1267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1984,\ (0:\ ),\ 0.494 + 0.869i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9030725308 + 0.5254545126i\)
\(L(\frac12)\) \(\approx\) \(0.9030725308 + 0.5254545126i\)
\(L(1)\) \(\approx\) \(0.8491343696 + 0.09791927735i\)
\(L(1)\) \(\approx\) \(0.8491343696 + 0.09791927735i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (-0.852 + 0.522i)T \)
5 \( 1 + (0.923 - 0.382i)T \)
7 \( 1 + (-0.453 - 0.891i)T \)
11 \( 1 + (-0.996 + 0.0784i)T \)
13 \( 1 + (0.522 + 0.852i)T \)
17 \( 1 + (0.951 - 0.309i)T \)
19 \( 1 + (-0.972 + 0.233i)T \)
23 \( 1 + (-0.453 + 0.891i)T \)
29 \( 1 + (0.233 + 0.972i)T \)
37 \( 1 + (0.923 - 0.382i)T \)
41 \( 1 + (-0.987 - 0.156i)T \)
43 \( 1 + (-0.852 - 0.522i)T \)
47 \( 1 + (-0.587 + 0.809i)T \)
53 \( 1 + (-0.760 + 0.649i)T \)
59 \( 1 + (0.972 + 0.233i)T \)
61 \( 1 + (-0.382 + 0.923i)T \)
67 \( 1 + (-0.382 + 0.923i)T \)
71 \( 1 + (0.891 + 0.453i)T \)
73 \( 1 + (0.891 - 0.453i)T \)
79 \( 1 + (0.951 - 0.309i)T \)
83 \( 1 + (0.972 - 0.233i)T \)
89 \( 1 + (0.453 + 0.891i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.649823920578218121303190343038, −18.69522058302643633067219498656, −18.45072772694437620917534214018, −17.84020437166020483639053398650, −16.97977903873334463181560566364, −16.37804482285314867189342922161, −15.46033777722572130345629797902, −14.85043437705788618704410687635, −13.730758212518048997350687882614, −13.00374851959257089361341365653, −12.70205291918717691083949016416, −11.75742057837002439202695720462, −10.852817424360858465900281001861, −10.23723252524046452127656861863, −9.68909848690736833925583548749, −8.32465365102640199708740329084, −7.943806651936243495639126447430, −6.42356573295674084668355161051, −6.379560337112537974438335380839, −5.44178656127471060045441527102, −4.918095520469134565987879074843, −3.36990983317891493300329813093, −2.480001410201165282017388903599, −1.83300311438724411276378634612, −0.477368861246206328861244925437, 0.933950174280274268805301239002, 1.82476843779486256888229418454, 3.15975713894113065982541137194, 4.03766693423490021405046333983, 4.86838064650119727605976011388, 5.58026640014066095672426256645, 6.32739281644004781673184799411, 7.013003038126863179143939629179, 8.069266468458970966941487937043, 9.1618078697698136394726045329, 9.8131529909583195255342980441, 10.397303875754616713610454320498, 10.97270157100312822446539905826, 11.998254988252809500472099750958, 12.767184228968717423604429470772, 13.41412227452113605640892715862, 14.11902256913178039410741432271, 15.04195584207279981070833757738, 16.13508402517247892236197195104, 16.421618913055689930879302077356, 17.04235092471194131689707847526, 17.83600085893630077036451952552, 18.41495936264556665102073645366, 19.302738577285559853136338124840, 20.44468507251780127785742967663

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