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Properties

Label 1-1984-1984.259-r0-0-0
Degree $1$
Conductor $1984$
Sign $-0.598 - 0.801i$
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  + (0.477 − 0.878i)3-s + (−0.991 − 0.130i)5-s + (0.544 − 0.838i)7-s + (−0.544 − 0.838i)9-s + (0.902 + 0.430i)11-s + (0.0261 + 0.999i)13-s + (−0.587 + 0.809i)15-s + (−0.743 − 0.669i)17-s + (0.725 − 0.688i)19-s + (−0.477 − 0.878i)21-s + (0.453 − 0.891i)23-s + (0.965 + 0.258i)25-s + (−0.996 + 0.0784i)27-s + (0.972 − 0.233i)29-s + (0.809 − 0.587i)33-s + ⋯
L(s)  = 1  + (0.477 − 0.878i)3-s + (−0.991 − 0.130i)5-s + (0.544 − 0.838i)7-s + (−0.544 − 0.838i)9-s + (0.902 + 0.430i)11-s + (0.0261 + 0.999i)13-s + (−0.587 + 0.809i)15-s + (−0.743 − 0.669i)17-s + (0.725 − 0.688i)19-s + (−0.477 − 0.878i)21-s + (0.453 − 0.891i)23-s + (0.965 + 0.258i)25-s + (−0.996 + 0.0784i)27-s + (0.972 − 0.233i)29-s + (0.809 − 0.587i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $-0.598 - 0.801i$
Analytic conductor: \(9.21365\)
Root analytic conductor: \(9.21365\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1984} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1984,\ (0:\ ),\ -0.598 - 0.801i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7322715688 - 1.461351590i\)
\(L(\frac12)\) \(\approx\) \(0.7322715688 - 1.461351590i\)
\(L(1)\) \(\approx\) \(1.008880935 - 0.5609242693i\)
\(L(1)\) \(\approx\) \(1.008880935 - 0.5609242693i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (0.477 - 0.878i)T \)
5 \( 1 + (-0.991 - 0.130i)T \)
7 \( 1 + (0.544 - 0.838i)T \)
11 \( 1 + (0.902 + 0.430i)T \)
13 \( 1 + (0.0261 + 0.999i)T \)
17 \( 1 + (-0.743 - 0.669i)T \)
19 \( 1 + (0.725 - 0.688i)T \)
23 \( 1 + (0.453 - 0.891i)T \)
29 \( 1 + (0.972 - 0.233i)T \)
37 \( 1 + (0.608 - 0.793i)T \)
41 \( 1 + (-0.629 + 0.777i)T \)
43 \( 1 + (-0.999 - 0.0261i)T \)
47 \( 1 + (-0.587 + 0.809i)T \)
53 \( 1 + (0.983 - 0.182i)T \)
59 \( 1 + (0.958 - 0.284i)T \)
61 \( 1 + (0.923 + 0.382i)T \)
67 \( 1 + (-0.793 + 0.608i)T \)
71 \( 1 + (0.838 - 0.544i)T \)
73 \( 1 + (0.0523 - 0.998i)T \)
79 \( 1 + (-0.743 - 0.669i)T \)
83 \( 1 + (0.958 + 0.284i)T \)
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

−20.053727450504259661002254828133, −19.72495776939731820023008609122, −18.92551217439026300201246216856, −18.11105676233010900872739994815, −17.21105392648310002984613718121, −16.42753890503372041827357752750, −15.59444811560865747064329737495, −15.20773730035904274391012278471, −14.64078943751049538675906700550, −13.82409740426603190038180453748, −12.84585624163649601256038136946, −11.74903759657259449800977293146, −11.50472616500297045246278485503, −10.56709574729514972038718557797, −9.81677548985014650576062258534, −8.70835008155685114262536959704, −8.46445029652876257433383441911, −7.70311825682162404847980904586, −6.59042652592421171564425586231, −5.50129452402244005605725144603, −4.906712783421080163883847919407, −3.84482974265718748304143182429, −3.3894413234060204862784700422, −2.46636627441627169816854525621, −1.21781004160931367920060128484, 0.60223325711001473151977891321, 1.398297076212252792320957147992, 2.446957831470771253004322622209, 3.45041570928614053655997632540, 4.33571691231624151162656067524, 4.83545101530314115343621103596, 6.501342025779935752234306143763, 6.95670662263984022563506500271, 7.495824414170708230016254928204, 8.4532557996000548261301086567, 8.96090909302737674583744408682, 9.89756229497046119756770096144, 11.27642990010555816187805153774, 11.507916482714384354941122362008, 12.24753640527230260062572628629, 13.18850235408784454360518451522, 13.84566006974989711323798378853, 14.54811478078250778224478947124, 15.08223119632061141062061446218, 16.20114657481352053060077126892, 16.767120929185605916088399056139, 17.756134299449127227077717451620, 18.177777266704860916799632122064, 19.27532920583504794218384058094, 19.613158638889253523401904495207

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