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Properties

Label 1-1984-1984.493-r0-0-0
Degree $1$
Conductor $1984$
Sign $-0.460 + 0.887i$
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  + (0.566 − 0.824i)3-s + (0.793 + 0.608i)5-s + (−0.358 + 0.933i)7-s + (−0.358 − 0.933i)9-s + (−0.958 + 0.284i)11-s + (−0.902 + 0.430i)13-s + (0.951 − 0.309i)15-s + (0.994 − 0.104i)17-s + (0.333 + 0.942i)19-s + (0.566 + 0.824i)21-s + (−0.987 − 0.156i)23-s + (0.258 + 0.965i)25-s + (−0.972 − 0.233i)27-s + (−0.760 − 0.649i)29-s + (−0.309 + 0.951i)33-s + ⋯
L(s)  = 1  + (0.566 − 0.824i)3-s + (0.793 + 0.608i)5-s + (−0.358 + 0.933i)7-s + (−0.358 − 0.933i)9-s + (−0.958 + 0.284i)11-s + (−0.902 + 0.430i)13-s + (0.951 − 0.309i)15-s + (0.994 − 0.104i)17-s + (0.333 + 0.942i)19-s + (0.566 + 0.824i)21-s + (−0.987 − 0.156i)23-s + (0.258 + 0.965i)25-s + (−0.972 − 0.233i)27-s + (−0.760 − 0.649i)29-s + (−0.309 + 0.951i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $-0.460 + 0.887i$
Analytic conductor: \(9.21365\)
Root analytic conductor: \(9.21365\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1984} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1984,\ (0:\ ),\ -0.460 + 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5167721735 + 0.8502067976i\)
\(L(\frac12)\) \(\approx\) \(0.5167721735 + 0.8502067976i\)
\(L(1)\) \(\approx\) \(1.071079159 + 0.09354765693i\)
\(L(1)\) \(\approx\) \(1.071079159 + 0.09354765693i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (0.566 - 0.824i)T \)
5 \( 1 + (0.793 + 0.608i)T \)
7 \( 1 + (-0.358 + 0.933i)T \)
11 \( 1 + (-0.958 + 0.284i)T \)
13 \( 1 + (-0.902 + 0.430i)T \)
17 \( 1 + (0.994 - 0.104i)T \)
19 \( 1 + (0.333 + 0.942i)T \)
23 \( 1 + (-0.987 - 0.156i)T \)
29 \( 1 + (-0.760 - 0.649i)T \)
37 \( 1 + (-0.130 + 0.991i)T \)
41 \( 1 + (-0.838 - 0.544i)T \)
43 \( 1 + (0.430 - 0.902i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (-0.0261 + 0.999i)T \)
59 \( 1 + (0.983 - 0.182i)T \)
61 \( 1 + (-0.382 + 0.923i)T \)
67 \( 1 + (0.991 - 0.130i)T \)
71 \( 1 + (-0.933 + 0.358i)T \)
73 \( 1 + (0.777 - 0.629i)T \)
79 \( 1 + (-0.994 + 0.104i)T \)
83 \( 1 + (-0.983 - 0.182i)T \)
89 \( 1 + (-0.987 + 0.156i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−19.94748312920899091131579094489, −19.28621528680323718700348883975, −18.14246904806821464882720816567, −17.4453845887851919291985271919, −16.585450335278227056719291959432, −16.27426766164000323755969142658, −15.426563635839008867065246541597, −14.425014639188039708045667325825, −13.9921607012885841519761773758, −13.09737909752892458500801377170, −12.73604027377163200755492317374, −11.39035134364451837797696047638, −10.53872249611365884632312904872, −9.80559176006536419512477627919, −9.65881464721106980630272896438, −8.46333278049063825306519670402, −7.83825204509040459134041208509, −7.005247947460434601968761546108, −5.67117734063478915719321647016, −5.19318749313395278696275393602, −4.395111279678384962747244695774, −3.38321290531319435126809931668, −2.697583344031715212868945060427, −1.66524545153361847861997564950, −0.27548793555206171949346767444, 1.55085260024396910060141205341, 2.28641533220791723625950349021, 2.807913666658204114072994163069, 3.71941832919444250326633313598, 5.25888608499699124669987146181, 5.79783865595593697997013346419, 6.58701194690626358896382186483, 7.45528727004758089848865807431, 8.03524581896326591787759168701, 9.022364900997226916784986271087, 9.87878864156135949751072027965, 10.15854918219318614591624397503, 11.63495293571981505424362670966, 12.184429187871400464468110520137, 12.856297471294033290278301494921, 13.65343467278722797763265420759, 14.325161917772838988434395846370, 14.89611066880866683230508423829, 15.64335699275460147182399285795, 16.700841734801922782279052829733, 17.48531981811619392024328311297, 18.32195198357648576494177426122, 18.71449608532459763627429027463, 19.12958832632402356818384056529, 20.26093659691636840341904750700

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