↑

Properties

Label 1-1984-1984.635-r0-0-0
Degree $1$
Conductor $1984$
Sign $-0.0359 - 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.760 + 0.649i)3-s + (0.923 + 0.382i)5-s + (−0.156 − 0.987i)7-s + (0.156 − 0.987i)9-s + (0.852 − 0.522i)11-s + (−0.649 − 0.760i)13-s + (−0.951 + 0.309i)15-s + (−0.587 − 0.809i)17-s + (−0.0784 − 0.996i)19-s + (0.760 + 0.649i)21-s + (−0.156 + 0.987i)23-s + (0.707 + 0.707i)25-s + (0.522 + 0.852i)27-s + (0.996 − 0.0784i)29-s + (−0.309 + 0.951i)33-s + ⋯
L(s)  = 1  + (−0.760 + 0.649i)3-s + (0.923 + 0.382i)5-s + (−0.156 − 0.987i)7-s + (0.156 − 0.987i)9-s + (0.852 − 0.522i)11-s + (−0.649 − 0.760i)13-s + (−0.951 + 0.309i)15-s + (−0.587 − 0.809i)17-s + (−0.0784 − 0.996i)19-s + (0.760 + 0.649i)21-s + (−0.156 + 0.987i)23-s + (0.707 + 0.707i)25-s + (0.522 + 0.852i)27-s + (0.996 − 0.0784i)29-s + (−0.309 + 0.951i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0359 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0359 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $-0.0359 - 0.999i$
Analytic conductor: \(9.21365\)
Root analytic conductor: \(9.21365\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1984} (635, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1984,\ (0:\ ),\ -0.0359 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7338783823 - 0.7607763922i\)
\(L(\frac12)\) \(\approx\) \(0.7338783823 - 0.7607763922i\)
\(L(1)\) \(\approx\) \(0.9027966028 - 0.09206957767i\)
\(L(1)\) \(\approx\) \(0.9027966028 - 0.09206957767i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−19.921220786975409100936300443324, −19.41611963633582439399707212657, −18.47701819554862430116361026540, −18.00312023104605834063290414483, −17.28597739154849685137154473546, −16.61700838867906656607364733447, −16.1377524347776239624591735036, −14.69106311312855822142087182070, −14.50026276737191301934831525416, −13.22802565999115913866061676005, −12.81017435352003391542106651046, −11.96626749357577953162269501529, −11.66471165841323489126375880976, −10.33672945963197828222002291489, −9.854487827521025371713969513505, −8.85922085708270500761584436645, −8.28606686947314494936722222294, −7.03808467576796215162107022088, −6.2748174017363211110845276, −6.00032627232294324908249731737, −4.88640042466013207311202826481, −4.33756204647241828582671069241, −2.69147026938838487496333585286, −1.91743542796561861172517139220, −1.35844361345988049858756635559, 0.40601913433811974182432878684, 1.40675565199103862505663553968, 2.80246076648741520501789348229, 3.51106423402734384970562645686, 4.585478834228737557126417388508, 5.16789717296697429171753770576, 6.17122913833850638532659470181, 6.71444544516980992557722013693, 7.450745182581566625434885404244, 8.82221472366934782956340673096, 9.59581683820179307064900638334, 10.05793412399662435446679263790, 10.88322003976604985442494120318, 11.39186032523472146397430034912, 12.309471326813519124531413724854, 13.39239456738851327437973091575, 13.78945674166798218204354376661, 14.70511025171141972398173998119, 15.4497399683244896957249204525, 16.27170730239195471629803837911, 17.04190432987081061161658652589, 17.512530075081430166417674016215, 17.93355667270740793280944239447, 19.06253077592711134294942598202, 20.012264998999147738905103930319

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