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Properties

Label 1-1984-1984.75-r0-0-0
Degree $1$
Conductor $1984$
Sign $-0.522 + 0.852i$
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  + (0.333 − 0.942i)3-s + (−0.608 − 0.793i)5-s + (0.777 − 0.629i)7-s + (−0.777 − 0.629i)9-s + (−0.999 + 0.0261i)11-s + (−0.182 − 0.983i)13-s + (−0.951 + 0.309i)15-s + (−0.406 + 0.913i)17-s + (0.566 + 0.824i)19-s + (−0.333 − 0.942i)21-s + (0.156 − 0.987i)23-s + (−0.258 + 0.965i)25-s + (−0.852 + 0.522i)27-s + (−0.0784 − 0.996i)29-s + (−0.309 + 0.951i)33-s + ⋯
L(s)  = 1  + (0.333 − 0.942i)3-s + (−0.608 − 0.793i)5-s + (0.777 − 0.629i)7-s + (−0.777 − 0.629i)9-s + (−0.999 + 0.0261i)11-s + (−0.182 − 0.983i)13-s + (−0.951 + 0.309i)15-s + (−0.406 + 0.913i)17-s + (0.566 + 0.824i)19-s + (−0.333 − 0.942i)21-s + (0.156 − 0.987i)23-s + (−0.258 + 0.965i)25-s + (−0.852 + 0.522i)27-s + (−0.0784 − 0.996i)29-s + (−0.309 + 0.951i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $-0.522 + 0.852i$
Analytic conductor: \(9.21365\)
Root analytic conductor: \(9.21365\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1984} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1984,\ (0:\ ),\ -0.522 + 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3079018421 - 0.5497004149i\)
\(L(\frac12)\) \(\approx\) \(-0.3079018421 - 0.5497004149i\)
\(L(1)\) \(\approx\) \(0.6969432250 - 0.5405007053i\)
\(L(1)\) \(\approx\) \(0.6969432250 - 0.5405007053i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (0.333 - 0.942i)T \)
5 \( 1 + (-0.608 - 0.793i)T \)
7 \( 1 + (0.777 - 0.629i)T \)
11 \( 1 + (-0.999 + 0.0261i)T \)
13 \( 1 + (-0.182 - 0.983i)T \)
17 \( 1 + (-0.406 + 0.913i)T \)
19 \( 1 + (0.566 + 0.824i)T \)
23 \( 1 + (0.156 - 0.987i)T \)
29 \( 1 + (-0.0784 - 0.996i)T \)
37 \( 1 + (0.991 - 0.130i)T \)
41 \( 1 + (-0.998 - 0.0523i)T \)
43 \( 1 + (-0.983 - 0.182i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (0.284 - 0.958i)T \)
59 \( 1 + (-0.430 - 0.902i)T \)
61 \( 1 + (-0.923 + 0.382i)T \)
67 \( 1 + (0.130 - 0.991i)T \)
71 \( 1 + (-0.629 + 0.777i)T \)
73 \( 1 + (-0.358 + 0.933i)T \)
79 \( 1 + (-0.406 + 0.913i)T \)
83 \( 1 + (-0.430 + 0.902i)T \)
89 \( 1 + (-0.156 - 0.987i)T \)
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

−20.376307407688898314834835612425, −19.89540274071177337411404796996, −18.953326282699177256080580619574, −18.29382710194591820649664414025, −17.731984569526914853541128507436, −16.520233016650890179778569687510, −15.96559950454696093371935164875, −15.192933322742580965027105386532, −14.90796821768423871766653708752, −13.943435777483045314919285963754, −13.42394373870257139534841389034, −11.95623606167281063308006140733, −11.43409898396719365007022925912, −10.954758214670644504723386102671, −10.040834533934707358189641600834, −9.2267757090378985359086926246, −8.56850397269639675612009474866, −7.66091112039098501325937879247, −7.08517969821221747405773791543, −5.841326393951449854801103592667, −4.87479687852560458430095055629, −4.55029207986706344431271392274, −3.22967250548669749277791239526, −2.81003829163327451691970332692, −1.821132128279093542241440407691, 0.2074050119538606513174283781, 1.181294474659645025347031239698, 2.038204745719348510148065555019, 3.10842850066952503654137626254, 4.02210390830070247847993330324, 4.95815763113878589821627129747, 5.68922959706364597061155526918, 6.7428552033481303508086543782, 7.79708339428355134983740083342, 8.02198027317017943013540215887, 8.537099972601988465207682188096, 9.80654553721519960348432317740, 10.64361281712796884075462665632, 11.490844782770683405244682714264, 12.20478245233291997422220200323, 13.07971680711191339713468627467, 13.233910235873211180017557219763, 14.38776982692731514696085276756, 15.00647551883775222642927559276, 15.75759006867384540882774981682, 16.822150366244055227333715294941, 17.268974172376541673796228146292, 18.178253636882868372849295048436, 18.643341135475259596560932626770, 19.68686535430551095036439153354

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