↑

Properties

Label 1-1792-1792.1427-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.534 - 0.844i$
Analytic cond. $8.32201$
Root an. cond. $8.32201$
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  + (0.881 + 0.471i)3-s + (−0.634 + 0.773i)5-s + (0.555 + 0.831i)9-s + (−0.956 − 0.290i)11-s + (0.773 − 0.634i)13-s + (−0.923 + 0.382i)15-s + (−0.923 − 0.382i)17-s + (0.0980 − 0.995i)19-s + (−0.980 − 0.195i)23-s + (−0.195 − 0.980i)25-s + (0.0980 + 0.995i)27-s + (−0.290 − 0.956i)29-s + (0.707 − 0.707i)31-s + (−0.707 − 0.707i)33-s + (−0.995 + 0.0980i)37-s + ⋯
L(s)  = 1  + (0.881 + 0.471i)3-s + (−0.634 + 0.773i)5-s + (0.555 + 0.831i)9-s + (−0.956 − 0.290i)11-s + (0.773 − 0.634i)13-s + (−0.923 + 0.382i)15-s + (−0.923 − 0.382i)17-s + (0.0980 − 0.995i)19-s + (−0.980 − 0.195i)23-s + (−0.195 − 0.980i)25-s + (0.0980 + 0.995i)27-s + (−0.290 − 0.956i)29-s + (0.707 − 0.707i)31-s + (−0.707 − 0.707i)33-s + (−0.995 + 0.0980i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.534 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.534 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.534 - 0.844i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.534 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.076742486 - 0.5926326638i\)
\(L(\frac12)\) \(\approx\) \(1.076742486 - 0.5926326638i\)
\(L(1)\) \(\approx\) \(1.091339330 + 0.08497907384i\)
\(L(1)\) \(\approx\) \(1.091339330 + 0.08497907384i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.881 + 0.471i)T \)
5 \( 1 + (-0.634 + 0.773i)T \)
11 \( 1 + (-0.956 - 0.290i)T \)
13 \( 1 + (0.773 - 0.634i)T \)
17 \( 1 + (-0.923 - 0.382i)T \)
19 \( 1 + (0.0980 - 0.995i)T \)
23 \( 1 + (-0.980 - 0.195i)T \)
29 \( 1 + (-0.290 - 0.956i)T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (-0.995 + 0.0980i)T \)
41 \( 1 + (0.195 - 0.980i)T \)
43 \( 1 + (0.881 - 0.471i)T \)
47 \( 1 + (0.382 - 0.923i)T \)
53 \( 1 + (-0.290 + 0.956i)T \)
59 \( 1 + (-0.773 - 0.634i)T \)
61 \( 1 + (0.471 - 0.881i)T \)
67 \( 1 + (-0.471 + 0.881i)T \)
71 \( 1 + (0.555 - 0.831i)T \)
73 \( 1 + (0.831 - 0.555i)T \)
79 \( 1 + (0.382 + 0.923i)T \)
83 \( 1 + (-0.995 - 0.0980i)T \)
89 \( 1 + (-0.980 + 0.195i)T \)
97 \( 1 + (0.707 - 0.707i)T \)
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

−20.25835413494783222539255832714, −19.65551787006520918873530393991, −18.96151422964752010863986652499, −18.21522437809340879036218701084, −17.59634631306362715101168855646, −16.36739113929956902228176067979, −15.86999344640000553728574873341, −15.25516170166536961351358513675, −14.27013304717146214594332516968, −13.641156588157561520874662278275, −12.795463528051987876967412691185, −12.40329928808963761470996768870, −11.46929308467181130522881503695, −10.517463031563029572495807818802, −9.56589898457431247484397352499, −8.74625845723451783222819475702, −8.22264561816877330798654464238, −7.57824088379575437779085171908, −6.67772606835658850970529644034, −5.72746503878598725579203549967, −4.57189347128157891490433269040, −3.941129634731884877231499161, −3.06873410237221430504767829758, −1.93176167429717167960274050216, −1.256089442979857652104401749433, 0.3810927948165034913604327116, 2.22068208059839786037786253503, 2.71102422831781331162461667882, 3.63426213634580249630187260718, 4.2850795581930360216159771828, 5.290327455019405812668600802170, 6.349595450930895541920349834705, 7.32079118214734015575303441209, 7.97366491896180802175428772756, 8.58468904358353389214245160144, 9.493664826612794034012842137517, 10.49386596170626670041175354430, 10.81970694166567933379846015646, 11.70379505547704044941792248028, 12.82413389599357909312898709323, 13.69871219860214059347282257216, 13.97008318115634739540940995515, 15.22095840546299781102691506834, 15.67025906547021390604007995424, 15.80957691689355990067234034109, 17.18300697993220234263439634463, 18.13799066002421329031824021925, 18.68270473180097091071451844031, 19.38364358509427868824022394238, 20.1590273127866904172433735187

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