Properties

Label 1-1792-1792.1019-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.156 + 0.987i$
Analytic cond. $192.577$
Root an. cond. $192.577$
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  + (−0.227 − 0.973i)3-s + (−0.582 + 0.812i)5-s + (−0.896 + 0.442i)9-s + (−0.999 + 0.0327i)11-s + (−0.0980 − 0.995i)13-s + (0.923 + 0.382i)15-s + (0.793 + 0.608i)17-s + (0.352 + 0.935i)19-s + (0.946 + 0.321i)23-s + (−0.321 − 0.946i)25-s + (0.634 + 0.773i)27-s + (0.881 − 0.471i)29-s + (−0.258 + 0.965i)31-s + (0.258 + 0.965i)33-s + (−0.162 − 0.986i)37-s + ⋯
L(s)  = 1  + (−0.227 − 0.973i)3-s + (−0.582 + 0.812i)5-s + (−0.896 + 0.442i)9-s + (−0.999 + 0.0327i)11-s + (−0.0980 − 0.995i)13-s + (0.923 + 0.382i)15-s + (0.793 + 0.608i)17-s + (0.352 + 0.935i)19-s + (0.946 + 0.321i)23-s + (−0.321 − 0.946i)25-s + (0.634 + 0.773i)27-s + (0.881 − 0.471i)29-s + (−0.258 + 0.965i)31-s + (0.258 + 0.965i)33-s + (−0.162 − 0.986i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.156 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3964582000 + 0.4643990031i\)
\(L(\frac12)\) \(\approx\) \(0.3964582000 + 0.4643990031i\)
\(L(1)\) \(\approx\) \(0.7661975924 - 0.09299004089i\)
\(L(1)\) \(\approx\) \(0.7661975924 - 0.09299004089i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.227 - 0.973i)T \)
5 \( 1 + (-0.582 + 0.812i)T \)
11 \( 1 + (-0.999 + 0.0327i)T \)
13 \( 1 + (-0.0980 - 0.995i)T \)
17 \( 1 + (0.793 + 0.608i)T \)
19 \( 1 + (0.352 + 0.935i)T \)
23 \( 1 + (0.946 + 0.321i)T \)
29 \( 1 + (0.881 - 0.471i)T \)
31 \( 1 + (-0.258 + 0.965i)T \)
37 \( 1 + (-0.162 - 0.986i)T \)
41 \( 1 + (0.980 - 0.195i)T \)
43 \( 1 + (-0.956 + 0.290i)T \)
47 \( 1 + (-0.991 - 0.130i)T \)
53 \( 1 + (-0.0327 - 0.999i)T \)
59 \( 1 + (-0.910 + 0.412i)T \)
61 \( 1 + (-0.683 - 0.729i)T \)
67 \( 1 + (0.973 - 0.227i)T \)
71 \( 1 + (0.831 - 0.555i)T \)
73 \( 1 + (-0.997 - 0.0654i)T \)
79 \( 1 + (-0.608 - 0.793i)T \)
83 \( 1 + (0.773 + 0.634i)T \)
89 \( 1 + (0.751 + 0.659i)T \)
97 \( 1 + (-0.707 - 0.707i)T \)
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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

−20.097758315905909142383337920293, −19.11769337240484302057852051335, −18.41069669577572874414068740474, −17.36061812479076335625841452429, −16.683126531775439918777308602303, −16.15459719671886169643836011445, −15.55285436617646351010996508226, −14.85125650984831721489011878248, −13.914612489603151272227189799798, −13.10269309109176478489154430629, −12.18144485212507475661164522618, −11.51756231300207653476693441323, −10.90983395146658697400826810641, −9.90051829392021421132965048696, −9.271826859611283047019351556533, −8.578132857226845285916632658726, −7.73544665705806727699842136158, −6.79922915712547673811369492983, −5.64191187535870156917096710926, −4.82811552706195249178106830584, −4.556294128334379210982012063310, −3.37524294631576099781439893559, −2.654693365998762764983101506885, −1.112451331886618946910957709907, −0.15725025455302430100587990562, 0.81886420617710804472409789832, 1.9392841319206793271322087504, 2.99826684255840877085773354276, 3.42611212271308177216427840913, 4.94191720441503778168877241605, 5.6833175043396327965663630781, 6.46699245056596023810437395472, 7.411875510863958589071281349817, 7.861682086175505603296835589348, 8.451691644810984226848545401447, 9.907578326623373966290227184536, 10.62404396795748176590275369495, 11.17602128249905034590992892277, 12.22793217837475552244770187885, 12.59750453748136624921028846893, 13.473222500155699873277695309639, 14.33158301039712809787482857124, 14.9302818175152847371823199658, 15.7972483889457774238398849515, 16.54739287294344091867986315439, 17.60420675709831160325906219566, 18.05069437258250064279721648942, 18.74248349567803525002145624155, 19.37006059922199214512120768158, 19.94560402478035054224858651014

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