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Properties

Label 1-1792-1792.51-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: $\chi_{1792} (51, \cdot )$
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

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

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