↑

Properties

Label 1-1792-1792.797-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.999 + 0.0245i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0980 − 0.995i)3-s + (−0.881 + 0.471i)5-s + (−0.980 − 0.195i)9-s + (−0.634 + 0.773i)11-s + (0.471 − 0.881i)13-s + (0.382 + 0.923i)15-s + (−0.382 + 0.923i)17-s + (−0.290 − 0.956i)19-s + (0.831 − 0.555i)23-s + (0.555 − 0.831i)25-s + (−0.290 + 0.956i)27-s + (−0.773 + 0.634i)29-s + (0.707 − 0.707i)31-s + (0.707 + 0.707i)33-s + (0.956 + 0.290i)37-s + ⋯
L(s)  = 1  + (0.0980 − 0.995i)3-s + (−0.881 + 0.471i)5-s + (−0.980 − 0.195i)9-s + (−0.634 + 0.773i)11-s + (0.471 − 0.881i)13-s + (0.382 + 0.923i)15-s + (−0.382 + 0.923i)17-s + (−0.290 − 0.956i)19-s + (0.831 − 0.555i)23-s + (0.555 − 0.831i)25-s + (−0.290 + 0.956i)27-s + (−0.773 + 0.634i)29-s + (0.707 − 0.707i)31-s + (0.707 + 0.707i)33-s + (0.956 + 0.290i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.999 + 0.0245i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.999 + 0.0245i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006626445253 - 0.5399442303i\)
\(L(\frac12)\) \(\approx\) \(0.006626445253 - 0.5399442303i\)
\(L(1)\) \(\approx\) \(0.7764966476 - 0.2297872587i\)
\(L(1)\) \(\approx\) \(0.7764966476 - 0.2297872587i\)

Euler product

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

−20.63623793616268431674556996204, −19.72724844477584244441371357671, −18.97686615666436655500197920706, −18.40611309399364071199716920992, −17.027047378278590330604585563619, −16.67433356269328632814816550330, −15.73890648838802990114645795413, −15.62660393438971099101009185908, −14.53964542333277970625586172715, −13.80658703577634911201629472608, −13.029743075942731352230439742076, −11.91525071353863393573161243678, −11.32814579810669188557335753122, −10.80153966410502706762303467342, −9.73564759825083357789706576225, −9.02302903751137471704419608799, −8.37318156259840768483902834577, −7.668406524466781565488773752822, −6.54046602676256141348322198407, −5.48949438626089704625163138801, −4.846009283122023978160208577334, −3.964945373299548825133102062186, −3.40505988584030743262769913907, −2.36947583200129748314961090063, −0.90891537682158202549463182994, 0.12838598892930341748738578639, 1.02617718608814027464208437469, 2.310650800793406421841393546076, 2.89735761207499796077873959987, 3.91741683127541638834224634622, 4.90998805302414775908306935652, 5.94645294936339952751173120880, 6.80386295547264298482158057294, 7.36843238443536406710018814805, 8.181118551748891120343418020296, 8.65931551253807727980874290751, 9.91692767277853421968484705560, 11.02020611244118787135550887575, 11.19161235253071768896755146355, 12.46282926344903425825315667141, 12.81460944595839401758476428407, 13.4911560606244655538145209631, 14.65403162139016438652151912849, 15.124225282700697985030164009284, 15.733513904742412757297029458190, 16.909131946467378500334059371013, 17.61505218556433292800283294296, 18.27325836570848229351695047024, 18.89751691024532485502796754108, 19.57812363803707054644196830157

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