↑

Properties

Label 1-2793-2793.125-r0-0-0
Degree $1$
Conductor $2793$
Sign $-0.433 + 0.901i$
Analytic cond. $12.9706$
Root an. cond. $12.9706$
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  + (0.988 + 0.149i)2-s + (0.955 + 0.294i)4-s + (0.0747 + 0.997i)5-s + (0.900 + 0.433i)8-s + (−0.0747 + 0.997i)10-s + (−0.623 + 0.781i)11-s + (−0.365 − 0.930i)13-s + (0.826 + 0.563i)16-s + (0.955 − 0.294i)17-s + (−0.222 + 0.974i)20-s + (−0.733 + 0.680i)22-s + (−0.955 − 0.294i)23-s + (−0.988 + 0.149i)25-s + (−0.222 − 0.974i)26-s + (0.733 + 0.680i)29-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)2-s + (0.955 + 0.294i)4-s + (0.0747 + 0.997i)5-s + (0.900 + 0.433i)8-s + (−0.0747 + 0.997i)10-s + (−0.623 + 0.781i)11-s + (−0.365 − 0.930i)13-s + (0.826 + 0.563i)16-s + (0.955 − 0.294i)17-s + (−0.222 + 0.974i)20-s + (−0.733 + 0.680i)22-s + (−0.955 − 0.294i)23-s + (−0.988 + 0.149i)25-s + (−0.222 − 0.974i)26-s + (0.733 + 0.680i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2793\)    =    \(3 \cdot 7^{2} \cdot 19\)
Sign: $-0.433 + 0.901i$
Analytic conductor: \(12.9706\)
Root analytic conductor: \(12.9706\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2793} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2793,\ (0:\ ),\ -0.433 + 0.901i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.554604680 + 2.471777635i\)
\(L(\frac12)\) \(\approx\) \(1.554604680 + 2.471777635i\)
\(L(1)\) \(\approx\) \(1.698780234 + 0.7671302297i\)
\(L(1)\) \(\approx\) \(1.698780234 + 0.7671302297i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.988 + 0.149i)T \)
5 \( 1 + (0.0747 + 0.997i)T \)
11 \( 1 + (-0.623 + 0.781i)T \)
13 \( 1 + (-0.365 - 0.930i)T \)
17 \( 1 + (0.955 - 0.294i)T \)
23 \( 1 + (-0.955 - 0.294i)T \)
29 \( 1 + (0.733 + 0.680i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 + (0.0747 + 0.997i)T \)
43 \( 1 + (0.826 + 0.563i)T \)
47 \( 1 + (0.365 + 0.930i)T \)
53 \( 1 + (-0.955 - 0.294i)T \)
59 \( 1 + (0.826 + 0.563i)T \)
61 \( 1 + (0.733 + 0.680i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.733 - 0.680i)T \)
73 \( 1 + (-0.365 + 0.930i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (-0.988 + 0.149i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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

−19.24234250544144925868874117530, −18.4868153534705992165472627404, −17.28334493884010728277172650925, −16.74115372475585609304397022585, −15.929542075889010636715472305725, −15.72542438731269651481045055121, −14.42307395378420982529268731447, −14.0496890120569732618754730314, −13.33305393177512477422691477951, −12.54625469105532861420367022545, −12.09067242937032482334461797152, −11.36531392703703096787070617539, −10.49429276856092703911034117976, −9.76116354660900760346925707162, −8.88467959687146003084097583993, −7.97955022823479324482560584105, −7.347108905343628749484944281748, −6.242632753490220127942478040334, −5.583588360761473730261682023304, −5.06504221291456510283084868041, −4.07134364533356728969658759516, −3.58817378025445098160350882473, −2.360567095670190557678386433124, −1.735626370277855097554075418842, −0.58348085774111694253315177396, 1.41140880462761739677111758103, 2.5401875999415748512414926249, 2.91479367021822438139630844516, 3.80623314814166405160297634500, 4.74543858098218735670107170287, 5.48930532215995211617580181592, 6.16738379820170038723192832602, 7.04538757011135305298924977745, 7.61573729803759810010708410477, 8.19988379412049230361545667276, 9.751512939943064923748258837705, 10.25201810664564611660511131808, 10.89189739818391179627969527766, 11.75970059526441428370312409036, 12.45870133990987542214667640061, 13.026270920146744068283177588863, 13.92774850343056913268690035940, 14.60217371970645920996641416713, 14.93884270175590281597036058778, 15.82956911161008078988734647219, 16.30765396211397030292922060996, 17.44669565049243378093522920330, 17.93118091069407591534084194244, 18.73303714467384925564629904580, 19.6156468568472894204844171413

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