L-function 1-2793-2793.125-r0-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	3	$1$
	7	$1$
	19	$1$
good	2	$1 + (0.988 + 0.149i)T$
	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	$1 + (-0.222 + 0.974i)T$

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