L-function 1-1053-1053.284-r0-0-0

• Label: 1-1053-1053.284-r0-0-0
• Degree: $1$
• Conductor: $1053$
• Sign: $-0.286 - 0.957i$
• Analytic cond.: $4.89011$
• Root an. cond.: $4.89011$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.918 − 0.396i)2^-s + (0.686 + 0.727i)4^-s + (0.448 − 0.893i)5^-s + (0.727 + 0.686i)7^-s + (−0.342 − 0.939i)8^-s + (−0.766 + 0.642i)10^-s + (0.549 − 0.835i)11^-s + (−0.396 − 0.918i)14^-s + (−0.0581 + 0.998i)16^-s + (−0.939 − 0.342i)17^-s + (0.984 + 0.173i)19^-s + (0.957 − 0.286i)20^-s + (−0.835 + 0.549i)22^-s + (−0.686 − 0.727i)23^-s + (−0.597 − 0.802i)25^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1053$    =    $3^{4} \cdot 13$
Sign:	$-0.286 - 0.957i$
Analytic conductor:	$4.89011$
Root analytic conductor:	$4.89011$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1053} (284, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1053,\ (0:\ ),\ -0.286 - 0.957i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.6239664141 - 0.8382822701i$
$L(1)$	$\approx$	$0.7618109872 - 0.3283255330i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	13	$1$
good	2	$1 + (-0.918 - 0.396i)T$
	5	$1 + (0.448 - 0.893i)T$
	7	$1 + (0.727 + 0.686i)T$
	11	$1 + (0.549 - 0.835i)T$
	17	$1 + (-0.939 - 0.342i)T$
	19	$1 + (0.984 + 0.173i)T$
	23	$1 + (-0.686 - 0.727i)T$
	29	$1 + (-0.597 - 0.802i)T$
	31	$1 + (0.727 - 0.686i)T$

Imaginary part of the first few zeros on the critical line

−21.69665651835446347387634424131, −20.80604314768702578430644477927, −19.90288280715810497839927143233, −19.54466632860136232013465791504, −18.26512878494594047599526603591, −17.804681180140853259167791249322, −17.42714852739715152662400769556, −16.40729160726008484265910372014, −15.491396210216531653668620341678, −14.702882652211938854362812574490, −14.202229389708272879688805379103, −13.31168440487850851346728113243, −11.83352752091673919723222004453, −11.22085339973899109656761402635, −10.421948005311409546217630446820, −9.779746409666491190121723147319, −8.95711566077261632674215680180, −7.8487945876242960107727115025, −7.17027075460633836842164340856, −6.59324339914786788582509099102, −5.55863748229217542481595860397, −4.52495414012034377673891007531, −3.24859276679031577800369996083, −1.99713375720893115285263521078, −1.37531472911322107179882175201, 0.60507871063697973787762256277, 1.70405921205225265351717396658, 2.417052232540971202464893196689, 3.68714203912146678283894367224, 4.78171300950069128183234865711, 5.80051989128182472540491088467, 6.635332872412110557995661900864, 7.96508212882840883918427499596, 8.47008183244211043255381150980, 9.189421985631393910327068620394, 9.86379017664597302715644708628, 10.973190312623471189843628415885, 11.787157638940271370501172693988, 12.177571313558574339067152320240, 13.37231025993185692397641941041, 14.01198008469043188644056571597, 15.367829047807922248007343612196, 15.92181879936879732131579202775, 16.97302209694439889515991545721, 17.25671868670348710904153931563, 18.37757028816112359513820739688, 18.68442831703071420580101996686, 19.918526689992005318157029798951, 20.39893287414427874924059031720, 21.110530774104284448237473697603

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