L-function 1-2652-2652.203-r1-0-0

• Label: 1-2652-2652.203-r1-0-0
• Degree: $1$
• Conductor: $2652$
• Sign: $0.289 + 0.957i$
• Analytic cond.: $284.996$
• Root an. cond.: $284.996$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ i·5^-s − i·7^-s + i·11^-s − i·19^-s + 23^-s − 25^-s + 29^-s + i·31^-s + 35^-s + i·37^-s + i·41^-s + 43^-s − i·47^-s − 49^-s − 53^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$2652$    =    $2^{2} \cdot 3 \cdot 13 \cdot 17$
Sign:	$0.289 + 0.957i$
Analytic conductor:	$284.996$
Root analytic conductor:	$284.996$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2652} (203, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 2652,\ (1:\ ),\ 0.289 + 0.957i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.643877838 + 1.219849362i$
$L(1)$	$\approx$	$1.075618898 + 0.1842456970i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	3	$1$
	13	$1$
	17	$1$
good	5	$1$
	7	$1$
	11	$1$
	19	$1 - iT$
	23	$1$
	29	$1$
	31	$1$
	37	$1 + iT$
	41	$1$

Imaginary part of the first few zeros on the critical line

−19.10956064636441266752044591556, −18.39839401862955188135216063870, −17.48981246494809953304747523198, −16.88276877422785777681644106888, −16.03568852323072417840652775504, −15.75098451281606064143654806189, −14.722255341152508721348282572454, −14.03566329615061796427885474951, −13.17834829732965333787803956383, −12.54009602222527183157285568826, −11.96164207785894910024052951095, −11.21429136814709594949187819434, −10.36122982571962732343073083333, −9.28034492782624357656800554962, −8.9417584090894607363851337144, −8.19557378970015794381274494239, −7.50161740104936835524152294297, −6.13157231923907737778152466677, −5.80090288172321568368146165004, −4.97879612694454144513622772018, −4.12869669708688481371458180387, −3.17473240890460697593794793333, −2.28826144508215462356608315416, −1.29488249763888711895309197462, −0.43725738136437255845062952910, 0.76413261241354197522231600514, 1.77795092247748605156493586096, 2.81891249739205050324702558859, 3.41414858487303379050400182490, 4.50768020148246360295445045797, 4.97644780078939820155912710355, 6.393761151498034007044642702302, 6.830356242139791647672047262835, 7.4254792294155592477579334814, 8.23054868929958096977486016744, 9.36123508126607548216047082043, 9.97855572221918606945473014626, 10.72496352159313116172129331176, 11.16976182411773541550636072146, 12.12986818959568642817900725196, 12.94303539004129030711808944684, 13.7427225502222050807743849393, 14.248082290083688128775216467057, 15.12415842868160741006042007330, 15.53129080935302200488606887593, 16.55814624498709684893815778434, 17.36149520085207346143707275365, 17.789749420702633933183960062933, 18.51039017039806790702231562002, 19.48641617190070176595273425397

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