LMFDB - L-function 1-2652-2652.1019-r1-0-0

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 + ⋯

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:	$\chi_{2652} (1019, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 2652,\ (1:\ ),\ 0.289 - 0.957i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.643877838 - 1.219849362i$
$L(\frac12)$	$\approx$	$1.643877838 - 1.219849362i$
$L(1)$	$\approx$	$1.075618898 - 0.1842456970i$
$L(1)$	$\approx$	$1.075618898 - 0.1842456970i$

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 $
	43	$ 1 $
	47	$ 1 $
	53	$ 1 $
	59	$ 1 $
	61	$ 1 $
	67	$ 1 $
	71	$ 1 + iT $
	73	$ 1 $
	79	$ 1 $
	83	$ 1 $
	89	$ 1 + T $
	97	$ 1 $
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$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

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

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(\frac{1}{2})\)	\(\approx\)	\(1.643877838 - 1.219849362i\)
\(L(\frac12)\)	\(\approx\)	\(1.643877838 - 1.219849362i\)
\(L(1)\)	\(\approx\)	\(1.075618898 - 0.1842456970i\)
\(L(1)\)	\(\approx\)	\(1.075618898 - 0.1842456970i\)

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