LMFDB - L-function 2-33e2-33.32-c1-0-16

Dirichlet series

L(s) = 1

− 1.73·2^-s + 0.999·4^-s + 3.34i·5^-s + 1.03i·7^-s + 1.73·8^-s − 5.79i·10^-s − 6.17i·13^-s − 1.79i·14^-s − 5·16^-s + 4.26·17^-s − 5.65i·19^-s + 3.34i·20^-s − 6.69i·23^-s − 6.19·25^-s + 10.6i·26^-s + ⋯

L(s) = 1

− 1.22·2^-s + 0.499·4^-s + 1.49i·5^-s + 0.391i·7^-s + 0.612·8^-s − 1.83i·10^-s − 1.71i·13^-s − 0.479i·14^-s − 1.25·16^-s + 1.03·17^-s − 1.29i·19^-s + 0.748i·20^-s − 1.39i·23^-s − 1.23·25^-s + 2.09i·26^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0659i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1089$ = $3^{2} \cdot 11^{2}$
Sign:	$0.997 + 0.0659i$
Analytic conductor:	$8.69570$
Root analytic conductor:	$2.94884$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1089} (1088, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1089,\ (\ :1/2),\ 0.997 + 0.0659i)$

Particular Values

$L(1)$	$\approx$	$0.7740424806$
$L(\frac12)$	$\approx$	$0.7740424806$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 $
	11	$ 1 $
good	2	$ 1 + 1.73T + 2T^{2} $
	5	$ 1 - 3.34iT - 5T^{2} $
	7	$ 1 - 1.03iT - 7T^{2} $
	13	$ 1 + 6.17iT - 13T^{2} $
	17	$ 1 - 4.26T + 17T^{2} $
	19	$ 1 + 5.65iT - 19T^{2} $
	23	$ 1 + 6.69iT - 23T^{2} $
	29	$ 1 - 0.464T + 29T^{2} $
	31	$ 1 - 9.46T + 31T^{2} $
	37	$ 1 + 3T + 37T^{2} $
	41	$ 1 + 0.464T + 41T^{2} $
	43	$ 1 + 5.93iT - 43T^{2} $
	47	$ 1 - 9.79iT - 47T^{2} $
	53	$ 1 - 0.896iT - 53T^{2} $
	59	$ 1 - 1.79iT - 59T^{2} $
	61	$ 1 - 1.41iT - 61T^{2} $
	67	$ 1 - 0.928T + 67T^{2} $
	71	$ 1 + 6.69iT - 71T^{2} $
	73	$ 1 + 3.48iT - 73T^{2} $
	79	$ 1 + 0.757iT - 79T^{2} $
	83	$ 1 + 9.46T + 83T^{2} $
	89	$ 1 + 10.6iT - 89T^{2} $
	97	$ 1 - T + 97T^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−10.14755863945870599208404061292, −9.014353581536813908533844396284, −8.212560225278728064910848743190, −7.54272707840599118530726201602, −6.78852667116849968258384100683, −5.87518275545777147718248524483, −4.68136264972373528801725023166, −3.13566021778699767677843421811, −2.50202572027507866532171620786, −0.68230968651203488465923674079, 1.05063964477033145930900136371, 1.75768412606207051868366068912, 3.83805922632564112381975736196, 4.61821961762401860156742153419, 5.57925922923502941948474451950, 6.81827449628234482064041230439, 7.76000328763721469962363578617, 8.361869721447544068278778562492, 9.054237354203499761264014542463, 9.760502355358582352762310535608

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

\(L(1)\)	\(\approx\)	\(0.7740424806\)
\(L(\frac12)\)	\(\approx\)	\(0.7740424806\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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