LMFDB - L-function 1-48-48.11-r0-0-0

Dirichlet series

L(s) = 1

− i·5^-s + 7^-s + i·11^-s − i·13^-s − 17^-s + i·19^-s − 23^-s − 25^-s + i·29^-s − 31^-s − i·35^-s + i·37^-s + 41^-s − i·43^-s + 47^-s + ⋯

L(s) = 1

− i·5^-s + 7^-s + i·11^-s − i·13^-s − 17^-s + i·19^-s − 23^-s − 25^-s + i·29^-s − 31^-s − i·35^-s + i·37^-s + 41^-s − i·43^-s + 47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$1$
Conductor:	$48$ = $2^{4} \cdot 3$
Sign:	$0.923 - 0.382i$
Analytic conductor:	$0.222911$
Root analytic conductor:	$0.222911$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{48} (11, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 48,\ (0:\ ),\ 0.923 - 0.382i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.8717045904 - 0.1733928237i$
$L(\frac12)$	$\approx$	$0.8717045904 - 0.1733928237i$
$L(1)$	$\approx$	$1.020168853 - 0.1259994663i$
$L(1)$	$\approx$	$1.020168853 - 0.1259994663i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
good	5	$ 1 $
	7	$ 1 $
	11	$ 1 $
	13	$ 1 - iT $
	17	$ 1 $
	19	$ 1 + T $
	23	$ 1 $
	29	$ 1 $
	31	$ 1 $
	37	$ 1 + iT $
	41	$ 1 $
	43	$ 1 - iT $
	47	$ 1 $
	53	$ 1 $
	59	$ 1 $
	61	$ 1 - T $
	67	$ 1 $
	71	$ 1 + iT $
	73	$ 1 $
	79	$ 1 $
	83	$ 1 $
	89	$ 1 - T $
	97	$ 1 $
show more
show less

$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−34.07157613187348546686696076571, −33.1064569788815915340320970165, −31.52782831194015653545716395537, −30.56630792509172516373790244024, −29.61489387186538137389796403422, −28.24025099518263954230937623053, −26.83860122179109652950169004029, −26.23834880429563446933101756048, −24.53866485801569517424613073297, −23.64976436104949759960903966951, −22.08599771811400200179247418824, −21.33893062000594098001209159786, −19.6880634972816490388960889348, −18.49602924764055737428529132403, −17.49379824529992687770081326112, −15.87836807923724293387222796553, −14.53786493282839845601741011695, −13.643666525876531529129030505082, −11.55299350201086034274972350221, −10.85584748909344450820592331202, −9.02495658118244187700793674788, −7.50958020377302481533582415431, −6.083971482044770317661908398033, −4.19414022755256984159923204976, −2.3034635391148037906869057256, 1.781012879698985036991061601368, 4.28674929978863646743381271242, 5.50451924769678764880895326870, 7.610903846836790593628450221260, 8.7659999755719292502196316342, 10.33389662102123585133413954652, 11.93494806825236707414784471852, 13.00822370664870217579803931959, 14.5610912721199171472968092355, 15.81829999763547334524480872993, 17.26348221080926405546689864202, 18.12603892526519624278503613558, 20.10826766858780377225970176470, 20.599348196665348037922352600946, 22.09933021448122534253160481272, 23.55144697311556623809667657457, 24.54836107410118062402862288257, 25.54378180805934922900677470957, 27.28504208150445842359856195294, 27.95578674194250931122874826180, 29.197281748281395876095205863079, 30.59570491619174589522255379120, 31.52053978295450363195863858800, 32.82592496398376289645925692085, 33.70967941767759775557811727044

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(0.8717045904 - 0.1733928237i\)
\(L(\frac12)\)	\(\approx\)	\(0.8717045904 - 0.1733928237i\)
\(L(1)\)	\(\approx\)	\(1.020168853 - 0.1259994663i\)
\(L(1)\)	\(\approx\)	\(1.020168853 - 0.1259994663i\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more