LMFDB - L-function 2-1584-44.43-c1-0-16

Dirichlet series

L(s) = 1

− 3·5^-s + 3.31i·11^-s − 3.31i·23^-s + 4·25^-s − 9.94i·31^-s + 7·37^-s − 6.63i·47^-s − 7·49^-s − 6·53^-s − 9.94i·55^-s − 3.31i·59^-s − 9.94i·67^-s − 16.5i·71^-s + 9·89^-s − 17·97^-s + ⋯

L(s) = 1

− 1.34·5^-s + 1.00i·11^-s − 0.691i·23^-s + 0.800·25^-s − 1.78i·31^-s + 1.15·37^-s − 0.967i·47^-s − 49^-s − 0.824·53^-s − 1.34i·55^-s − 0.431i·59^-s − 1.21i·67^-s − 1.96i·71^-s + 0.953·89^-s − 1.72·97^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1584$ = $2^{4} \cdot 3^{2} \cdot 11$
Sign:	$i$
Analytic conductor:	$12.6483$
Root analytic conductor:	$3.55644$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1584} (703, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1584,\ (\ :1/2),\ i)$

Particular Values

$L(1)$	$\approx$	$0.7612397610$
$L(\frac12)$	$\approx$	$0.7612397610$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	11	$ 1 - 3.31iT $
good	5	$ 1 + 3T + 5T^{2} $
	7	$ 1 + 7T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 - 17T^{2} $
	19	$ 1 + 19T^{2} $
	23	$ 1 + 3.31iT - 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 + 9.94iT - 31T^{2} $
	37	$ 1 - 7T + 37T^{2} $
	41	$ 1 - 41T^{2} $
	43	$ 1 + 43T^{2} $
	47	$ 1 + 6.63iT - 47T^{2} $
	53	$ 1 + 6T + 53T^{2} $
	59	$ 1 + 3.31iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 + 9.94iT - 67T^{2} $
	71	$ 1 + 16.5iT - 71T^{2} $
	73	$ 1 - 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 - 9T + 89T^{2} $
	97	$ 1 + 17T + 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

−9.268837417625597732973522039058, −8.158487958521229707799719767839, −7.76427541762664832793694490961, −6.95320088838568664856591127885, −6.06369749379060009767879803291, −4.77413391941721413371555392182, −4.24757181404877892307860964303, −3.28772959447658657385014592230, −2.05847254630549191612834740248, −0.34302960194193152541892960370, 1.12974241868404701799265172265, 2.89569895439985644400896794507, 3.63200033997258241259008051521, 4.50280814318955923080287379462, 5.48711054168476884362949155134, 6.47058745551317505987186758689, 7.35385081331080536077665418146, 8.065230724546502233192443461282, 8.636712062768920253185706800042, 9.527298031399968695009072617038

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7612397610\)
\(L(\frac12)\)	\(\approx\)	\(0.7612397610\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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