LMFDB - L-function 2-1584-44.43-c1-0-10

Dirichlet series

L(s) = 1

− 2.73·5^-s − 5.13·7^-s + (−1.88 + 2.73i)11^-s − 5.13i·13^-s + 3.76i·17^-s + 3.76·19^-s + 1.26i·23^-s + 2.46·25^-s − 2i·31^-s + 14.0·35^-s + 0.535·37^-s − 10.2i·41^-s + 3.76·43^-s + 4.19i·47^-s + 19.3·49^-s + ⋯

L(s) = 1

− 1.22·5^-s − 1.94·7^-s + (−0.566 + 0.823i)11^-s − 1.42i·13^-s + 0.912i·17^-s + 0.862·19^-s + 0.264i·23^-s + 0.492·25^-s − 0.359i·31^-s + 2.37·35^-s + 0.0881·37^-s − 1.60i·41^-s + 0.573·43^-s + 0.612i·47^-s + 2.77·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0791i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1584$ = $2^{4} \cdot 3^{2} \cdot 11$
Sign:	$0.996 + 0.0791i$
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:	$(2,\ 1584,\ (\ :1/2),\ 0.996 + 0.0791i)$

Particular Values

$L(1)$	$\approx$	$0.6915463662$
$L(\frac12)$	$\approx$	$0.6915463662$
$L(\frac{3}{2})$		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 + (1.88 - 2.73i)T $
good	5	$ 1 + 2.73T + 5T^{2} $
	7	$ 1 + 5.13T + 7T^{2} $
	13	$ 1 + 5.13iT - 13T^{2} $
	17	$ 1 - 3.76iT - 17T^{2} $
	19	$ 1 - 3.76T + 19T^{2} $
	23	$ 1 - 1.26iT - 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 + 2iT - 31T^{2} $
	37	$ 1 - 0.535T + 37T^{2} $
	41	$ 1 + 10.2iT - 41T^{2} $
	43	$ 1 - 3.76T + 43T^{2} $
	47	$ 1 - 4.19iT - 47T^{2} $
	53	$ 1 - 10.7T + 53T^{2} $
	59	$ 1 - 9.46iT - 59T^{2} $
	61	$ 1 - 2.38iT - 61T^{2} $
	67	$ 1 - 10.3iT - 67T^{2} $
	71	$ 1 + 10.7iT - 71T^{2} $
	73	$ 1 + 10.2iT - 73T^{2} $
	79	$ 1 - 5.13T + 79T^{2} $
	83	$ 1 + 10.2T + 83T^{2} $
	89	$ 1 - 10.3T + 89T^{2} $
	97	$ 1 - 5.46T + 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

−9.475118209314111554973841736861, −8.598277221303195924618230929176, −7.53385070333210197779980223463, −7.32489936962704426752280907022, −6.13463901697323520316804110426, −5.42157485564327696847866596917, −4.08522439675853833574654485847, −3.45603162113302516051925250032, −2.63733891581698118175310761670, −0.53541363683619127872032832978, 0.57339477833496142559792334339, 2.69976076683283500719891765888, 3.42278131420198372643253919985, 4.16481302312180365604462608312, 5.32932665255692698316246617778, 6.41957272127299413311823283827, 6.95885846283835998094679823177, 7.75147351997956101716184493330, 8.725617524795713462793757011267, 9.426473281374356046142004029780

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6915463662\)
\(L(\frac12)\)	\(\approx\)	\(0.6915463662\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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