LMFDB - L-function 2-864-24.11-c1-0-9

Dirichlet series

L(s) = 1

− 3.07·5^-s + 3.99i·7^-s − 1.89i·11^-s − 1.06i·13^-s − 7.08i·17^-s − 3.73·19^-s − 0.824·23^-s + 4.46·25^-s + 4.50·29^-s − 5.84i·31^-s − 12.2i·35^-s − 6.91i·37^-s − 3.79i·41^-s − 2·43^-s − 6.97·47^-s + ⋯

L(s) = 1

− 1.37·5^-s + 1.50i·7^-s − 0.572i·11^-s − 0.296i·13^-s − 1.71i·17^-s − 0.856·19^-s − 0.171·23^-s + 0.892·25^-s + 0.836·29^-s − 1.04i·31^-s − 2.07i·35^-s − 1.13i·37^-s − 0.593i·41^-s − 0.304·43^-s − 1.01·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.398 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$864$ = $2^{5} \cdot 3^{3}$
Sign:	$-0.398 + 0.917i$
Analytic conductor:	$6.89907$
Root analytic conductor:	$2.62660$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{864} (431, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 864,\ (\ :1/2),\ -0.398 + 0.917i)$

Particular Values

$L(1)$	$\approx$	$0.270599 - 0.412420i$
$L(\frac12)$	$\approx$	$0.270599 - 0.412420i$
$L(\frac{3}{2})$		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 $
good	5	$ 1 + 3.07T + 5T^{2} $
	7	$ 1 - 3.99iT - 7T^{2} $
	11	$ 1 + 1.89iT - 11T^{2} $
	13	$ 1 + 1.06iT - 13T^{2} $
	17	$ 1 + 7.08iT - 17T^{2} $
	19	$ 1 + 3.73T + 19T^{2} $
	23	$ 1 + 0.824T + 23T^{2} $
	29	$ 1 - 4.50T + 29T^{2} $
	31	$ 1 + 5.84iT - 31T^{2} $
	37	$ 1 + 6.91iT - 37T^{2} $
	41	$ 1 + 3.79iT - 41T^{2} $
	43	$ 1 + 2T + 43T^{2} $
	47	$ 1 + 6.97T + 47T^{2} $
	53	$ 1 + 10.6T + 53T^{2} $
	59	$ 1 + 1.89iT - 59T^{2} $
	61	$ 1 - 1.06iT - 61T^{2} $
	67	$ 1 - 9.19T + 67T^{2} $
	71	$ 1 + 10.6T + 71T^{2} $
	73	$ 1 - 5.92T + 73T^{2} $
	79	$ 1 + 6.12iT - 79T^{2} $
	83	$ 1 - 10.3iT - 83T^{2} $
	89	$ 1 + 13.6iT - 89T^{2} $
	97	$ 1 + 11.3T + 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

−9.753868706879189637100872329865, −8.895280660137932144343952516795, −8.265989881118638899327899455586, −7.50812635528284922901639708224, −6.42341808974145418262124301990, −5.44159350837571738116361173269, −4.52605534554818714567998922708, −3.34511029879566128483568629553, −2.42626453747348053072670578524, −0.24726067690025425058691021133, 1.44489562777165457114826293861, 3.35303581159794576460311244158, 4.15423042265553982984305965198, 4.69080046851218350005800373081, 6.44955703966816708358881980654, 7.00005384304301149855038776770, 8.067374063990042777312500957540, 8.316172303984689158207519295535, 9.789283347821775099439575030609, 10.56411116604606671451065272602

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.270599 - 0.412420i\)
\(L(\frac12)\)	\(\approx\)	\(0.270599 - 0.412420i\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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