LMFDB - L-function 2-1584-11.10-c2-0-13

Dirichlet series

L(s) = 1

− 2.38·5^-s − 10.1i·7^-s + (−6.00 + 9.21i)11^-s + 1.76i·13^-s + 6.88i·17^-s − 5.12i·19^-s − 14.1·23^-s − 19.3·25^-s + 17.1i·29^-s + 37.8·31^-s + 24.0i·35^-s − 33.8·37^-s + 11.5i·41^-s + 1.30i·43^-s + 73.8·47^-s + ⋯

L(s) = 1

− 0.476·5^-s − 1.44i·7^-s + (−0.546 + 0.837i)11^-s + 0.135i·13^-s + 0.405i·17^-s − 0.269i·19^-s − 0.616·23^-s − 0.772·25^-s + 0.592i·29^-s + 1.21·31^-s + 0.688i·35^-s − 0.913·37^-s + 0.281i·41^-s + 0.0304i·43^-s + 1.57·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1584$ = $2^{4} \cdot 3^{2} \cdot 11$
Sign:	$0.546 - 0.837i$
Analytic conductor:	$43.1608$
Root analytic conductor:	$6.56969$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1584} (1297, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1584,\ (\ :1),\ 0.546 - 0.837i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.141565497$
$L(\frac12)$	$\approx$	$1.141565497$
$L(2)$		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 + (6.00 - 9.21i)T $
good	5	$ 1 + 2.38T + 25T^{2} $
	7	$ 1 + 10.1iT - 49T^{2} $
	13	$ 1 - 1.76iT - 169T^{2} $
	17	$ 1 - 6.88iT - 289T^{2} $
	19	$ 1 + 5.12iT - 361T^{2} $
	23	$ 1 + 14.1T + 529T^{2} $
	29	$ 1 - 17.1iT - 841T^{2} $
	31	$ 1 - 37.8T + 961T^{2} $
	37	$ 1 + 33.8T + 1.36e3T^{2} $
	41	$ 1 - 11.5iT - 1.68e3T^{2} $
	43	$ 1 - 1.30iT - 1.84e3T^{2} $
	47	$ 1 - 73.8T + 2.20e3T^{2} $
	53	$ 1 - 2.81T + 2.80e3T^{2} $
	59	$ 1 + 30.6T + 3.48e3T^{2} $
	61	$ 1 - 76.1iT - 3.72e3T^{2} $
	67	$ 1 - 70.4T + 4.48e3T^{2} $
	71	$ 1 - 102.T + 5.04e3T^{2} $
	73	$ 1 - 64.4iT - 5.32e3T^{2} $
	79	$ 1 + 128. iT - 6.24e3T^{2} $
	83	$ 1 - 47.4iT - 6.88e3T^{2} $
	89	$ 1 - 157.T + 7.92e3T^{2} $
	97	$ 1 + 36.1T + 9.40e3T^{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

−9.480985593494070976236144430932, −8.394740735262426407676586691588, −7.64342433960702557153999390884, −7.14835472460161225806369588841, −6.26680989131155112453479001975, −5.04221841042236510021741184115, −4.24647534777548406482242313825, −3.59289166905011724094144072880, −2.23505338065096218617413407034, −0.919108122850998432788298763328, 0.38240089707180249777230067190, 2.09832598015774105126453034500, 2.94882573278538782479222227239, 3.95506255058069625274292863753, 5.15268256536386696828578239152, 5.76568353126030698914574910047, 6.54566114411849847714266783726, 7.80578146165243441230099037594, 8.243519967251289945513642921165, 9.020409257316093623179069517334

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 $\F_{q}$
Belyi maps

\(L(\frac{3}{2})\)	\(\approx\)	\(1.141565497\)
\(L(\frac12)\)	\(\approx\)	\(1.141565497\)
\(L(2)\)		not available
\(L(1)\)		not available

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