LMFDB - L-function 2-1584-11.10-c2-0-19

Dirichlet series

L(s) = 1

− 1.84·5^-s − 3.78i·7^-s + (−9.00 + 6.31i)11^-s − 19.1i·13^-s + 25.7i·17^-s + 10.1i·19^-s + 30.5·23^-s − 21.6·25^-s − 18.8i·29^-s − 52.3·31^-s + 6.96i·35^-s + 56.3·37^-s − 13.1i·41^-s + 76.2i·43^-s − 41.2·47^-s + ⋯

L(s) = 1

− 0.368·5^-s − 0.540i·7^-s + (−0.818 + 0.573i)11^-s − 1.47i·13^-s + 1.51i·17^-s + 0.533i·19^-s + 1.32·23^-s − 0.864·25^-s − 0.648i·29^-s − 1.68·31^-s + 0.199i·35^-s + 1.52·37^-s − 0.320i·41^-s + 1.77i·43^-s − 0.878·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.573i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.818 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1584$ = $2^{4} \cdot 3^{2} \cdot 11$
Sign:	$0.818 - 0.573i$
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:	$(2,\ 1584,\ (\ :1),\ 0.818 - 0.573i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.420905799$
$L(\frac12)$	$\approx$	$1.420905799$
$L(2)$		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 + (9.00 - 6.31i)T $
good	5	$ 1 + 1.84T + 25T^{2} $
	7	$ 1 + 3.78iT - 49T^{2} $
	13	$ 1 + 19.1iT - 169T^{2} $
	17	$ 1 - 25.7iT - 289T^{2} $
	19	$ 1 - 10.1iT - 361T^{2} $
	23	$ 1 - 30.5T + 529T^{2} $
	29	$ 1 + 18.8iT - 841T^{2} $
	31	$ 1 + 52.3T + 961T^{2} $
	37	$ 1 - 56.3T + 1.36e3T^{2} $
	41	$ 1 + 13.1iT - 1.68e3T^{2} $
	43	$ 1 - 76.2iT - 1.84e3T^{2} $
	47	$ 1 + 41.2T + 2.20e3T^{2} $
	53	$ 1 - 102.T + 2.80e3T^{2} $
	59	$ 1 - 68.0T + 3.48e3T^{2} $
	61	$ 1 + 54.9iT - 3.72e3T^{2} $
	67	$ 1 + 15.1T + 4.48e3T^{2} $
	71	$ 1 - 81.6T + 5.04e3T^{2} $
	73	$ 1 + 110. iT - 5.32e3T^{2} $
	79	$ 1 - 101. iT - 6.24e3T^{2} $
	83	$ 1 - 145. iT - 6.88e3T^{2} $
	89	$ 1 + 75.2T + 7.92e3T^{2} $
	97	$ 1 - 51.7T + 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

−9.430534599816371491916135176903, −8.140760812398677835682813748423, −7.920621189350692703140968477377, −7.06517747085893050766681470156, −5.94392018067899939347403296511, −5.26367189820884576637678086285, −4.16105090749950061883327025764, −3.41029688814083549656508222704, −2.23145844206869137066378954330, −0.838715155009292343685593328635, 0.51861557292620957322786169561, 2.12286151645510695561603191075, 3.00897936605509359084472571353, 4.11350664602985011511429931628, 5.09807727283429827716176559448, 5.72969624824221055408698388317, 7.06847470569184532699259453698, 7.27859493948876834789187453309, 8.594737560928228113898031303930, 9.029510171986147850183063998336

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

\(L(\frac{3}{2})\)	\(\approx\)	\(1.420905799\)
\(L(\frac12)\)	\(\approx\)	\(1.420905799\)
\(L(2)\)		not available
\(L(1)\)		not available

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