LMFDB - L-function 2-864-12.11-c3-0-20

Dirichlet series

L(s) = 1

+ 4.19i·5^-s + 15.8i·7^-s + 59.8·11^-s + 44.5·13^-s + 40.8i·17^-s − 121. i·19^-s − 61.5·23^-s + 107.·25^-s − 154. i·29^-s + 120. i·31^-s − 66.6·35^-s − 164.·37^-s − 311. i·41^-s + 331. i·43^-s − 30.1·47^-s + ⋯

L(s) = 1

+ 0.375i·5^-s + 0.857i·7^-s + 1.64·11^-s + 0.949·13^-s + 0.582i·17^-s − 1.46i·19^-s − 0.557·23^-s + 0.859·25^-s − 0.990i·29^-s + 0.695i·31^-s − 0.321·35^-s − 0.731·37^-s − 1.18i·41^-s + 1.17i·43^-s − 0.0935·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$864$ = $2^{5} \cdot 3^{3}$
Sign:	$0.707 - 0.707i$
Analytic conductor:	$50.9776$
Root analytic conductor:	$7.13986$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{864} (863, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 864,\ (\ :3/2),\ 0.707 - 0.707i)$

Particular Values

$L(2)$	$\approx$	$2.436072296$
$L(\frac12)$	$\approx$	$2.436072296$
$L(\frac{5}{2})$		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 - 4.19iT - 125T^{2} $
	7	$ 1 - 15.8iT - 343T^{2} $
	11	$ 1 - 59.8T + 1.33e3T^{2} $
	13	$ 1 - 44.5T + 2.19e3T^{2} $
	17	$ 1 - 40.8iT - 4.91e3T^{2} $
	19	$ 1 + 121. iT - 6.85e3T^{2} $
	23	$ 1 + 61.5T + 1.21e4T^{2} $
	29	$ 1 + 154. iT - 2.43e4T^{2} $
	31	$ 1 - 120. iT - 2.97e4T^{2} $
	37	$ 1 + 164.T + 5.06e4T^{2} $
	41	$ 1 + 311. iT - 6.89e4T^{2} $
	43	$ 1 - 331. iT - 7.95e4T^{2} $
	47	$ 1 + 30.1T + 1.03e5T^{2} $
	53	$ 1 - 466. iT - 1.48e5T^{2} $
	59	$ 1 - 384.T + 2.05e5T^{2} $
	61	$ 1 - 568.T + 2.26e5T^{2} $
	67	$ 1 - 808. iT - 3.00e5T^{2} $
	71	$ 1 + 164.T + 3.57e5T^{2} $
	73	$ 1 - 936.T + 3.89e5T^{2} $
	79	$ 1 - 47.6iT - 4.93e5T^{2} $
	83	$ 1 + 323.T + 5.71e5T^{2} $
	89	$ 1 + 649. iT - 7.04e5T^{2} $
	97	$ 1 + 249.T + 9.12e5T^{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

−9.783601026170017719023962711838, −8.851302579928747448700828205055, −8.554752894282809964592619136245, −7.10643291600698221654130690618, −6.44295038400976925238462723617, −5.67462663425585786143798652843, −4.40284758484174738304947840431, −3.47926768403462794202542988615, −2.30821036344559748415565829418, −1.04962555890396641352545112413, 0.817407098678759306939125952721, 1.68357963360543830157714266604, 3.55489437837747534345638233087, 4.02392737897643425208735833023, 5.23251285311901968344829805107, 6.34906066398963564000497735686, 6.97130964456668387253185965609, 8.082037872529630233544901616206, 8.821974693129861944384709808909, 9.655065043368820947152657060340

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(2.436072296\)
\(L(\frac12)\)	\(\approx\)	\(2.436072296\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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