LMFDB - L-function 2-6048-8.5-c1-0-23

Dirichlet series

L(s) = 1

− 2.52i·5^-s − 7^-s − 5.70i·11^-s + 3.06i·13^-s + 5.49·17^-s + 7.28i·19^-s + 0.539·23^-s − 1.38·25^-s + 8.35i·29^-s − 6.74·31^-s + 2.52i·35^-s + 10.2i·37^-s − 5.58·41^-s + 3.98i·43^-s + 4.83·47^-s + ⋯

L(s) = 1

− 1.12i·5^-s − 0.377·7^-s − 1.71i·11^-s + 0.848i·13^-s + 1.33·17^-s + 1.67i·19^-s + 0.112·23^-s − 0.276·25^-s + 1.55i·29^-s − 1.21·31^-s + 0.427i·35^-s + 1.68i·37^-s − 0.872·41^-s + 0.607i·43^-s + 0.705·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$0.826 - 0.563i$
Analytic conductor:	$48.2935$
Root analytic conductor:	$6.94935$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6048} (3025, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6048,\ (\ :1/2),\ 0.826 - 0.563i)$

Particular Values

$L(1)$	$\approx$	$1.516464153$
$L(\frac12)$	$\approx$	$1.516464153$
$L(\frac{3}{2})$		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 $
	7	$ 1 + T $
good	5	$ 1 + 2.52iT - 5T^{2} $
	11	$ 1 + 5.70iT - 11T^{2} $
	13	$ 1 - 3.06iT - 13T^{2} $
	17	$ 1 - 5.49T + 17T^{2} $
	19	$ 1 - 7.28iT - 19T^{2} $
	23	$ 1 - 0.539T + 23T^{2} $
	29	$ 1 - 8.35iT - 29T^{2} $
	31	$ 1 + 6.74T + 31T^{2} $
	37	$ 1 - 10.2iT - 37T^{2} $
	41	$ 1 + 5.58T + 41T^{2} $
	43	$ 1 - 3.98iT - 43T^{2} $
	47	$ 1 - 4.83T + 47T^{2} $
	53	$ 1 - 11.1iT - 53T^{2} $
	59	$ 1 + 10.1iT - 59T^{2} $
	61	$ 1 - 4.14iT - 61T^{2} $
	67	$ 1 + 5.96iT - 67T^{2} $
	71	$ 1 + 4.93T + 71T^{2} $
	73	$ 1 - 8.66T + 73T^{2} $
	79	$ 1 + 12.0T + 79T^{2} $
	83	$ 1 - 5.83iT - 83T^{2} $
	89	$ 1 + 9.28T + 89T^{2} $
	97	$ 1 - 12.3T + 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

−8.306996214517805362167609838171, −7.58477099085768736622942250885, −6.62655387647608498711922819375, −5.82850833331372921753789458913, −5.43297954673078668075550185149, −4.56451931049401484866674843430, −3.53409010237672926878073241785, −3.20166362894334016700763578369, −1.60261862329206593572192102879, −1.00884246225050178491521901746, 0.44197698104252919093145469429, 2.00463264809297039919460520844, 2.68715171124867381224553322027, 3.45688891452991902737456556585, 4.28915882766614765970098036795, 5.23381938785630537591751684229, 5.85575035759980581407251579116, 6.85487355124074903773365994512, 7.25318655589556196804863220682, 7.66749769681163317033624574066

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.516464153\)
\(L(\frac12)\)	\(\approx\)	\(1.516464153\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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