LMFDB - L-function 2-6048-24.11-c1-0-3

Dirichlet series

L(s) = 1

− 2.85·5^-s − i·7^-s + 4.98i·11^-s + 2.35i·13^-s + 8.19i·17^-s − 5.90·19^-s + 4.43·23^-s + 3.17·25^-s − 6.41·29^-s + 3.31i·31^-s + 2.85i·35^-s − 5.51i·37^-s − 3.41i·41^-s − 1.37·43^-s + 2.60·47^-s + ⋯

L(s) = 1

− 1.27·5^-s − 0.377i·7^-s + 1.50i·11^-s + 0.652i·13^-s + 1.98i·17^-s − 1.35·19^-s + 0.925·23^-s + 0.634·25^-s − 1.19·29^-s + 0.595i·31^-s + 0.483i·35^-s − 0.905i·37^-s − 0.533i·41^-s − 0.209·43^-s + 0.379·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$-0.755 + 0.654i$
Analytic conductor:	$48.2935$
Root analytic conductor:	$6.94935$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6048} (5615, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6048,\ (\ :1/2),\ -0.755 + 0.654i)$

Particular Values

$L(1)$	$\approx$	$0.1944309066$
$L(\frac12)$	$\approx$	$0.1944309066$
$L(\frac{3}{2})$		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 + iT $
good	5	$ 1 + 2.85T + 5T^{2} $
	11	$ 1 - 4.98iT - 11T^{2} $
	13	$ 1 - 2.35iT - 13T^{2} $
	17	$ 1 - 8.19iT - 17T^{2} $
	19	$ 1 + 5.90T + 19T^{2} $
	23	$ 1 - 4.43T + 23T^{2} $
	29	$ 1 + 6.41T + 29T^{2} $
	31	$ 1 - 3.31iT - 31T^{2} $
	37	$ 1 + 5.51iT - 37T^{2} $
	41	$ 1 + 3.41iT - 41T^{2} $
	43	$ 1 + 1.37T + 43T^{2} $
	47	$ 1 - 2.60T + 47T^{2} $
	53	$ 1 - 13.3T + 53T^{2} $
	59	$ 1 - 10.6iT - 59T^{2} $
	61	$ 1 + 1.41iT - 61T^{2} $
	67	$ 1 + 0.221T + 67T^{2} $
	71	$ 1 + 0.398T + 71T^{2} $
	73	$ 1 + 10.7T + 73T^{2} $
	79	$ 1 - 9.76iT - 79T^{2} $
	83	$ 1 + 2.06iT - 83T^{2} $
	89	$ 1 + 17.6iT - 89T^{2} $
	97	$ 1 + 3.62T + 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

−8.678627168588501357666132001511, −7.55397882988649379421343754495, −7.29740867865866514904159042472, −6.57887126177945967694135612727, −5.67606885057738842299548438619, −4.58898835098931235974398076772, −4.08155501699619583129260542966, −3.70182752425059965333430727863, −2.30568565568410475465715517028, −1.51413382529173861148662732682, 0.06511277867035971586365332615, 0.835501587957087443117857669692, 2.48189162973743229319333360666, 3.17587741669727444361252707849, 3.83685221568063036428738973076, 4.75962647481907952134318578939, 5.43016008864169350920372327280, 6.22774314608448062370651914895, 7.07243390916032594610119010528, 7.68596626046361310783606097485

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.1944309066\)
\(L(\frac12)\)	\(\approx\)	\(0.1944309066\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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