LMFDB - L-function 2-6048-24.11-c1-0-36

Dirichlet series

L(s) = 1

− 2.90·5^-s − i·7^-s + 1.28i·11^-s + 1.86i·13^-s − 3.87i·17^-s + 2.04·19^-s − 0.934·23^-s + 3.45·25^-s − 2.98·29^-s + 1.85i·31^-s + 2.90i·35^-s − 3.85i·37^-s + 8.72i·41^-s + 1.19·43^-s − 11.1·47^-s + ⋯

L(s) = 1

− 1.30·5^-s − 0.377i·7^-s + 0.388i·11^-s + 0.518i·13^-s − 0.939i·17^-s + 0.468·19^-s − 0.194·23^-s + 0.691·25^-s − 0.553·29^-s + 0.333i·31^-s + 0.491i·35^-s − 0.634i·37^-s + 1.36i·41^-s + 0.181·43^-s − 1.62·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0726i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$0.997 - 0.0726i$
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:	$(2,\ 6048,\ (\ :1/2),\ 0.997 - 0.0726i)$

Particular Values

$L(1)$	$\approx$	$1.107814304$
$L(\frac12)$	$\approx$	$1.107814304$
$L(\frac{3}{2})$		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 + 2.90T + 5T^{2} $
	11	$ 1 - 1.28iT - 11T^{2} $
	13	$ 1 - 1.86iT - 13T^{2} $
	17	$ 1 + 3.87iT - 17T^{2} $
	19	$ 1 - 2.04T + 19T^{2} $
	23	$ 1 + 0.934T + 23T^{2} $
	29	$ 1 + 2.98T + 29T^{2} $
	31	$ 1 - 1.85iT - 31T^{2} $
	37	$ 1 + 3.85iT - 37T^{2} $
	41	$ 1 - 8.72iT - 41T^{2} $
	43	$ 1 - 1.19T + 43T^{2} $
	47	$ 1 + 11.1T + 47T^{2} $
	53	$ 1 - 4.43T + 53T^{2} $
	59	$ 1 + 10.9iT - 59T^{2} $
	61	$ 1 - 4.70iT - 61T^{2} $
	67	$ 1 + 4.41T + 67T^{2} $
	71	$ 1 + 9.72T + 71T^{2} $
	73	$ 1 - 9.40T + 73T^{2} $
	79	$ 1 - 16.4iT - 79T^{2} $
	83	$ 1 + 3.72iT - 83T^{2} $
	89	$ 1 + 12.2iT - 89T^{2} $
	97	$ 1 - 15.8T + 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

−7.987784933879810082990095120595, −7.32745272703215099363542540133, −6.96756473725090573371080699137, −6.01088927410428222362172073178, −4.95708247002924207533866350488, −4.46223848046850841124479861526, −3.66853771985040648596779928148, −3.00285964316588691230909348116, −1.79870857970520998743530053372, −0.57181651928980280830723857314, 0.52392351764112644112895090051, 1.80693300165186365255147772401, 2.99387432664040430268337653643, 3.63155787217996986449724919924, 4.28689960659680930828877562934, 5.21790028786421809883680490349, 5.90279048209183522253065203455, 6.69366254580728825153051554934, 7.61037630095099738338556282994, 7.949390614157186415232639655655

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.107814304\)
\(L(\frac12)\)	\(\approx\)	\(1.107814304\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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