LMFDB - L-function 2-6048-12.11-c1-0-53

Dirichlet series

L(s) = 1

− 2.64i·5^-s − i·7^-s + 4.12·11^-s − 4.86·13^-s + 2.03i·17^-s + 4.16i·19^-s + 9.48·23^-s − 1.99·25^-s − 1.54i·29^-s + 0.248i·31^-s − 2.64·35^-s + 7.88·37^-s − 2.00i·41^-s − 0.547i·43^-s + 2.59·47^-s + ⋯

L(s) = 1

− 1.18i·5^-s − 0.377i·7^-s + 1.24·11^-s − 1.35·13^-s + 0.494i·17^-s + 0.956i·19^-s + 1.97·23^-s − 0.398·25^-s − 0.287i·29^-s + 0.0446i·31^-s − 0.446·35^-s + 1.29·37^-s − 0.313i·41^-s − 0.0835i·43^-s + 0.378·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$48.2935$
Root analytic conductor:	$6.94935$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6048} (2591, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6048,\ (\ :1/2),\ 0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$2.137287986$
$L(\frac12)$	$\approx$	$2.137287986$
$L(\frac{3}{2})$		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 + 2.64iT - 5T^{2} $
	11	$ 1 - 4.12T + 11T^{2} $
	13	$ 1 + 4.86T + 13T^{2} $
	17	$ 1 - 2.03iT - 17T^{2} $
	19	$ 1 - 4.16iT - 19T^{2} $
	23	$ 1 - 9.48T + 23T^{2} $
	29	$ 1 + 1.54iT - 29T^{2} $
	31	$ 1 - 0.248iT - 31T^{2} $
	37	$ 1 - 7.88T + 37T^{2} $
	41	$ 1 + 2.00iT - 41T^{2} $
	43	$ 1 + 0.547iT - 43T^{2} $
	47	$ 1 - 2.59T + 47T^{2} $
	53	$ 1 - 11.5iT - 53T^{2} $
	59	$ 1 + 12.4T + 59T^{2} $
	61	$ 1 - 14.2T + 61T^{2} $
	67	$ 1 - 6.52iT - 67T^{2} $
	71	$ 1 - 5.95T + 71T^{2} $
	73	$ 1 - 11.7T + 73T^{2} $
	79	$ 1 + 7.47iT - 79T^{2} $
	83	$ 1 - 3.94T + 83T^{2} $
	89	$ 1 - 12.9iT - 89T^{2} $
	97	$ 1 + 5.64T + 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

−8.007886939447452603896785690864, −7.30822562347185393236494183965, −6.63080081211503814223904111028, −5.77184070112609239664579347644, −4.99409025305640252894224937309, −4.41227260212813690106511718385, −3.73295728182039510306918095358, −2.62953484174176012983876429367, −1.47535502903929700656508948929, −0.78210144882727260029422571181, 0.831494375140135948996653399023, 2.23808657607683580958223795366, 2.84250630676006989671616221552, 3.54621368689525952962549086031, 4.67662526704622821681074581076, 5.16429586647966585104969046831, 6.28218566396175511304974728186, 6.90284985453768955709598563781, 7.13588875716192694312852962838, 8.064876662030712570076796037695

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.137287986\)
\(L(\frac12)\)	\(\approx\)	\(2.137287986\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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