LMFDB - L-function 2-6048-24.11-c1-0-80

Dirichlet series

L(s) = 1

+ 3.47·5^-s + i·7^-s − 5.11i·11^-s + 0.268i·13^-s + 1.96i·17^-s − 0.909·19^-s − 5.86·23^-s + 7.04·25^-s − 7.19·29^-s − 4.57i·31^-s + 3.47i·35^-s − 6.98i·37^-s − 8.34i·41^-s + 9.25·43^-s + 6.49·47^-s + ⋯

L(s) = 1

+ 1.55·5^-s + 0.377i·7^-s − 1.54i·11^-s + 0.0744i·13^-s + 0.476i·17^-s − 0.208·19^-s − 1.22·23^-s + 1.40·25^-s − 1.33·29^-s − 0.821i·31^-s + 0.586i·35^-s − 1.14i·37^-s − 1.30i·41^-s + 1.41·43^-s + 0.947·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$0.331 + 0.943i$
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:	$(2,\ 6048,\ (\ :1/2),\ 0.331 + 0.943i)$

Particular Values

$L(1)$	$\approx$	$2.336923913$
$L(\frac12)$	$\approx$	$2.336923913$
$L(\frac{3}{2})$		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 - 3.47T + 5T^{2} $
	11	$ 1 + 5.11iT - 11T^{2} $
	13	$ 1 - 0.268iT - 13T^{2} $
	17	$ 1 - 1.96iT - 17T^{2} $
	19	$ 1 + 0.909T + 19T^{2} $
	23	$ 1 + 5.86T + 23T^{2} $
	29	$ 1 + 7.19T + 29T^{2} $
	31	$ 1 + 4.57iT - 31T^{2} $
	37	$ 1 + 6.98iT - 37T^{2} $
	41	$ 1 + 8.34iT - 41T^{2} $
	43	$ 1 - 9.25T + 43T^{2} $
	47	$ 1 - 6.49T + 47T^{2} $
	53	$ 1 - 4.54T + 53T^{2} $
	59	$ 1 + 3.50iT - 59T^{2} $
	61	$ 1 + 1.96iT - 61T^{2} $
	67	$ 1 - 11.0T + 67T^{2} $
	71	$ 1 + 1.20T + 71T^{2} $
	73	$ 1 + 9.01T + 73T^{2} $
	79	$ 1 + 13.5iT - 79T^{2} $
	83	$ 1 - 4.96iT - 83T^{2} $
	89	$ 1 + 18.2iT - 89T^{2} $
	97	$ 1 - 3.39T + 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

−8.012215192472246057696012676755, −7.19485072726364839570167581078, −6.14362849296650178206881372942, −5.79170619811245771021539635940, −5.52779452876710739416908412711, −4.21023217846903590130260966782, −3.45150567927644784982454399148, −2.34081810493461459394888487292, −1.90178841952352067613315707185, −0.56020638836161763193599954516, 1.24474889543280377245687268840, 2.04759227458208773932243794723, 2.66439047583172196799141234196, 3.92270871603351107504592699145, 4.67209641071654296983720762384, 5.40669745777288526758644085900, 6.05818850538427299330315398198, 6.80552684618022429831428311193, 7.36629900974614057918586949678, 8.167916326802079465281468054700

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.336923913\)
\(L(\frac12)\)	\(\approx\)	\(2.336923913\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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