LMFDB - L-function 2-6048-24.11-c1-0-58

Dirichlet series

L(s) = 1

− 2.09·5^-s + i·7^-s + 0.961i·11^-s + 4.60i·13^-s + 3.28i·17^-s − 3.66·19^-s + 2.40·23^-s − 0.618·25^-s − 9.31·29^-s − 1.84i·31^-s − 2.09i·35^-s + 2.29i·37^-s − 0.314i·41^-s + 8.64·43^-s + 2.34·47^-s + ⋯

L(s) = 1

− 0.936·5^-s + 0.377i·7^-s + 0.289i·11^-s + 1.27i·13^-s + 0.795i·17^-s − 0.841·19^-s + 0.501·23^-s − 0.123·25^-s − 1.73·29^-s − 0.332i·31^-s − 0.353i·35^-s + 0.377i·37^-s − 0.0491i·41^-s + 1.31·43^-s + 0.341·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$-0.562 + 0.826i$
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:	$(2,\ 6048,\ (\ :1/2),\ -0.562 + 0.826i)$

Particular Values

$L(1)$	$\approx$	$0.05045854468$
$L(\frac12)$	$\approx$	$0.05045854468$
$L(\frac{3}{2})$		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 + 2.09T + 5T^{2} $
	11	$ 1 - 0.961iT - 11T^{2} $
	13	$ 1 - 4.60iT - 13T^{2} $
	17	$ 1 - 3.28iT - 17T^{2} $
	19	$ 1 + 3.66T + 19T^{2} $
	23	$ 1 - 2.40T + 23T^{2} $
	29	$ 1 + 9.31T + 29T^{2} $
	31	$ 1 + 1.84iT - 31T^{2} $
	37	$ 1 - 2.29iT - 37T^{2} $
	41	$ 1 + 0.314iT - 41T^{2} $
	43	$ 1 - 8.64T + 43T^{2} $
	47	$ 1 - 2.34T + 47T^{2} $
	53	$ 1 + 7.73T + 53T^{2} $
	59	$ 1 + 7.99iT - 59T^{2} $
	61	$ 1 - 10.6iT - 61T^{2} $
	67	$ 1 + 4.12T + 67T^{2} $
	71	$ 1 + 6.79T + 71T^{2} $
	73	$ 1 - 16.0T + 73T^{2} $
	79	$ 1 + 1.26iT - 79T^{2} $
	83	$ 1 - 2.99iT - 83T^{2} $
	89	$ 1 + 12.8iT - 89T^{2} $
	97	$ 1 - 4.98T + 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

−7.74280938952273675694757483772, −7.25874262817603995743416596287, −6.41268430222454003623898022183, −5.80036490252801530133971081740, −4.75533577970390840817355011051, −4.12212175220871405955403408337, −3.57314619817223057054010462066, −2.37293023103861520232011772982, −1.60213959991849731375978425502, −0.01567717841978473363710566037, 0.890233237215982089063002958698, 2.27968491216751601530798235296, 3.26391853765694894273534718258, 3.81951495600073655318089110244, 4.65964494283675564962531982481, 5.43950671821804569219644569199, 6.16562875231066709764205002569, 7.13203695544260913028983407339, 7.64956095291694491639816464261, 8.110866495327526567762285970120

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.05045854468\)
\(L(\frac12)\)	\(\approx\)	\(0.05045854468\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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