LMFDB - L-function 2-6048-24.11-c1-0-27

Dirichlet series

L(s) = 1

− 1.27·5^-s + i·7^-s + 6.57i·11^-s + 4.05i·13^-s + 6.09i·17^-s + 2.08·19^-s + 6.85·23^-s − 3.37·25^-s + 6.53·29^-s + 3.26i·31^-s − 1.27i·35^-s + 2.95i·37^-s − 3.35i·41^-s − 10.9·43^-s + 7.12·47^-s + ⋯

L(s) = 1

− 0.570·5^-s + 0.377i·7^-s + 1.98i·11^-s + 1.12i·13^-s + 1.47i·17^-s + 0.477·19^-s + 1.42·23^-s − 0.674·25^-s + 1.21·29^-s + 0.587i·31^-s − 0.215i·35^-s + 0.485i·37^-s − 0.523i·41^-s − 1.67·43^-s + 1.03·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$-0.888 - 0.458i$
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:	$(2,\ 6048,\ (\ :1/2),\ -0.888 - 0.458i)$

Particular Values

$L(1)$	$\approx$	$1.421336273$
$L(\frac12)$	$\approx$	$1.421336273$
$L(\frac{3}{2})$		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 + 1.27T + 5T^{2} $
	11	$ 1 - 6.57iT - 11T^{2} $
	13	$ 1 - 4.05iT - 13T^{2} $
	17	$ 1 - 6.09iT - 17T^{2} $
	19	$ 1 - 2.08T + 19T^{2} $
	23	$ 1 - 6.85T + 23T^{2} $
	29	$ 1 - 6.53T + 29T^{2} $
	31	$ 1 - 3.26iT - 31T^{2} $
	37	$ 1 - 2.95iT - 37T^{2} $
	41	$ 1 + 3.35iT - 41T^{2} $
	43	$ 1 + 10.9T + 43T^{2} $
	47	$ 1 - 7.12T + 47T^{2} $
	53	$ 1 + 2.87T + 53T^{2} $
	59	$ 1 + 7.75iT - 59T^{2} $
	61	$ 1 - 12.0iT - 61T^{2} $
	67	$ 1 - 3.01T + 67T^{2} $
	71	$ 1 - 3.48T + 71T^{2} $
	73	$ 1 - 2.76T + 73T^{2} $
	79	$ 1 + 0.849iT - 79T^{2} $
	83	$ 1 + 15.8iT - 83T^{2} $
	89	$ 1 - 4.06iT - 89T^{2} $
	97	$ 1 + 4.90T + 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

−8.399809910227191654204032453099, −7.60002275663721084625803341203, −6.91825070842223244587053212304, −6.51145054819298568469842484584, −5.38449782755665959330113029090, −4.63726992818476901876558715144, −4.14832837623549770695583833948, −3.20173810945897665328463085952, −2.11411613722860564928786729116, −1.43419478822141387370926786388, 0.43956092862542266239444849789, 0.990808915414971665643945458747, 2.80164062031246244868119022231, 3.15576938080059727742820187560, 3.96692489634411990921589066039, 5.06588363946822699002741032600, 5.47707655336834624195138372751, 6.42004621423627613080285786946, 7.10290082147748824918382084518, 7.960927074092118967494214354004

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.421336273\)
\(L(\frac12)\)	\(\approx\)	\(1.421336273\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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