LMFDB - L-function 2-6048-24.11-c1-0-93

Dirichlet series

L(s) = 1

− 0.162·5^-s − i·7^-s − 2.40i·11^-s − 4.76i·13^-s − 2.74i·17^-s + 1.86·19^-s − 1.32·23^-s − 4.97·25^-s − 3.96·29^-s + 8.01i·31^-s + 0.162i·35^-s − 6.09i·37^-s − 3.16i·41^-s + 1.70·43^-s − 12.1·47^-s + ⋯

L(s) = 1

− 0.0724·5^-s − 0.377i·7^-s − 0.724i·11^-s − 1.32i·13^-s − 0.665i·17^-s + 0.426·19^-s − 0.277·23^-s − 0.994·25^-s − 0.735·29^-s + 1.43i·31^-s + 0.0273i·35^-s − 1.00i·37^-s − 0.494i·41^-s + 0.260·43^-s − 1.77·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$-0.994 - 0.108i$
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:	$(2,\ 6048,\ (\ :1/2),\ -0.994 - 0.108i)$

Particular Values

$L(1)$	$\approx$	$0.5650731520$
$L(\frac12)$	$\approx$	$0.5650731520$
$L(\frac{3}{2})$		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 + 0.162T + 5T^{2} $
	11	$ 1 + 2.40iT - 11T^{2} $
	13	$ 1 + 4.76iT - 13T^{2} $
	17	$ 1 + 2.74iT - 17T^{2} $
	19	$ 1 - 1.86T + 19T^{2} $
	23	$ 1 + 1.32T + 23T^{2} $
	29	$ 1 + 3.96T + 29T^{2} $
	31	$ 1 - 8.01iT - 31T^{2} $
	37	$ 1 + 6.09iT - 37T^{2} $
	41	$ 1 + 3.16iT - 41T^{2} $
	43	$ 1 - 1.70T + 43T^{2} $
	47	$ 1 + 12.1T + 47T^{2} $
	53	$ 1 - 8.22T + 53T^{2} $
	59	$ 1 + 5.45iT - 59T^{2} $
	61	$ 1 + 4.28iT - 61T^{2} $
	67	$ 1 - 4.61T + 67T^{2} $
	71	$ 1 + 13.9T + 71T^{2} $
	73	$ 1 + 6.37T + 73T^{2} $
	79	$ 1 - 9.75iT - 79T^{2} $
	83	$ 1 - 16.4iT - 83T^{2} $
	89	$ 1 - 2.35iT - 89T^{2} $
	97	$ 1 - 14.4T + 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

−7.70807826885304009556524918736, −7.13524015399172033717695680868, −6.23657423966478219592796078047, −5.48772629901710437624762131248, −5.00340687002863745401813787983, −3.81059995086622059276105095622, −3.34100687572114034031108300446, −2.39346105864594494281275289366, −1.15892986908857978587009040857, −0.14572212169402424235894592357, 1.58445878275771928526532611161, 2.15408847156672865822450928861, 3.27535600568013267637577697051, 4.20565085947978584011092499310, 4.64500930872610302849800091837, 5.76819025108907843618684544379, 6.20438389078143442888373793075, 7.12467812198161735987913546695, 7.65142297663997960629279508377, 8.456657106822548441806062232282

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.5650731520\)
\(L(\frac12)\)	\(\approx\)	\(0.5650731520\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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