LMFDB - L-function 2-6048-24.11-c1-0-6

Dirichlet series

L(s) = 1

− 0.381·5^-s − i·7^-s + 2.13i·11^-s − 6.49i·13^-s − 6.97i·17^-s − 6.19·19^-s − 1.82·23^-s − 4.85·25^-s − 0.694·29^-s + 1.67i·31^-s + 0.381i·35^-s + 8.06i·37^-s + 1.31i·41^-s − 7.67·43^-s + 6.82·47^-s + ⋯

L(s) = 1

− 0.170·5^-s − 0.377i·7^-s + 0.643i·11^-s − 1.80i·13^-s − 1.69i·17^-s − 1.42·19^-s − 0.380·23^-s − 0.970·25^-s − 0.129·29^-s + 0.301i·31^-s + 0.0644i·35^-s + 1.32i·37^-s + 0.204i·41^-s − 1.17·43^-s + 0.995·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.482 - 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$-0.482 - 0.875i$
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:	$(2,\ 6048,\ (\ :1/2),\ -0.482 - 0.875i)$

Particular Values

$L(1)$	$\approx$	$0.3254716192$
$L(\frac12)$	$\approx$	$0.3254716192$
$L(\frac{3}{2})$		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 + 0.381T + 5T^{2} $
	11	$ 1 - 2.13iT - 11T^{2} $
	13	$ 1 + 6.49iT - 13T^{2} $
	17	$ 1 + 6.97iT - 17T^{2} $
	19	$ 1 + 6.19T + 19T^{2} $
	23	$ 1 + 1.82T + 23T^{2} $
	29	$ 1 + 0.694T + 29T^{2} $
	31	$ 1 - 1.67iT - 31T^{2} $
	37	$ 1 - 8.06iT - 37T^{2} $
	41	$ 1 - 1.31iT - 41T^{2} $
	43	$ 1 + 7.67T + 43T^{2} $
	47	$ 1 - 6.82T + 47T^{2} $
	53	$ 1 - 0.954T + 53T^{2} $
	59	$ 1 - 12.8iT - 59T^{2} $
	61	$ 1 - 12.4iT - 61T^{2} $
	67	$ 1 + 0.634T + 67T^{2} $
	71	$ 1 - 5.79T + 71T^{2} $
	73	$ 1 - 8.13T + 73T^{2} $
	79	$ 1 - 14.1iT - 79T^{2} $
	83	$ 1 + 3.36iT - 83T^{2} $
	89	$ 1 + 15.9iT - 89T^{2} $
	97	$ 1 - 10.8T + 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

−8.215167158658897391728773353267, −7.58290739918991035864162871688, −7.03515590620972252069266053190, −6.18568061182603640450932029334, −5.38577621038828299020065908761, −4.71915340432445668840027366022, −3.94335602495095979940465802277, −3.01873680010343787186163192561, −2.29704557327303454797600487326, −0.992862875220433885757262665387, 0.089036021360895998987872826137, 1.82939759112669168863148186484, 2.14957882588864389608731473706, 3.71217089031242948133520706833, 3.95079492138542271230321181503, 4.88666111096454457099713174590, 5.96646901245524098469143168681, 6.28832574371740939064522204136, 7.02253522807383789021281349043, 8.093514566469523719991890622201

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.3254716192\)
\(L(\frac12)\)	\(\approx\)	\(0.3254716192\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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