LMFDB - L-function 2-6048-8.5-c1-0-54

Dirichlet series

L(s) = 1

− 2.99i·5^-s − 7^-s + 2.13i·11^-s − 0.665i·13^-s + 7.04·17^-s − 1.39i·19^-s − 0.184·23^-s − 3.94·25^-s − 1.27i·29^-s + 7.62·31^-s + 2.99i·35^-s + 6.69i·37^-s − 0.274·41^-s + 2.63i·43^-s + 5.77·47^-s + ⋯

L(s) = 1

− 1.33i·5^-s − 0.377·7^-s + 0.645i·11^-s − 0.184i·13^-s + 1.70·17^-s − 0.320i·19^-s − 0.0384·23^-s − 0.788·25^-s − 0.236i·29^-s + 1.36·31^-s + 0.505i·35^-s + 1.10i·37^-s − 0.0428·41^-s + 0.402i·43^-s + 0.842·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$0.626 + 0.779i$
Analytic conductor:	$48.2935$
Root analytic conductor:	$6.94935$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6048} (3025, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6048,\ (\ :1/2),\ 0.626 + 0.779i)$

Particular Values

$L(1)$	$\approx$	$2.003475355$
$L(\frac12)$	$\approx$	$2.003475355$
$L(\frac{3}{2})$		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 + T $
good	5	$ 1 + 2.99iT - 5T^{2} $
	11	$ 1 - 2.13iT - 11T^{2} $
	13	$ 1 + 0.665iT - 13T^{2} $
	17	$ 1 - 7.04T + 17T^{2} $
	19	$ 1 + 1.39iT - 19T^{2} $
	23	$ 1 + 0.184T + 23T^{2} $
	29	$ 1 + 1.27iT - 29T^{2} $
	31	$ 1 - 7.62T + 31T^{2} $
	37	$ 1 - 6.69iT - 37T^{2} $
	41	$ 1 + 0.274T + 41T^{2} $
	43	$ 1 - 2.63iT - 43T^{2} $
	47	$ 1 - 5.77T + 47T^{2} $
	53	$ 1 - 7.48iT - 53T^{2} $
	59	$ 1 - 9.03iT - 59T^{2} $
	61	$ 1 + 9.80iT - 61T^{2} $
	67	$ 1 + 10.8iT - 67T^{2} $
	71	$ 1 + 7.93T + 71T^{2} $
	73	$ 1 + 2.80T + 73T^{2} $
	79	$ 1 + 7.98T + 79T^{2} $
	83	$ 1 + 6.76iT - 83T^{2} $
	89	$ 1 - 16.7T + 89T^{2} $
	97	$ 1 + 0.107T + 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

−7.86772380199606774665955110825, −7.54254077236962991865146388286, −6.42295138167600924574844427853, −5.81305045009247915992904050152, −4.93795749258125988581588212579, −4.56622415065818540931991330403, −3.54760409144485001829523903442, −2.69233524668410266219486408830, −1.46624948454731568505786896014, −0.72280886894750874773610417294, 0.839560169370459686868119857809, 2.16027212800779361971493136966, 3.12228601882092501471623785900, 3.43424464473231738454768200312, 4.42865428519699249337176031876, 5.68198411114730377628975319010, 5.92127920016920906394654063410, 6.89076540075556353923699180389, 7.30587151140404382308133176884, 8.103940090646370558721041318322

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.003475355\)
\(L(\frac12)\)	\(\approx\)	\(2.003475355\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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