LMFDB - L-function 2-6048-24.11-c1-0-46

Dirichlet series

L(s) = 1

− 2.35·5^-s − i·7^-s − 1.09i·11^-s − 4.10i·13^-s + 4.74i·17^-s + 7.10·19^-s − 3.99·23^-s + 0.523·25^-s − 4.09·29^-s + 10.8i·31^-s + 2.35i·35^-s + 2.27i·37^-s + 1.29i·41^-s + 8.59·43^-s + 6.79·47^-s + ⋯

L(s) = 1

− 1.05·5^-s − 0.377i·7^-s − 0.328i·11^-s − 1.13i·13^-s + 1.15i·17^-s + 1.62·19^-s − 0.832·23^-s + 0.104·25^-s − 0.761·29^-s + 1.95i·31^-s + 0.397i·35^-s + 0.373i·37^-s + 0.201i·41^-s + 1.31·43^-s + 0.991·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$0.738 + 0.673i$
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:	$(2,\ 6048,\ (\ :1/2),\ 0.738 + 0.673i)$

Particular Values

$L(1)$	$\approx$	$1.322285160$
$L(\frac12)$	$\approx$	$1.322285160$
$L(\frac{3}{2})$		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 + 2.35T + 5T^{2} $
	11	$ 1 + 1.09iT - 11T^{2} $
	13	$ 1 + 4.10iT - 13T^{2} $
	17	$ 1 - 4.74iT - 17T^{2} $
	19	$ 1 - 7.10T + 19T^{2} $
	23	$ 1 + 3.99T + 23T^{2} $
	29	$ 1 + 4.09T + 29T^{2} $
	31	$ 1 - 10.8iT - 31T^{2} $
	37	$ 1 - 2.27iT - 37T^{2} $
	41	$ 1 - 1.29iT - 41T^{2} $
	43	$ 1 - 8.59T + 43T^{2} $
	47	$ 1 - 6.79T + 47T^{2} $
	53	$ 1 - 0.688T + 53T^{2} $
	59	$ 1 + 7.72iT - 59T^{2} $
	61	$ 1 + 9.70iT - 61T^{2} $
	67	$ 1 + 8.23T + 67T^{2} $
	71	$ 1 - 6.46T + 71T^{2} $
	73	$ 1 - 8.44T + 73T^{2} $
	79	$ 1 + 5.70iT - 79T^{2} $
	83	$ 1 - 10.0iT - 83T^{2} $
	89	$ 1 + 2.54iT - 89T^{2} $
	97	$ 1 + 15.6T + 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

−7.949934134446479422696814338029, −7.48720593883005848808280884178, −6.68208722457436300162046843789, −5.74429455379729574437481079400, −5.19729267713379743512240245137, −4.15797003540947517386724929495, −3.55647101179091243918600829094, −2.97376548193653044857026724585, −1.55129339441423012451648881709, −0.51884005113862071979435041433, 0.72231588585870146577850858501, 2.06106146989572748911984590793, 2.87013931214986180606832834728, 4.00187273609041343256605992156, 4.24359240559278725820011933317, 5.35032443847790980205911521870, 5.91785317740669948830660240622, 7.00642126003364570374944410644, 7.51021568420347135688112224746, 7.894538230290211852110445835109

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.322285160\)
\(L(\frac12)\)	\(\approx\)	\(1.322285160\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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