LMFDB - L-function 2-6048-12.11-c1-0-25

Dirichlet series

L(s) = 1

− 3.34i·5^-s − i·7^-s − 0.896·11^-s + 4·13^-s + 6.31i·17^-s + i·19^-s + 7.20·23^-s − 6.19·25^-s + 7.72i·29^-s + 3.73i·31^-s − 3.34·35^-s − 11.7·37^-s + 7.86i·41^-s + 7.46i·43^-s + 8.76·47^-s + ⋯

L(s) = 1

− 1.49i·5^-s − 0.377i·7^-s − 0.270·11^-s + 1.10·13^-s + 1.53i·17^-s + 0.229i·19^-s + 1.50·23^-s − 1.23·25^-s + 1.43i·29^-s + 0.670i·31^-s − 0.565·35^-s − 1.92·37^-s + 1.22i·41^-s + 1.13i·43^-s + 1.27·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$0.707 - 0.707i$
Analytic conductor:	$48.2935$
Root analytic conductor:	$6.94935$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6048} (2591, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$

Particular Values

$L(1)$	$\approx$	$1.547950357$
$L(\frac12)$	$\approx$	$1.547950357$
$L(\frac{3}{2})$		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 + 3.34iT - 5T^{2} $
	11	$ 1 + 0.896T + 11T^{2} $
	13	$ 1 - 4T + 13T^{2} $
	17	$ 1 - 6.31iT - 17T^{2} $
	19	$ 1 - iT - 19T^{2} $
	23	$ 1 - 7.20T + 23T^{2} $
	29	$ 1 - 7.72iT - 29T^{2} $
	31	$ 1 - 3.73iT - 31T^{2} $
	37	$ 1 + 11.7T + 37T^{2} $
	41	$ 1 - 7.86iT - 41T^{2} $
	43	$ 1 - 7.46iT - 43T^{2} $
	47	$ 1 - 8.76T + 47T^{2} $
	53	$ 1 - 2.82iT - 53T^{2} $
	59	$ 1 + 3.86T + 59T^{2} $
	61	$ 1 + 12.3T + 61T^{2} $
	67	$ 1 - 10.9iT - 67T^{2} $
	71	$ 1 - 5.79T + 71T^{2} $
	73	$ 1 + 14.3T + 73T^{2} $
	79	$ 1 + 7.46iT - 79T^{2} $
	83	$ 1 + 6.96T + 83T^{2} $
	89	$ 1 - 13.2iT - 89T^{2} $
	97	$ 1 - 2.92T + 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

−8.353575154844122393278062133500, −7.58477670247894055667596838513, −6.70695194077267369393908921154, −5.94158642394182080313751609991, −5.21411569479535282617968189953, −4.61491206370951779287718162364, −3.82107964610672629080455813401, −3.06421778394097865855517736340, −1.40236039253164226727185466096, −1.28024240140347935555932253455, 0.41146556034045590459938346929, 1.94634508671348713731728004051, 2.81883859585133722905702375271, 3.28695563499783577684956514838, 4.23581105479123405276433585109, 5.28251310554289529768107945348, 5.87849002355043217041885810103, 6.73552442376573283537729297557, 7.15823581941050439872568558551, 7.78999066568145179700250283907

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.547950357\)
\(L(\frac12)\)	\(\approx\)	\(1.547950357\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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