LMFDB - L-function 2-6048-8.5-c1-0-15

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

$L(1)$	$\approx$	$1.076458196$
$L(\frac12)$	$\approx$	$1.076458196$
$L(\frac{3}{2})$		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

−8.375822944062922134788917511136, −7.45695703894193856053412920067, −6.77768581567085229313155690569, −6.00647252457115133187145465321, −5.24607741144970297110282251183, −4.33822088886532796865711107708, −4.20794448922653639888849555763, −2.77748397368066467123686594195, −1.92932394663155921621564397087, −0.894234483912562051179571043720, 0.32235762301354009281713925685, 1.89529099931313700663338453438, 2.86569521845771541064653433314, 3.24550969050668296566540516173, 4.28350392871463377282347002819, 5.06866641586695547547907603608, 6.15771819766520238973512245869, 6.62283803375149706464890016894, 6.96491110036188830044117060166, 8.070812519712291687336774521234

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.076458196\)
\(L(\frac12)\)	\(\approx\)	\(1.076458196\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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