LMFDB - L-function 2-6336-12.11-c1-0-60

Dirichlet series

L(s) = 1

− 2.05i·5^-s + 4.18i·7^-s + 11^-s − 3.27·13^-s − 0.332i·17^-s − 8.47i·19^-s − 3.52·23^-s + 0.777·25^-s + 9.52i·29^-s − 3.95i·31^-s + 8.60·35^-s + 1.37·37^-s + 5.42i·41^-s + 1.53i·43^-s + 6.93·47^-s + ⋯

L(s) = 1

− 0.918i·5^-s + 1.58i·7^-s + 0.301·11^-s − 0.908·13^-s − 0.0806i·17^-s − 1.94i·19^-s − 0.734·23^-s + 0.155·25^-s + 1.76i·29^-s − 0.711i·31^-s + 1.45·35^-s + 0.226·37^-s + 0.847i·41^-s + 0.234i·43^-s + 1.01·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6336$ = $2^{6} \cdot 3^{2} \cdot 11$
Sign:	$-0.577 + 0.816i$
Analytic conductor:	$50.5932$
Root analytic conductor:	$7.11289$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6336} (3455, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6336,\ (\ :1/2),\ -0.577 + 0.816i)$

Particular Values

$L(1)$	$\approx$	$0.8170046961$
$L(\frac12)$	$\approx$	$0.8170046961$
$L(\frac{3}{2})$		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 $
	11	$ 1 - T $
good	5	$ 1 + 2.05iT - 5T^{2} $
	7	$ 1 - 4.18iT - 7T^{2} $
	13	$ 1 + 3.27T + 13T^{2} $
	17	$ 1 + 0.332iT - 17T^{2} $
	19	$ 1 + 8.47iT - 19T^{2} $
	23	$ 1 + 3.52T + 23T^{2} $
	29	$ 1 - 9.52iT - 29T^{2} $
	31	$ 1 + 3.95iT - 31T^{2} $
	37	$ 1 - 1.37T + 37T^{2} $
	41	$ 1 - 5.42iT - 41T^{2} $
	43	$ 1 - 1.53iT - 43T^{2} $
	47	$ 1 - 6.93T + 47T^{2} $
	53	$ 1 + 8.83iT - 53T^{2} $
	59	$ 1 - 3.70T + 59T^{2} $
	61	$ 1 + 14.8T + 61T^{2} $
	67	$ 1 + 6.32iT - 67T^{2} $
	71	$ 1 + 5.77T + 71T^{2} $
	73	$ 1 + 8.72T + 73T^{2} $
	79	$ 1 + 13.2iT - 79T^{2} $
	83	$ 1 + 0.984T + 83T^{2} $
	89	$ 1 + 1.46iT - 89T^{2} $
	97	$ 1 - 5.99T + 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

−7.893223821203731923226346055063, −7.07391842224974353997738227346, −6.31279236917589878086386597714, −5.51144385376860003158303444379, −4.93196487748131279353022380195, −4.45934207753903246962745447325, −3.09650459707162994986109349563, −2.48683015954762650627320912012, −1.54404296213257329214461444295, −0.21191329553960134136275002167, 1.12334622928603039487948193596, 2.19094046689362566703588126988, 3.16339181617819992743394390393, 4.02707399243959785958606044935, 4.32213568406200701489669286449, 5.59419742137317257122887898231, 6.25710594220819640342635289978, 6.97435606392473650354764890260, 7.59548701173519781242022077678, 7.905137973081746021351185737802

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8170046961\)
\(L(\frac12)\)	\(\approx\)	\(0.8170046961\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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