LMFDB - L-function 2-2736-57.56-c1-0-6

Dirichlet series

L(s) = 1

− 1.41i·5^-s − 2·7^-s − 4.24i·11^-s + 2.82i·13^-s + 7.07i·17^-s + (1 + 4.24i)19^-s − 1.41i·23^-s + 2.99·25^-s − 10·29^-s − 2.82i·31^-s + 2.82i·35^-s + 5.65i·37^-s + 10·41^-s − 12·43^-s + 1.41i·47^-s + ⋯

L(s) = 1

− 0.632i·5^-s − 0.755·7^-s − 1.27i·11^-s + 0.784i·13^-s + 1.71i·17^-s + (0.229 + 0.973i)19^-s − 0.294i·23^-s + 0.599·25^-s − 1.85·29^-s − 0.508i·31^-s + 0.478i·35^-s + 0.929i·37^-s + 1.56·41^-s − 1.82·43^-s + 0.206i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$0.662 - 0.749i$
Analytic conductor:	$21.8470$
Root analytic conductor:	$4.67408$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2736} (1025, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2736,\ (\ :1/2),\ 0.662 - 0.749i)$

Particular Values

$L(1)$	$\approx$	$1.280665781$
$L(\frac12)$	$\approx$	$1.280665781$
$L(\frac{3}{2})$		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 $
	19	$ 1 + (-1 - 4.24i)T $
good	5	$ 1 + 1.41iT - 5T^{2} $
	7	$ 1 + 2T + 7T^{2} $
	11	$ 1 + 4.24iT - 11T^{2} $
	13	$ 1 - 2.82iT - 13T^{2} $
	17	$ 1 - 7.07iT - 17T^{2} $
	23	$ 1 + 1.41iT - 23T^{2} $
	29	$ 1 + 10T + 29T^{2} $
	31	$ 1 + 2.82iT - 31T^{2} $
	37	$ 1 - 5.65iT - 37T^{2} $
	41	$ 1 - 10T + 41T^{2} $
	43	$ 1 + 12T + 43T^{2} $
	47	$ 1 - 1.41iT - 47T^{2} $
	53	$ 1 - 10T + 53T^{2} $
	59	$ 1 - 12T + 59T^{2} $
	61	$ 1 - 8T + 61T^{2} $
	67	$ 1 - 14.1iT - 67T^{2} $
	71	$ 1 - 8T + 71T^{2} $
	73	$ 1 - 6T + 73T^{2} $
	79	$ 1 + 11.3iT - 79T^{2} $
	83	$ 1 + 4.24iT - 83T^{2} $
	89	$ 1 - 6T + 89T^{2} $
	97	$ 1 - 8.48iT - 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

−8.704162875101944710790112934832, −8.453744924674234598200545653154, −7.47618608457496133074704355911, −6.40831483395080228282414480273, −5.97859252613812734879612858794, −5.14355431780296368447505403770, −3.89347823598642255882414816154, −3.56230926850487348522793878462, −2.16636562265307172176770250809, −1.02033610835468270570295208608, 0.48466762490716200556038009173, 2.19305119089552562109018607698, 2.95036380937654436880837791093, 3.78551198529710502008390937481, 4.96257444134477495286624829039, 5.49906396392519123990776075042, 6.79080499329829796541042616820, 7.04931526000035031761977189596, 7.71865709717719150320283949744, 8.906418544850366507911922198744

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.280665781\)
\(L(\frac12)\)	\(\approx\)	\(1.280665781\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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