LMFDB - L-function 2-336-28.27-c3-0-23

Dirichlet series

L(s) = 1

− 3·3^-s − 20.9i·5^-s + (−17.6 + 5.46i)7^-s + 9·9^-s − 23.8i·11^-s − 74.6i·13^-s + 62.7i·15^-s + 68.6i·17^-s − 26.6·19^-s + (53.0 − 16.4i)21^-s + 74.6i·23^-s − 313.·25^-s − 27·27^-s + 128.·29^-s + 212.·31^-s + ⋯

L(s) = 1

− 0.577·3^-s − 1.87i·5^-s + (−0.955 + 0.295i)7^-s + 0.333·9^-s − 0.653i·11^-s − 1.59i·13^-s + 1.08i·15^-s + 0.979i·17^-s − 0.321·19^-s + (0.551 − 0.170i)21^-s + 0.677i·23^-s − 2.50·25^-s − 0.192·27^-s + 0.822·29^-s + 1.22·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$336$ = $2^{4} \cdot 3 \cdot 7$
Sign:	$-0.733 - 0.679i$
Analytic conductor:	$19.8246$
Root analytic conductor:	$4.45248$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{336} (223, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 336,\ (\ :3/2),\ -0.733 - 0.679i)$

Particular Values

$L(2)$	$\approx$	$0.3097686574$
$L(\frac12)$	$\approx$	$0.3097686574$
$L(\frac{5}{2})$		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 + 3T $
	7	$ 1 + (17.6 - 5.46i)T $
good	5	$ 1 + 20.9iT - 125T^{2} $
	11	$ 1 + 23.8iT - 1.33e3T^{2} $
	13	$ 1 + 74.6iT - 2.19e3T^{2} $
	17	$ 1 - 68.6iT - 4.91e3T^{2} $
	19	$ 1 + 26.6T + 6.85e3T^{2} $
	23	$ 1 - 74.6iT - 1.21e4T^{2} $
	29	$ 1 - 128.T + 2.43e4T^{2} $
	31	$ 1 - 212.T + 2.97e4T^{2} $
	37	$ 1 + 329.T + 5.06e4T^{2} $
	41	$ 1 - 182. iT - 6.89e4T^{2} $
	43	$ 1 - 260. iT - 7.95e4T^{2} $
	47	$ 1 + 401.T + 1.03e5T^{2} $
	53	$ 1 + 76.7T + 1.48e5T^{2} $
	59	$ 1 + 901.T + 2.05e5T^{2} $
	61	$ 1 + 271. iT - 2.26e5T^{2} $
	67	$ 1 - 499. iT - 3.00e5T^{2} $
	71	$ 1 - 299. iT - 3.57e5T^{2} $
	73	$ 1 - 452. iT - 3.89e5T^{2} $
	79	$ 1 + 347. iT - 4.93e5T^{2} $
	83	$ 1 - 775.T + 5.71e5T^{2} $
	89	$ 1 - 48.7iT - 7.04e5T^{2} $
	97	$ 1 + 1.05e3iT - 9.12e5T^{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

−10.42289792774492653530613226212, −9.654575118779189358435029299498, −8.578940957149999053735731818669, −8.016354655256485847487123985898, −6.28114240315378803919426590599, −5.59792042540338060314688946989, −4.67055301435595551250045506519, −3.30316278952309992712604465797, −1.23966943708225325965871612758, −0.12758573228161346390308600919, 2.23811778346898725610760873930, 3.40911008751076188453989196978, 4.62463350674462104232151641453, 6.42627824969000761327152062070, 6.64639720833638875514870511781, 7.44948623623674897598299923785, 9.200199908267379383251060053355, 10.12661029926269211817102781907, 10.63634585688051439673419426233, 11.66955766036395376612647633707

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 $\F_{q}$
Belyi maps

\(L(2)\)	\(\approx\)	\(0.3097686574\)
\(L(\frac12)\)	\(\approx\)	\(0.3097686574\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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