LMFDB - L-function 2-12e3-8.5-c1-0-28

Dirichlet series

L(s) = 1

− 3.46i·5^-s − 1.73·7^-s − 6i·11^-s − 5.19i·13^-s + 6·17^-s + 5i·19^-s − 3.46·23^-s − 6.99·25^-s + 6.92i·29^-s − 3.46·31^-s + 5.99i·35^-s − 1.73i·37^-s + 4i·43^-s − 3.46·47^-s − 4·49^-s + ⋯

L(s) = 1

− 1.54i·5^-s − 0.654·7^-s − 1.80i·11^-s − 1.44i·13^-s + 1.45·17^-s + 1.14i·19^-s − 0.722·23^-s − 1.39·25^-s + 1.28i·29^-s − 0.622·31^-s + 1.01i·35^-s − 0.284i·37^-s + 0.609i·43^-s − 0.505·47^-s − 0.571·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$-0.965 + 0.258i$
Analytic conductor:	$13.7981$
Root analytic conductor:	$3.71458$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (865, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :1/2),\ -0.965 + 0.258i)$

Particular Values

$L(1)$	$\approx$	$1.130269718$
$L(\frac12)$	$\approx$	$1.130269718$
$L(\frac{3}{2})$		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 $
good	5	$ 1 + 3.46iT - 5T^{2} $
	7	$ 1 + 1.73T + 7T^{2} $
	11	$ 1 + 6iT - 11T^{2} $
	13	$ 1 + 5.19iT - 13T^{2} $
	17	$ 1 - 6T + 17T^{2} $
	19	$ 1 - 5iT - 19T^{2} $
	23	$ 1 + 3.46T + 23T^{2} $
	29	$ 1 - 6.92iT - 29T^{2} $
	31	$ 1 + 3.46T + 31T^{2} $
	37	$ 1 + 1.73iT - 37T^{2} $
	41	$ 1 + 41T^{2} $
	43	$ 1 - 4iT - 43T^{2} $
	47	$ 1 + 3.46T + 47T^{2} $
	53	$ 1 + 6.92iT - 53T^{2} $
	59	$ 1 - 6iT - 59T^{2} $
	61	$ 1 + 12.1iT - 61T^{2} $
	67	$ 1 + 5iT - 67T^{2} $
	71	$ 1 - 13.8T + 71T^{2} $
	73	$ 1 + 7T + 73T^{2} $
	79	$ 1 + 5.19T + 79T^{2} $
	83	$ 1 - 83T^{2} $
	89	$ 1 + 6T + 89T^{2} $
	97	$ 1 - 7T + 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

−8.841487991733294603771408084393, −8.170264591772674755971764090112, −7.78509924682066668540267328549, −6.23358761697295366910190316794, −5.60320509588613989111622321803, −5.13612961535080512695514452825, −3.64358237834676755224027042162, −3.23441629224107350828913679595, −1.37896630562254092107009099997, −0.43822075308657445277402382742, 1.92150545283313508485068719855, 2.73154157921338314643139219174, 3.76913434873480545149505351272, 4.58970158480119669996468861125, 5.86403618205160996918257304332, 6.74370623169598198460332529628, 7.10206238315820967406578431553, 7.80600089835819909304314900649, 9.172799470875636972687369660761, 9.895110927879994285943569027535

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.130269718\)
\(L(\frac12)\)	\(\approx\)	\(1.130269718\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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