LMFDB - L-function 2-24e2-24.5-c2-0-5

Dirichlet series

L(s) = 1

− 7.34·5^-s − 10.3·7^-s + 8.48·11^-s − 10.3i·13^-s + 21.2i·17^-s − 20i·19^-s + 14.6i·23^-s + 29·25^-s + 36.7·29^-s + 51.9·31^-s + 76.3·35^-s + 41.5i·37^-s + 72.1i·41^-s − 40i·43^-s − 73.4i·47^-s + ⋯

L(s) = 1

− 1.46·5^-s − 1.48·7^-s + 0.771·11^-s − 0.799i·13^-s + 1.24i·17^-s − 1.05i·19^-s + 0.638i·23^-s + 1.15·25^-s + 1.26·29^-s + 1.67·31^-s + 2.18·35^-s + 1.12i·37^-s + 1.75i·41^-s − 0.930i·43^-s − 1.56i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$576$ = $2^{6} \cdot 3^{2}$
Sign:	$0.985 - 0.169i$
Analytic conductor:	$15.6948$
Root analytic conductor:	$3.96167$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{576} (161, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 576,\ (\ :1),\ 0.985 - 0.169i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.9494232706$
$L(\frac12)$	$\approx$	$0.9494232706$
$L(2)$		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 + 7.34T + 25T^{2} $
	7	$ 1 + 10.3T + 49T^{2} $
	11	$ 1 - 8.48T + 121T^{2} $
	13	$ 1 + 10.3iT - 169T^{2} $
	17	$ 1 - 21.2iT - 289T^{2} $
	19	$ 1 + 20iT - 361T^{2} $
	23	$ 1 - 14.6iT - 529T^{2} $
	29	$ 1 - 36.7T + 841T^{2} $
	31	$ 1 - 51.9T + 961T^{2} $
	37	$ 1 - 41.5iT - 1.36e3T^{2} $
	41	$ 1 - 72.1iT - 1.68e3T^{2} $
	43	$ 1 + 40iT - 1.84e3T^{2} $
	47	$ 1 + 73.4iT - 2.20e3T^{2} $
	53	$ 1 + 36.7T + 2.80e3T^{2} $
	59	$ 1 - 33.9T + 3.48e3T^{2} $
	61	$ 1 - 3.72e3T^{2} $
	67	$ 1 + 100iT - 4.48e3T^{2} $
	71	$ 1 - 73.4iT - 5.04e3T^{2} $
	73	$ 1 - 20T + 5.32e3T^{2} $
	79	$ 1 + 51.9T + 6.24e3T^{2} $
	83	$ 1 - 127.T + 6.88e3T^{2} $
	89	$ 1 - 12.7iT - 7.92e3T^{2} $
	97	$ 1 - 40T + 9.40e3T^{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

−10.47662194123569167819652120521, −9.755409483273306391992992072206, −8.628144432454986274144391981554, −7.987089538928343177652950189619, −6.81646880734088137261264225559, −6.28524833503062817616224271470, −4.72606239938324589286425987796, −3.68034939044616539051347417974, −3.01038932760130879744590799196, −0.71546652267543075100127325975, 0.62889199641935170677189688218, 2.82868782579970734328022415390, 3.79691655759987322606253498045, 4.56342282991477425423819386117, 6.21671302282777782049054096237, 6.86474108636536514547359867455, 7.75528284489210313312170945447, 8.799723988782662373714518593229, 9.550688968796974416534195933696, 10.46557733850247897435079715567

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

\(L(\frac{3}{2})\)	\(\approx\)	\(0.9494232706\)
\(L(\frac12)\)	\(\approx\)	\(0.9494232706\)
\(L(2)\)		not available
\(L(1)\)		not available

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