LMFDB - L-function 2-1440-5.4-c3-0-35

Dirichlet series

L(s) = 1

+ (−8.58 + 7.16i)5^-s + 28.3i·7^-s − 65.2·11^-s − 33.6i·13^-s + 73.3i·17^-s − 134.·19^-s − 14.7i·23^-s + (22.3 − 122. i)25^-s − 224.·29^-s + 68.8·31^-s + (−202. − 243. i)35^-s + 196. i·37^-s + 143.·41^-s − 15.0i·43^-s − 134. i·47^-s + ⋯

L(s) = 1

+ (−0.767 + 0.640i)5^-s + 1.52i·7^-s − 1.78·11^-s − 0.718i·13^-s + 1.04i·17^-s − 1.61·19^-s − 0.133i·23^-s + (0.178 − 0.983i)25^-s − 1.43·29^-s + 0.398·31^-s + (−0.979 − 1.17i)35^-s + 0.872i·37^-s + 0.545·41^-s − 0.0534i·43^-s − 0.417i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1440$ = $2^{5} \cdot 3^{2} \cdot 5$
Sign:	$0.640 + 0.767i$
Analytic conductor:	$84.9627$
Root analytic conductor:	$9.21752$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1440} (289, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1440,\ (\ :3/2),\ 0.640 + 0.767i)$

Particular Values

$L(2)$	$\approx$	$0.2133027140$
$L(\frac12)$	$\approx$	$0.2133027140$
$L(\frac{5}{2})$		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 $
	5	$ 1 + (8.58 - 7.16i)T $
good	7	$ 1 - 28.3iT - 343T^{2} $
	11	$ 1 + 65.2T + 1.33e3T^{2} $
	13	$ 1 + 33.6iT - 2.19e3T^{2} $
	17	$ 1 - 73.3iT - 4.91e3T^{2} $
	19	$ 1 + 134.T + 6.85e3T^{2} $
	23	$ 1 + 14.7iT - 1.21e4T^{2} $
	29	$ 1 + 224.T + 2.43e4T^{2} $
	31	$ 1 - 68.8T + 2.97e4T^{2} $
	37	$ 1 - 196. iT - 5.06e4T^{2} $
	41	$ 1 - 143.T + 6.89e4T^{2} $
	43	$ 1 + 15.0iT - 7.95e4T^{2} $
	47	$ 1 + 134. iT - 1.03e5T^{2} $
	53	$ 1 + 262. iT - 1.48e5T^{2} $
	59	$ 1 + 119.T + 2.05e5T^{2} $
	61	$ 1 - 16.5T + 2.26e5T^{2} $
	67	$ 1 - 545. iT - 3.00e5T^{2} $
	71	$ 1 - 199.T + 3.57e5T^{2} $
	73	$ 1 - 43.2iT - 3.89e5T^{2} $
	79	$ 1 + 438.T + 4.93e5T^{2} $
	83	$ 1 - 1.22e3iT - 5.71e5T^{2} $
	89	$ 1 - 723.T + 7.04e5T^{2} $
	97	$ 1 - 1.13e3iT - 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

−8.774501743158948927463335103034, −8.217432859473693569382829346952, −7.69361352531403975220095544580, −6.48829352191198104696526328857, −5.75875672533679156260793192356, −4.98340555476351959143917376105, −3.79600812882450303422918906820, −2.73755814472205670228263065151, −2.17281117315157023519357208012, −0.07780191894708539126008273004, 0.59567698692088710112109830710, 2.05191732384790960838506587604, 3.35921251951596533814274831958, 4.36036256071345604689355814008, 4.77911645237154714207633435991, 5.95205360793739278971880172566, 7.31592024869474992954608052595, 7.43914309409567799129160327710, 8.358712787118157551001533682965, 9.233032312644463213680426203523

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.2133027140\)
\(L(\frac12)\)	\(\approx\)	\(0.2133027140\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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