LMFDB - L-function 2-1280-40.29-c1-0-16

Dirichlet series

L(s) = 1

− 3.23·3^-s − 2.23i·5^-s + 0.763i·7^-s + 7.47·9^-s + 7.23i·15^-s − 2.47i·21^-s + 5.70i·23^-s − 5.00·25^-s − 14.4·27^-s + 6i·29^-s + 1.70·35^-s + 4.47·41^-s + 11.2·43^-s − 16.7i·45^-s − 13.7i·47^-s + ⋯

L(s) = 1

− 1.86·3^-s − 0.999i·5^-s + 0.288i·7^-s + 2.49·9^-s + 1.86i·15^-s − 0.539i·21^-s + 1.19i·23^-s − 1.00·25^-s − 2.78·27^-s + 1.11i·29^-s + 0.288·35^-s + 0.698·41^-s + 1.71·43^-s − 2.49i·45^-s − 1.99i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1280$ = $2^{8} \cdot 5$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$10.2208$
Root analytic conductor:	$3.19700$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1280} (129, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1280,\ (\ :1/2),\ 0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$0.7659180767$
$L(\frac12)$	$\approx$	$0.7659180767$
$L(\frac{3}{2})$		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 $
	5	$ 1 + 2.23iT $
good	3	$ 1 + 3.23T + 3T^{2} $
	7	$ 1 - 0.763iT - 7T^{2} $
	11	$ 1 - 11T^{2} $
	13	$ 1 + 13T^{2} $
	17	$ 1 - 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 - 5.70iT - 23T^{2} $
	29	$ 1 - 6iT - 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 + 37T^{2} $
	41	$ 1 - 4.47T + 41T^{2} $
	43	$ 1 - 11.2T + 43T^{2} $
	47	$ 1 + 13.7iT - 47T^{2} $
	53	$ 1 + 53T^{2} $
	59	$ 1 - 59T^{2} $
	61	$ 1 + 13.4iT - 61T^{2} $
	67	$ 1 + 8.18T + 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 - 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 - 17.7T + 83T^{2} $
	89	$ 1 - 6T + 89T^{2} $
	97	$ 1 - 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

−9.637771173776759655248199109867, −8.958395746535551150577455536959, −7.75606208813945452281563551238, −6.96323535993980912001408834720, −5.94927020370902942254644882972, −5.42167068402821246228966967119, −4.75840812015071141314183607097, −3.79899227438733933686813987936, −1.75874994722527744116461720097, −0.62899967962144220574872994114, 0.823694067411163150394965490612, 2.45782777237964079927509745783, 4.00863332317167002079233108185, 4.68744777663236629113160696692, 5.98366795293723770576581380274, 6.15152175551460638753145340114, 7.17238056832804135831044722659, 7.71142356036044901794464097212, 9.245652047565780460086391217321, 10.19457275378008959528434429184

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7659180767\)
\(L(\frac12)\)	\(\approx\)	\(0.7659180767\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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