LMFDB - L-function 2-1280-8.3-c2-0-14

Dirichlet series

L(s) = 1

− 2.35·3^-s + 2.23i·5^-s − 5.25i·7^-s − 3.47·9^-s − 19.9·11^-s − 8.47i·13^-s − 5.25i·15^-s + 11.8·17^-s − 15.2·19^-s + 12.3i·21^-s − 0.555i·23^-s − 5.00·25^-s + 29.3·27^-s − 10.9i·29^-s + 8.29i·31^-s + ⋯

L(s) = 1

− 0.783·3^-s + 0.447i·5^-s − 0.751i·7^-s − 0.385·9^-s − 1.81·11^-s − 0.651i·13^-s − 0.350i·15^-s + 0.699·17^-s − 0.800·19^-s + 0.588i·21^-s − 0.0241i·23^-s − 0.200·25^-s + 1.08·27^-s − 0.377i·29^-s + 0.267i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]

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

Invariants

Degree:	$2$
Conductor:	$1280$ = $2^{8} \cdot 5$
Sign:	$0.707 - 0.707i$
Analytic conductor:	$34.8774$
Root analytic conductor:	$5.90571$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1280} (1151, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1280,\ (\ :1),\ 0.707 - 0.707i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.7061956250$
$L(\frac12)$	$\approx$	$0.7061956250$
$L(2)$		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 $
	5	$ 1 - 2.23iT $
good	3	$ 1 + 2.35T + 9T^{2} $
	7	$ 1 + 5.25iT - 49T^{2} $
	11	$ 1 + 19.9T + 121T^{2} $
	13	$ 1 + 8.47iT - 169T^{2} $
	17	$ 1 - 11.8T + 289T^{2} $
	19	$ 1 + 15.2T + 361T^{2} $
	23	$ 1 + 0.555iT - 529T^{2} $
	29	$ 1 + 10.9iT - 841T^{2} $
	31	$ 1 - 8.29iT - 961T^{2} $
	37	$ 1 - 18.3iT - 1.36e3T^{2} $
	41	$ 1 - 14.5T + 1.68e3T^{2} $
	43	$ 1 + 22.2T + 1.84e3T^{2} $
	47	$ 1 + 53.3iT - 2.20e3T^{2} $
	53	$ 1 - 66.3iT - 2.80e3T^{2} $
	59	$ 1 + 17.4T + 3.48e3T^{2} $
	61	$ 1 - 90.1iT - 3.72e3T^{2} $
	67	$ 1 - 50.2T + 4.48e3T^{2} $
	71	$ 1 + 80.7iT - 5.04e3T^{2} $
	73	$ 1 - 5.55T + 5.32e3T^{2} $
	79	$ 1 - 13.8iT - 6.24e3T^{2} $
	83	$ 1 + 76.2T + 6.88e3T^{2} $
	89	$ 1 - 111.T + 7.92e3T^{2} $
	97	$ 1 + 92.8T + 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

−9.998403240240747111394495012900, −8.591038073550839543936095976591, −7.85189273721787369248082642018, −7.16971297502448948930971385531, −6.11254702985665826326832410806, −5.46465121438748384550632588620, −4.64473376554256242256283661402, −3.35336970125939363198900325535, −2.45067407774836597713109783579, −0.67311497935157409497180207184, 0.35921341730030584926211462065, 2.04589752746125652668820769505, 3.03857918352473653928909796275, 4.50249562869545873611153801891, 5.35596205930893543186809233924, 5.75705241002834577730211059312, 6.77184790408668683262846504398, 7.920553316836412601312316618718, 8.474521740596217388795082584533, 9.388910394053087635767176925280

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.7061956250\)
\(L(\frac12)\)	\(\approx\)	\(0.7061956250\)
\(L(2)\)		not available
\(L(1)\)		not available

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