LMFDB - L-function 2-800-5.4-c3-0-8

Dirichlet series

L(s) = 1

− 7.21i·3^-s + 7.21i·7^-s − 24.9·9^-s − 43.2·11^-s + 34i·13^-s − 114i·17^-s + 51.9·21^-s + 209. i·23^-s − 14.4i·27^-s + 26·29^-s − 100.·31^-s + 312i·33^-s + 150i·37^-s + 245.·39^-s + 342·41^-s + ⋯

L(s) = 1

− 1.38i·3^-s + 0.389i·7^-s − 0.925·9^-s − 1.18·11^-s + 0.725i·13^-s − 1.62i·17^-s + 0.540·21^-s + 1.89i·23^-s − 0.102i·27^-s + 0.166·29^-s − 0.584·31^-s + 1.64i·33^-s + 0.666i·37^-s + 1.00·39^-s + 1.30·41^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$800$ = $2^{5} \cdot 5^{2}$
Sign:	$0.894 - 0.447i$
Analytic conductor:	$47.2015$
Root analytic conductor:	$6.87033$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{800} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 800,\ (\ :3/2),\ 0.894 - 0.447i)$

Particular Values

$L(2)$	$\approx$	$1.194181137$
$L(\frac12)$	$\approx$	$1.194181137$
$L(\frac{5}{2})$		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 $
	5	$ 1 $
good	3	$ 1 + 7.21iT - 27T^{2} $
	7	$ 1 - 7.21iT - 343T^{2} $
	11	$ 1 + 43.2T + 1.33e3T^{2} $
	13	$ 1 - 34iT - 2.19e3T^{2} $
	17	$ 1 + 114iT - 4.91e3T^{2} $
	19	$ 1 + 6.85e3T^{2} $
	23	$ 1 - 209. iT - 1.21e4T^{2} $
	29	$ 1 - 26T + 2.43e4T^{2} $
	31	$ 1 + 100.T + 2.97e4T^{2} $
	37	$ 1 - 150iT - 5.06e4T^{2} $
	41	$ 1 - 342T + 6.89e4T^{2} $
	43	$ 1 - 454. iT - 7.95e4T^{2} $
	47	$ 1 - 584. iT - 1.03e5T^{2} $
	53	$ 1 + 262iT - 1.48e5T^{2} $
	59	$ 1 + 490.T + 2.05e5T^{2} $
	61	$ 1 + 262T + 2.26e5T^{2} $
	67	$ 1 + 497. iT - 3.00e5T^{2} $
	71	$ 1 - 1.05e3T + 3.57e5T^{2} $
	73	$ 1 - 682iT - 3.89e5T^{2} $
	79	$ 1 + 201.T + 4.93e5T^{2} $
	83	$ 1 + 151. iT - 5.71e5T^{2} $
	89	$ 1 - 630T + 7.04e5T^{2} $
	97	$ 1 - 966iT - 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

−9.721158504659701292002600309470, −9.108334762914652067321396786676, −7.74257722824944976889985450998, −7.61493553087425545292083958595, −6.56790798161080948142092437747, −5.66602121809587128829935309799, −4.73244565037062254835258461224, −3.05730898651425979489855334745, −2.19082098171038456967365281350, −1.03848011832632045960713317062, 0.35838593777487912805694782091, 2.33389482330051400440407788869, 3.54692558895123923305297859825, 4.30495715744782972816984553624, 5.22702807906520805963327621027, 6.03589780721043224458436720510, 7.35584611855702110395088324291, 8.317960656626223762930139324684, 8.974672039193081665437386434501, 10.18358261225445468691064223952

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.194181137\)
\(L(\frac12)\)	\(\approx\)	\(1.194181137\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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