LMFDB - L-function 2-800-5.4-c3-0-20

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

$L(2)$	$\approx$	$2.156771044$
$L(\frac12)$	$\approx$	$2.156771044$
$L(\frac{5}{2})$		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

−9.670449010060725288843512602760, −8.830362606817138514055992778730, −7.892122030949732304939883681648, −7.36540599481791095388946506822, −6.11552646932547076415093125880, −5.95350985926991549707500484809, −4.28815604580371402362190468848, −3.10299030172198125957693878967, −1.78697655525800948989094874240, −1.04470883928351224473879465820, 0.71381845431506083813645043350, 2.50245042212019882946339593310, 3.78768803799983053548539296967, 4.39149373134623777934327201002, 5.19059664545849985676010933386, 6.50803207029225268712066845463, 7.19590224590867659844842486979, 8.695581684043758028730497626857, 9.099331960591596692209773161049, 10.01711501870987653707870779997

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(2.156771044\)
\(L(\frac12)\)	\(\approx\)	\(2.156771044\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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