LMFDB - L-function 2-800-40.29-c3-0-32

Dirichlet series

L(s) = 1

− 5.29·3^-s − 8i·7^-s + 1.00·9^-s − 15.8i·11^-s + 52.9·13^-s + 14i·17^-s + 37.0i·19^-s + 42.3i·21^-s + 152i·23^-s + 137.·27^-s − 158. i·29^-s − 224·31^-s + 84.0i·33^-s − 243.·37^-s − 280·39^-s + ⋯

L(s) = 1

− 1.01·3^-s − 0.431i·7^-s + 0.0370·9^-s − 0.435i·11^-s + 1.12·13^-s + 0.199i·17^-s + 0.447i·19^-s + 0.439i·21^-s + 1.37i·23^-s + 0.980·27^-s − 1.01i·29^-s − 1.29·31^-s + 0.443i·33^-s − 1.08·37^-s − 1.14·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$800$ = $2^{5} \cdot 5^{2}$
Sign:	$0.102 + 0.994i$
Analytic conductor:	$47.2015$
Root analytic conductor:	$6.87033$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{800} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 800,\ (\ :3/2),\ 0.102 + 0.994i)$

Particular Values

$L(2)$	$\approx$	$0.9496272166$
$L(\frac12)$	$\approx$	$0.9496272166$
$L(\frac{5}{2})$		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 + 5.29T + 27T^{2} $
	7	$ 1 + 8iT - 343T^{2} $
	11	$ 1 + 15.8iT - 1.33e3T^{2} $
	13	$ 1 - 52.9T + 2.19e3T^{2} $
	17	$ 1 - 14iT - 4.91e3T^{2} $
	19	$ 1 - 37.0iT - 6.85e3T^{2} $
	23	$ 1 - 152iT - 1.21e4T^{2} $
	29	$ 1 + 158. iT - 2.43e4T^{2} $
	31	$ 1 + 224T + 2.97e4T^{2} $
	37	$ 1 + 243.T + 5.06e4T^{2} $
	41	$ 1 + 70T + 6.89e4T^{2} $
	43	$ 1 - 439.T + 7.95e4T^{2} $
	47	$ 1 - 336iT - 1.03e5T^{2} $
	53	$ 1 - 31.7T + 1.48e5T^{2} $
	59	$ 1 + 534. iT - 2.05e5T^{2} $
	61	$ 1 + 95.2iT - 2.26e5T^{2} $
	67	$ 1 - 174.T + 3.00e5T^{2} $
	71	$ 1 - 72T + 3.57e5T^{2} $
	73	$ 1 + 294iT - 3.89e5T^{2} $
	79	$ 1 + 464T + 4.93e5T^{2} $
	83	$ 1 - 545.T + 5.71e5T^{2} $
	89	$ 1 + 266T + 7.04e5T^{2} $
	97	$ 1 + 994iT - 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

−9.798512971834631815591503148829, −8.838026309335254388000170037793, −7.921021267897717473831415791697, −6.95077472833546796839622286443, −5.91045455918512433975217632639, −5.55816532628798091969986071775, −4.21271568023846572187135029494, −3.30796262804687754249537137948, −1.56991285209321257267586183809, −0.38064227888644822913381425094, 0.926054461713609691854858707721, 2.39041388130288991323458643593, 3.74250904508244024707706308840, 4.92819707693045636223678365886, 5.64973021155916286298368602451, 6.47939273986663838402271312475, 7.27033039826661768108933264326, 8.619223883229021924358799862763, 9.040825266605525847049817172787, 10.44881032542427841225111485431

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.9496272166\)
\(L(\frac12)\)	\(\approx\)	\(0.9496272166\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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