LMFDB - L-function 2-800-5.4-c3-0-52

Dirichlet series

L(s) = 1

− 8.19i·3^-s − 21.7i·7^-s − 40.0·9^-s − 37.8·11^-s − 52.7i·13^-s − 99.9i·17^-s + 116.·19^-s − 177.·21^-s + 37.7i·23^-s + 107. i·27^-s − 218.·29^-s + 67.3·31^-s + 310. i·33^-s − 362. i·37^-s − 431.·39^-s + ⋯

L(s) = 1

− 1.57i·3^-s − 1.17i·7^-s − 1.48·9^-s − 1.03·11^-s − 1.12i·13^-s − 1.42i·17^-s + 1.40·19^-s − 1.84·21^-s + 0.342i·23^-s + 0.764i·27^-s − 1.40·29^-s + 0.390·31^-s + 1.63i·33^-s − 1.61i·37^-s − 1.77·39^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$800$ = $2^{5} \cdot 5^{2}$
Sign:	$-0.447 - 0.894i$
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:	$(2,\ 800,\ (\ :3/2),\ -0.447 - 0.894i)$

Particular Values

$L(2)$	$\approx$	$1.181151185$
$L(\frac12)$	$\approx$	$1.181151185$
$L(\frac{5}{2})$		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 + 8.19iT - 27T^{2} $
	7	$ 1 + 21.7iT - 343T^{2} $
	11	$ 1 + 37.8T + 1.33e3T^{2} $
	13	$ 1 + 52.7iT - 2.19e3T^{2} $
	17	$ 1 + 99.9iT - 4.91e3T^{2} $
	19	$ 1 - 116.T + 6.85e3T^{2} $
	23	$ 1 - 37.7iT - 1.21e4T^{2} $
	29	$ 1 + 218.T + 2.43e4T^{2} $
	31	$ 1 - 67.3T + 2.97e4T^{2} $
	37	$ 1 + 362. iT - 5.06e4T^{2} $
	41	$ 1 + 291.T + 6.89e4T^{2} $
	43	$ 1 - 183. iT - 7.95e4T^{2} $
	47	$ 1 - 443. iT - 1.03e5T^{2} $
	53	$ 1 - 416. iT - 1.48e5T^{2} $
	59	$ 1 - 828.T + 2.05e5T^{2} $
	61	$ 1 - 442.T + 2.26e5T^{2} $
	67	$ 1 + 570. iT - 3.00e5T^{2} $
	71	$ 1 + 341.T + 3.57e5T^{2} $
	73	$ 1 + 506. iT - 3.89e5T^{2} $
	79	$ 1 - 426.T + 4.93e5T^{2} $
	83	$ 1 + 982. iT - 5.71e5T^{2} $
	89	$ 1 + 926.T + 7.04e5T^{2} $
	97	$ 1 - 1.88e3iT - 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

−9.278095624516661068907089401215, −7.892934063024697504855487458449, −7.56227078498362959569574422748, −7.06985324323973303966532351864, −5.81493696725411369563310169498, −5.05093189709394774470216649072, −3.42378774097888648338711154720, −2.46843800263408243349692115452, −1.06733013463849606922839094729, −0.35060287253539888030017114947, 2.02615698110263831086645565274, 3.21264591883700515587046527044, 4.10733248566027376108221923412, 5.23191700732941212283306746644, 5.59576697277347327607754188392, 6.91487137359977685954729814118, 8.383217586377164049076655290021, 8.736209475016375841112912285210, 9.913090538300906763288610807391, 10.04710487273998710065948439040

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.181151185\)
\(L(\frac12)\)	\(\approx\)	\(1.181151185\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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