LMFDB - L-function 2-1800-5.4-c3-0-22

Dirichlet series

L(s) = 1

− 34i·7^-s + 18·11^-s + 12i·13^-s + 106i·17^-s + 44·19^-s + 56i·23^-s − 270·29^-s + 204·31^-s − 120i·37^-s + 80·41^-s + 536i·43^-s + 536i·47^-s − 813·49^-s + 542i·53^-s + 174·59^-s + ⋯

L(s) = 1

− 1.83i·7^-s + 0.493·11^-s + 0.256i·13^-s + 1.51i·17^-s + 0.531·19^-s + 0.507i·23^-s − 1.72·29^-s + 1.18·31^-s − 0.533i·37^-s + 0.304·41^-s + 1.90i·43^-s + 1.66i·47^-s − 2.37·49^-s + 1.40i·53^-s + 0.383·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1800$ = $2^{3} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.894 - 0.447i$
Analytic conductor:	$106.203$
Root analytic conductor:	$10.3055$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1800} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1800,\ (\ :3/2),\ 0.894 - 0.447i)$

Particular Values

$L(2)$	$\approx$	$1.947221727$
$L(\frac12)$	$\approx$	$1.947221727$
$L(\frac{5}{2})$		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 $
	3	$ 1 $
	5	$ 1 $
good	7	$ 1 + 34iT - 343T^{2} $
	11	$ 1 - 18T + 1.33e3T^{2} $
	13	$ 1 - 12iT - 2.19e3T^{2} $
	17	$ 1 - 106iT - 4.91e3T^{2} $
	19	$ 1 - 44T + 6.85e3T^{2} $
	23	$ 1 - 56iT - 1.21e4T^{2} $
	29	$ 1 + 270T + 2.43e4T^{2} $
	31	$ 1 - 204T + 2.97e4T^{2} $
	37	$ 1 + 120iT - 5.06e4T^{2} $
	41	$ 1 - 80T + 6.89e4T^{2} $
	43	$ 1 - 536iT - 7.95e4T^{2} $
	47	$ 1 - 536iT - 1.03e5T^{2} $
	53	$ 1 - 542iT - 1.48e5T^{2} $
	59	$ 1 - 174T + 2.05e5T^{2} $
	61	$ 1 - 186T + 2.26e5T^{2} $
	67	$ 1 + 332iT - 3.00e5T^{2} $
	71	$ 1 + 132T + 3.57e5T^{2} $
	73	$ 1 + 602iT - 3.89e5T^{2} $
	79	$ 1 - 548T + 4.93e5T^{2} $
	83	$ 1 + 492iT - 5.71e5T^{2} $
	89	$ 1 - 1.05e3T + 7.04e5T^{2} $
	97	$ 1 + 482iT - 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

−9.121091760259913526958591331207, −7.82867037058984113271337397311, −7.62057833335618610260785362622, −6.57443129548385460467229991912, −5.95051393878716710892201758588, −4.61329442210488183291415704558, −4.01703180594417217520602339697, −3.27238513572213657660252204929, −1.67482093976018764185638575613, −0.904165732238119488846219762494, 0.50017868248959179554983759389, 2.00529514628344239180590293947, 2.72487251535657876013795802685, 3.72401742020554166657651920925, 5.11848642772783412598426395849, 5.42489017013283801769948095745, 6.43784443858947596901707446601, 7.22222036331432147257107961472, 8.260706100496240387526849479736, 8.901496402463590853982671167754

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.947221727\)
\(L(\frac12)\)	\(\approx\)	\(1.947221727\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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