LMFDB - L-function 2-1800-5.4-c3-0-66

Dirichlet series

L(s) = 1

− 34i·7^-s − 16·11^-s − 58i·13^-s + 70i·17^-s − 4·19^-s − 134i·23^-s − 242·29^-s + 100·31^-s − 438i·37^-s + 138·41^-s − 178i·43^-s − 22i·47^-s − 813·49^-s + 162i·53^-s − 268·59^-s + ⋯

L(s) = 1

− 1.83i·7^-s − 0.438·11^-s − 1.23i·13^-s + 0.998i·17^-s − 0.0482·19^-s − 1.21i·23^-s − 1.54·29^-s + 0.579·31^-s − 1.94i·37^-s + 0.525·41^-s − 0.631i·43^-s − 0.0682i·47^-s − 2.37·49^-s + 0.419i·53^-s − 0.591·59^-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

$L(2)$	$\approx$	$0.8023436708$
$L(\frac12)$	$\approx$	$0.8023436708$
$L(\frac{5}{2})$		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 + 16T + 1.33e3T^{2} $
	13	$ 1 + 58iT - 2.19e3T^{2} $
	17	$ 1 - 70iT - 4.91e3T^{2} $
	19	$ 1 + 4T + 6.85e3T^{2} $
	23	$ 1 + 134iT - 1.21e4T^{2} $
	29	$ 1 + 242T + 2.43e4T^{2} $
	31	$ 1 - 100T + 2.97e4T^{2} $
	37	$ 1 + 438iT - 5.06e4T^{2} $
	41	$ 1 - 138T + 6.89e4T^{2} $
	43	$ 1 + 178iT - 7.95e4T^{2} $
	47	$ 1 + 22iT - 1.03e5T^{2} $
	53	$ 1 - 162iT - 1.48e5T^{2} $
	59	$ 1 + 268T + 2.05e5T^{2} $
	61	$ 1 - 250T + 2.26e5T^{2} $
	67	$ 1 - 422iT - 3.00e5T^{2} $
	71	$ 1 - 852T + 3.57e5T^{2} $
	73	$ 1 + 306iT - 3.89e5T^{2} $
	79	$ 1 - 456T + 4.93e5T^{2} $
	83	$ 1 - 434iT - 5.71e5T^{2} $
	89	$ 1 + 726T + 7.04e5T^{2} $
	97	$ 1 - 1.37e3iT - 9.12e5T^{2} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−8.216653917690370161414310019503, −7.70824885489226415366625975491, −7.01702403633827120924261003847, −6.07891544147464450692316539687, −5.18062215942209140810155891597, −4.11095301543375202653994383317, −3.60834072190163192861077403549, −2.33241406423316551513256639311, −0.986169811466296005768752808220, −0.18587571312521825345759844268, 1.60949101075104289735804464423, 2.45697838985817334358055238736, 3.32107028601643397210609178351, 4.64639228900083435063532685245, 5.34619753767099391865582756977, 6.08439767770628909758181768335, 6.93885057789568231693330367550, 7.904493480950619413955930623874, 8.655594404750326920676172206107, 9.497600651252117366081922348247

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.8023436708\)
\(L(\frac12)\)	\(\approx\)	\(0.8023436708\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more