LMFDB - L-function 2-60e2-20.19-c2-0-33

Dirichlet series

L(s) = 1

− 1.73·7^-s − 3.46i·11^-s − 11i·13^-s + 6i·17^-s + 19.0i·19^-s − 17.3·23^-s + 30·29^-s + 5.19i·31^-s − 14i·37^-s − 36·41^-s − 5.19·43^-s + 45.0·47^-s − 46·49^-s + 72i·53^-s − 38.1i·59^-s + ⋯

L(s) = 1

− 0.247·7^-s − 0.314i·11^-s − 0.846i·13^-s + 0.352i·17^-s + 1.00i·19^-s − 0.753·23^-s + 1.03·29^-s + 0.167i·31^-s − 0.378i·37^-s − 0.878·41^-s − 0.120·43^-s + 0.958·47^-s − 0.938·49^-s + 1.35i·53^-s − 0.645i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3600$ = $2^{4} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.834 - 0.550i$
Analytic conductor:	$98.0928$
Root analytic conductor:	$9.90418$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3600} (1999, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3600,\ (\ :1),\ 0.834 - 0.550i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.685515974$
$L(\frac12)$	$\approx$	$1.685515974$
$L(2)$		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 + 1.73T + 49T^{2} $
	11	$ 1 + 3.46iT - 121T^{2} $
	13	$ 1 + 11iT - 169T^{2} $
	17	$ 1 - 6iT - 289T^{2} $
	19	$ 1 - 19.0iT - 361T^{2} $
	23	$ 1 + 17.3T + 529T^{2} $
	29	$ 1 - 30T + 841T^{2} $
	31	$ 1 - 5.19iT - 961T^{2} $
	37	$ 1 + 14iT - 1.36e3T^{2} $
	41	$ 1 + 36T + 1.68e3T^{2} $
	43	$ 1 + 5.19T + 1.84e3T^{2} $
	47	$ 1 - 45.0T + 2.20e3T^{2} $
	53	$ 1 - 72iT - 2.80e3T^{2} $
	59	$ 1 + 38.1iT - 3.48e3T^{2} $
	61	$ 1 - 35T + 3.72e3T^{2} $
	67	$ 1 + 29.4T + 4.48e3T^{2} $
	71	$ 1 - 90.0iT - 5.04e3T^{2} $
	73	$ 1 + 62iT - 5.32e3T^{2} $
	79	$ 1 - 76.2iT - 6.24e3T^{2} $
	83	$ 1 - 72.7T + 6.88e3T^{2} $
	89	$ 1 - 144T + 7.92e3T^{2} $
	97	$ 1 + 181iT - 9.40e3T^{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

−8.305484571761284930808851776396, −7.88677183521428746614297289043, −6.94665021541271453308870876163, −6.10566537770386559910544666507, −5.60123611798990454638520754203, −4.61733122849422531587435199175, −3.70283520851719500716941461187, −2.99683446162355510567446935059, −1.90040069374203479891061898841, −0.75230876676792442455312679868, 0.49244938311742954311558013436, 1.78783505625559019139549907199, 2.67332048813011186453698147214, 3.64901579931128479106789092060, 4.55756000215471912977094271916, 5.14689785992628541075918217673, 6.28438573847987966301603150629, 6.74507848723217620680773522893, 7.51992057027617506457970480319, 8.365345675717317492763661401899

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.685515974\)
\(L(\frac12)\)	\(\approx\)	\(1.685515974\)
\(L(2)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more