LMFDB - L-function 2-42e2-7.6-c2-0-15

Dirichlet series

L(s) = 1

+ 4.38i·5^-s + 19.5·11^-s − 6.11i·13^-s − 8.76i·17^-s + 30.3i·19^-s + 24·23^-s + 5.78·25^-s − 13.5·29^-s − 28.0i·31^-s − 48.5·37^-s − 7.14i·41^-s + 53.7·43^-s − 39.9i·47^-s + 61.5·53^-s + 85.8i·55^-s + ⋯

L(s) = 1

+ 0.876i·5^-s + 1.78·11^-s − 0.470i·13^-s − 0.515i·17^-s + 1.59i·19^-s + 1.04·23^-s + 0.231·25^-s − 0.468·29^-s − 0.904i·31^-s − 1.31·37^-s − 0.174i·41^-s + 1.25·43^-s − 0.849i·47^-s + 1.16·53^-s + 1.56i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1764$ = $2^{2} \cdot 3^{2} \cdot 7^{2}$
Sign:	$0.755 - 0.654i$
Analytic conductor:	$48.0655$
Root analytic conductor:	$6.93293$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1764} (685, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1764,\ (\ :1),\ 0.755 - 0.654i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.355756269$
$L(\frac12)$	$\approx$	$2.355756269$
$L(2)$		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 $
	7	$ 1 $
good	5	$ 1 - 4.38iT - 25T^{2} $
	11	$ 1 - 19.5T + 121T^{2} $
	13	$ 1 + 6.11iT - 169T^{2} $
	17	$ 1 + 8.76iT - 289T^{2} $
	19	$ 1 - 30.3iT - 361T^{2} $
	23	$ 1 - 24T + 529T^{2} $
	29	$ 1 + 13.5T + 841T^{2} $
	31	$ 1 + 28.0iT - 961T^{2} $
	37	$ 1 + 48.5T + 1.36e3T^{2} $
	41	$ 1 + 7.14iT - 1.68e3T^{2} $
	43	$ 1 - 53.7T + 1.84e3T^{2} $
	47	$ 1 + 39.9iT - 2.20e3T^{2} $
	53	$ 1 - 61.5T + 2.80e3T^{2} $
	59	$ 1 + 77.1iT - 3.48e3T^{2} $
	61	$ 1 + 0.431iT - 3.72e3T^{2} $
	67	$ 1 + 56.9T + 4.48e3T^{2} $
	71	$ 1 - 123.T + 5.04e3T^{2} $
	73	$ 1 - 35.8iT - 5.32e3T^{2} $
	79	$ 1 + 52.1T + 6.24e3T^{2} $
	83	$ 1 - 136. iT - 6.88e3T^{2} $
	89	$ 1 - 15.2iT - 7.92e3T^{2} $
	97	$ 1 + 34.1iT - 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

−9.248311134608381635041376101625, −8.477170677953135499167259102756, −7.44989512732185554754619194005, −6.83922450734427148828868598204, −6.12151797581128092623335910944, −5.22842258423072833223954604427, −3.92963554592816514517888787965, −3.42699980448201111041797743362, −2.19915311749716856525432044457, −0.990314654495647411382131251969, 0.815025384034264472307507300434, 1.67359547770289489340340626888, 3.09854387516345170495380595859, 4.19034380837962249929046958912, 4.76784762398889158911777897511, 5.77688359503502679700417396490, 6.77772447326506209061889484559, 7.21974140822332698252377171637, 8.675492110359903974445603290899, 8.909085092575065100670283199681

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.355756269\)
\(L(\frac12)\)	\(\approx\)	\(2.355756269\)
\(L(2)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more