LMFDB - L-function 2-4032-21.20-c1-0-29

Dirichlet series

L(s) = 1

− 0.841·5^-s + (1.16 − 2.37i)7^-s − 3.64i·11^-s + 5.53i·13^-s + 0.841·17^-s + 3.06i·19^-s − 3.64i·23^-s − 4.29·25^-s + 8.89i·29^-s + 7.82i·31^-s + (−0.979 + 1.99i)35^-s + 3.29·37^-s + 8.66·41^-s + 2.32·43^-s + 9.10·47^-s + ⋯

L(s) = 1

− 0.376·5^-s + (0.439 − 0.898i)7^-s − 1.09i·11^-s + 1.53i·13^-s + 0.204·17^-s + 0.704i·19^-s − 0.760i·23^-s − 0.858·25^-s + 1.65i·29^-s + 1.40i·31^-s + (−0.165 + 0.338i)35^-s + 0.541·37^-s + 1.35·41^-s + 0.354·43^-s + 1.32·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$0.987 - 0.159i$
Analytic conductor:	$32.1956$
Root analytic conductor:	$5.67412$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (3905, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1/2),\ 0.987 - 0.159i)$

Particular Values

$L(1)$	$\approx$	$1.751001038$
$L(\frac12)$	$\approx$	$1.751001038$
$L(\frac{3}{2})$		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 + (-1.16 + 2.37i)T $
good	5	$ 1 + 0.841T + 5T^{2} $
	11	$ 1 + 3.64iT - 11T^{2} $
	13	$ 1 - 5.53iT - 13T^{2} $
	17	$ 1 - 0.841T + 17T^{2} $
	19	$ 1 - 3.06iT - 19T^{2} $
	23	$ 1 + 3.64iT - 23T^{2} $
	29	$ 1 - 8.89iT - 29T^{2} $
	31	$ 1 - 7.82iT - 31T^{2} $
	37	$ 1 - 3.29T + 37T^{2} $
	41	$ 1 - 8.66T + 41T^{2} $
	43	$ 1 - 2.32T + 43T^{2} $
	47	$ 1 - 9.10T + 47T^{2} $
	53	$ 1 + 4.24iT - 53T^{2} $
	59	$ 1 - 11.0T + 59T^{2} $
	61	$ 1 - 9.10iT - 61T^{2} $
	67	$ 1 + 8.48T + 67T^{2} $
	71	$ 1 - 0.354iT - 71T^{2} $
	73	$ 1 + 14.6iT - 73T^{2} $
	79	$ 1 + 8.48T + 79T^{2} $
	83	$ 1 - 9.10T + 83T^{2} $
	89	$ 1 - 6.97T + 89T^{2} $
	97	$ 1 + 3.57iT - 97T^{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

−8.562288954750370836128793683201, −7.64700779170934422331369802547, −7.09712657837820844429955219498, −6.32853704695565503178005637416, −5.50166754217180192167473890732, −4.49129088815036672363389840593, −3.96220067659998448588288108063, −3.14338547092745995054514211523, −1.84893433858918409160809992537, −0.866442049867925827860308625456, 0.67709157418497518662197374365, 2.15434612987800444157698518671, 2.72526788857460370489099138708, 3.94762896495353320108906669433, 4.60152215380605291165642979373, 5.66875989319618240544635004002, 5.87382480895280683130714028045, 7.19601626649244051523195113077, 7.82378622586218422762659736987, 8.150095821550301558120675293370

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.751001038\)
\(L(\frac12)\)	\(\approx\)	\(1.751001038\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more